Theorem r1tr 4654

Description: The cumulative hierarchy of sets is transitive. Lemma 7T of [Enderton] p. 202.

Assertion

Ref	Expression
r1tr	|- Tr (R1` A)

Proof of Theorem r1tr

Step	Hyp	Ref	Expression
1		fveq2 3724	. . . 4 |- (x = (/) -> (R1` x) = (R1` (/)))
2		treq 2686	. . . 4 |- ((R1` x) = (R1` (/)) -> (Tr (R1` x) <-> Tr (R1` (/))))
3	1, 2	syl 10	. . 3 |- (x = (/) -> (Tr (R1` x) <-> Tr (R1` (/))))
4		fveq2 3724	. . . 4 |- (x = y -> (R1` x) = (R1` y))
5		treq 2686	. . . 4 |- ((R1` x) = (R1` y) -> (Tr (R1` x) <-> Tr (R1` y)))
6	4, 5	syl 10	. . 3 |- (x = y -> (Tr (R1` x) <-> Tr (R1` y)))
7		fveq2 3724	. . . 4 |- (x = suc y -> (R1` x) = (R1` suc y))
8		treq 2686	. . . 4 |- ((R1` x) = (R1` suc y) -> (Tr (R1` x) <-> Tr (R1` suc y)))
9	7, 8	syl 10	. . 3 |- (x = suc y -> (Tr (R1` x) <-> Tr (R1` suc y)))
10		fveq2 3724	. . . 4 |- (x = A -> (R1` x) = (R1` A))
11		treq 2686	. . . 4 |- ((R1` x) = (R1` A) -> (Tr (R1` x) <-> Tr (R1` A)))
12	10, 11	syl 10	. . 3 |- (x = A -> (Tr (R1` x) <-> Tr (R1` A)))
13		tr0 2691	. . . 4 |- Tr (/)
14		r10 4651	. . . . 5 |- (R1` (/)) = (/)
15		treq 2686	. . . . 5 |- ((R1` (/)) = (/) -> (Tr (R1` (/)) <-> Tr (/)))
16	14, 15	ax-mp 7	. . . 4 |- (Tr (R1` (/)) <-> Tr (/))
17	13, 16	mpbir 190	. . 3 |- Tr (R1` (/))
18		r1suc 4652	. . . . . . . . . 10 |- (y e. On -> (R1` suc y) = P~(R1` y))
19	18	eleq2d 1541	. . . . . . . . 9 |- (y e. On -> (x e. (R1` suc y) <-> x e. P~(R1` y)))
20		visset 1813	. . . . . . . . . 10 |- x e. V
21	20	elpw 2404	. . . . . . . . 9 |- (x e. P~(R1` y) <-> x (_ (R1` y))
22	19, 21	syl6bb 536	. . . . . . . 8 |- (y e. On -> (x e. (R1` suc y) <-> x (_ (R1` y)))
23	22	adantr 389	. . . . . . 7 |- ((y e. On /\ Tr (R1` y)) -> (x e. (R1` suc y) <-> x (_ (R1` y)))
24		ssel 2063	. . . . . . . . . 10 |- (x (_ (R1` y) -> (z e. x -> z e. (R1` y)))
25		dftr4 2685	. . . . . . . . . . . 12 |- (Tr (R1` y) <-> (R1` y) (_ P~(R1` y))
26		ssel 2063	. . . . . . . . . . . 12 |- ((R1` y) (_ P~(R1` y) -> (z e. (R1` y) -> z e. P~(R1` y)))
27	25, 26	sylbi 199	. . . . . . . . . . 11 |- (Tr (R1` y) -> (z e. (R1` y) -> z e. P~(R1` y)))
28	18	eleq2d 1541	. . . . . . . . . . . 12 |- (y e. On -> (z e. (R1` suc y) <-> z e. P~(R1` y)))
29	28	biimprd 154	. . . . . . . . . . 11 |- (y e. On -> (z e. P~(R1` y) -> z e. (R1` suc y)))
30	27, 29	sylan9r 469	. . . . . . . . . 10 |- ((y e. On /\ Tr (R1` y)) -> (z e. (R1` y) -> z e. (R1` suc y)))
31	24, 30	sylan9r 469	. . . . . . . . 9 |- (((y e. On /\ Tr (R1` y)) /\ x (_ (R1` y)) -> (z e. x -> z e. (R1` suc y)))
32	31	ssrdv 2070	. . . . . . . 8 |- (((y e. On /\ Tr (R1` y)) /\ x (_ (R1` y)) -> x (_ (R1` suc y))
33	32	ex 373	. . . . . . 7 |- ((y e. On /\ Tr (R1` y)) -> (x (_ (R1` y) -> x (_ (R1` suc y)))
34	23, 33	sylbid 203	. . . . . 6 |- ((y e. On /\ Tr (R1` y)) -> (x e. (R1` suc y) -> x (_ (R1` suc y)))
35	34	r19.21aiv 1713	. . . . 5 |- ((y e. On /\ Tr (R1` y)) -> A.x e. (R1` suc y)x (_ (R1` suc y))
36		dftr3 2684	. . . . 5 |- (Tr (R1` suc y) <-> A.x e. (R1` suc y)x (_ (R1` suc y))
37	35, 36	sylibr 200	. . . 4 |- ((y e. On /\ Tr (R1` y)) -> Tr (R1` suc y))
38	37	ex 373	. . 3 |- (y e. On -> (Tr (R1` y) -> Tr (R1` suc y)))
39		r1lim 4653	. . . . . . . . . . 11 |- ((x e. V /\ Lim x) -> (R1` x) = U_y e. x (R1` y))
40	20, 39	mpan 695	. . . . . . . . . 10 |- (Lim x -> (R1` x) = U_y e. x (R1` y))
41	40	eleq2d 1541	. . . . . . . . 9 |- (Lim x -> (z e. (R1` x) <-> z e. U_y e. x (R1` y)))
42		eliun 2570	. . . . . . . . . 10 |- (z e. U_y e. x (R1` y) <-> E.y e. x z e. (R1` y))
43	42	biimp 151	. . . . . . . . 9 |- (z e. U_y e. x (R1` y) -> E.y e. x z e. (R1` y))
44	41, 43	syl6bi 214	. . . . . . . 8 |- (Lim x -> (z e. (R1` x) -> E.y e. x z e. (R1` y)))
45		hbra1 1687	. . . . . . . . 9 |- (A.y e. x Tr (R1` y) -> A.yA.y e. x Tr (R1` y))
46		ra4 1694	. . . . . . . . . 10 |- (A.y e. x Tr (R1` y) -> (y e. x -> Tr (R1` y)))
47		trss 2689	. . . . . . . . . 10 |- (Tr (R1` y) -> (z e. (R1` y) -> z (_ (R1` y)))
48	46, 47	syl6 22	. . . . . . . . 9 |- (A.y e. x Tr (R1` y) -> (y e. x -> (z e. (R1` y) -> z (_ (R1` y))))
49	45, 48	r19.22d 1735	. . . . . . . 8 |- (A.y e. x Tr (R1` y) -> (E.y e. x z e. (R1` y) -> E.y e. x z (_ (R1` y)))
50	44, 49	sylan9 468	. . . . . . 7 |- ((Lim x /\ A.y e. x Tr (R1` y)) -> (z e. (R1` x) -> E.y e. x z (_ (R1` y)))
51	40	sseq2d 2089	. . . . . . . . 9 |- (Lim x -> (z (_ (R1` x) <-> z (_ U_y e. x (R1` y)))
52		ssiun 2592	. . . . . . . . 9 |- (E.y e. x z (_ (R1` y) -> z (_ U_y e. x (R1` y))
53	51, 52	syl5bir 210	. . . . . . . 8 |- (Lim x -> (E.y e. x z (_ (R1` y) -> z (_ (R1` x)))
54	53	adantr 389	. . . . . . 7 |- ((Lim x /\ A.y e. x Tr (R1` y)) -> (E.y e. x z (_ (R1` y) -> z (_ (R1` x)))
55	50, 54	syld 27	. . . . . 6 |- ((Lim x /\ A.y e. x Tr (R1` y)) -> (z e. (R1` x) -> z (_ (R1` x)))
56	55	r19.21aiv 1713	. . . . 5 |- ((Lim x /\ A.y e. x Tr (R1` y)) -> A.z e. (R1` x)z (_ (R1` x))
57		dftr3 2684	. . . . 5 |- (Tr (R1` x) <-> A.z e. (R1` x)z (_ (R1` x))
58	56, 57	sylibr 200	. . . 4 |- ((Lim x /\ A.y e. x Tr (R1` y)) -> Tr (R1` x))
59	58	ex 373	. . 3 |- (Lim x -> (A.y e. x Tr (R1` y) -> Tr (R1` x)))
60	3, 6, 9, 12, 17, 38, 59	tfinds 3161	. 2 |- (A e. On -> Tr (R1` A))
61		r1fnon 4650	. . . . . . . 8 |- R1 Fn On
62		fndm 3587	. . . . . . . 8 |- (R1 Fn On -> dom R1 = On)
63	61, 62	ax-mp 7	. . . . . . 7 |- dom R1 = On
64	63	eleq2i 1538	. . . . . 6 |- (A e. dom R1 <-> A e. On)
65	64	negbii 187	. . . . 5 |- (-. A e. dom R1 <-> -. A e. On)
66		ndmfv 3745	. . . . 5 |- (-. A e. dom R1 -> (R1` A) = (/))
67	65, 66	sylbir 201	. . . 4 |- (-. A e. On -> (R1` A) = (/))
68		treq 2686	. . . 4 |- ((R1` A) = (/) -> (Tr (R1` A) <-> Tr (/)))
69	67, 68	syl 10	. . 3 |- (-. A e. On -> (Tr (R1` A) <-> Tr (/)))
70	13, 69	mpbiri 194	. 2 |- (-. A e. On -> Tr (R1` A))
71	60, 70	pm2.61i 126	1 |- Tr (R1` A)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645  E.wrex 1646  Vcvv 1811   (_ wss 2047  (/)c0 2280  P~cpw 2401  U_ciun 2566  Tr wtr 2680  Oncon0 2948  Lim wlim 2949  suc csuc 2950  dom cdm 3170   Fn wfn 3177  ` cfv 3182  R1cr1 4641

This theorem is referenced by:  r1ord 4655  r1ord2 4656

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-rdg 3932  df-r1 4643

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