Theorem rdgsucopabn 3945

Description: The value of the recursive definition generator at a successor (special case where the characteristic function is an ordered-pair class abstraction and where the mapping class D is a proper class). This is a technical lemma that can be used together with rdgsucopab 3944 to help eliminate redundant sethood antecedents.

Hypotheses

Ref	Expression
rdgsucopab.1	|- (z e. A -> A.x z e. A)
rdgsucopab.2	|- (z e. B -> A.x z e. B)
rdgsucopab.3	|- (z e. D -> A.x z e. D)
rdgsucopab.4	|- F = rec({<.x, y>. | y = C}, A)
rdgsucopab.5	|- (x = (F` B) -> C = D)

Assertion

Ref	Expression
rdgsucopabn	|- (-. D e. V -> (F` suc B) = (/))

Distinct variable groups:   z,D   y,z,C   z,A   z,B   x,y,z

Proof of Theorem rdgsucopabn

Step	Hyp	Ref	Expression
1		rdgsuct 3943	. . . . 5 |- (B e. On -> (rec({<.x, y>. | y = C}, A)` suc B) = ({<.x, y>. | y = C}` (rec({<.x, y>. | y = C}, A)` B)))
2		rdgsucopab.4	. . . . . 6 |- F = rec({<.x, y>. | y = C}, A)
3	2	fveq1i 3723	. . . . 5 |- (F` suc B) = (rec({<.x, y>. | y = C}, A)` suc B)
4	1, 3	syl5eq 1518	. . . 4 |- (B e. On -> (F` suc B) = ({<.x, y>. | y = C}` (rec({<.x, y>. | y = C}, A)` B)))
5		hbopab1 2811	. . . . . . 7 |- (z e. {<.x, y>. | y = C} -> A.x z e. {<.x, y>. | y = C})
6		rdgsucopab.1	. . . . . . 7 |- (z e. A -> A.x z e. A)
7	5, 6	hbrdg 3934	. . . . . 6 |- (z e. rec({<.x, y>. | y = C}, A) -> A.x z e. rec({<.x, y>. | y = C}, A))
8		rdgsucopab.2	. . . . . 6 |- (z e. B -> A.x z e. B)
9	7, 8	hbfv 3727	. . . . 5 |- (z e. (rec({<.x, y>. | y = C}, A)` B) -> A.x z e. (rec({<.x, y>. | y = C}, A)` B))
10		rdgsucopab.3	. . . . 5 |- (z e. D -> A.x z e. D)
11	2	fveq1i 3723	. . . . . . 7 |- (F` B) = (rec({<.x, y>. | y = C}, A)` B)
12	11	eqeq2i 1484	. . . . . 6 |- (x = (F` B) <-> x = (rec({<.x, y>. | y = C}, A)` B))
13		rdgsucopab.5	. . . . . 6 |- (x = (F` B) -> C = D)
14	12, 13	sylbir 201	. . . . 5 |- (x = (rec({<.x, y>. | y = C}, A)` B) -> C = D)
15	9, 10, 14	fvopabnf 3786	. . . 4 |- (-. D e. V -> ({<.x, y>. | y = C}` (rec({<.x, y>. | y = C}, A)` B)) = (/))
16	4, 15	sylan9eq 1526	. . 3 |- ((B e. On /\ -. D e. V) -> (F` suc B) = (/))
17	16	ex 373	. 2 |- (B e. On -> (-. D e. V -> (F` suc B) = (/)))
18		sucelon 3066	. . . . . 6 |- (B e. On <-> suc B e. On)
19	2	dmeqi 3310	. . . . . . . 8 |- dom F = dom rec({<.x, y>. | y = C}, A)
20		rdgfnon 3937	. . . . . . . . 9 |- rec({<.x, y>. | y = C}, A) Fn On
21		fndm 3585	. . . . . . . . 9 |- (rec({<.x, y>. | y = C}, A) Fn On -> dom rec({<.x, y>. | y = C}, A) = On)
22	20, 21	ax-mp 7	. . . . . . . 8 |- dom rec({<.x, y>. | y = C}, A) = On
23	19, 22	eqtr 1494	. . . . . . 7 |- dom F = On
24	23	eleq2i 1537	. . . . . 6 |- (suc B e. dom F <-> suc B e. On)
25	18, 24	bitr4 176	. . . . 5 |- (B e. On <-> suc B e. dom F)
26	25	negbii 187	. . . 4 |- (-. B e. On <-> -. suc B e. dom F)
27		ndmfv 3743	. . . 4 |- (-. suc B e. dom F -> (F` suc B) = (/))
28	26, 27	sylbi 199	. . 3 |- (-. B e. On -> (F` suc B) = (/))
29	28	a1d 12	. 2 |- (-. B e. On -> (-. D e. V -> (F` suc B) = (/)))
30	17, 29	pm2.61i 126	1 |- (-. D e. V -> (F` suc B) = (/))

Syntax hints:  -. wn 2   -> wi 3  A.wal 954   = wceq 956   e. wcel 958  Vcvv 1809  (/)c0 2278  {copab 2664  Oncon0 2946  suc csuc 2948  dom cdm 3168   Fn wfn 3175  ` cfv 3180  reccrdg 3929

This theorem is referenced by:  alephon 4853

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 2691  ax-sep 2701  ax-nul 2708  ax-pow 2740  ax-pr 2777  ax-un 2864

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 1586  df-ral 1648  df-rex 1649  df-rab 1651  df-v 1810  df-sbc 1940  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-if 2360  df-pw 2400  df-sn 2410  df-pr 2411  df-tp 2413  df-op 2414  df-uni 2502  df-iun 2566  df-br 2618  df-opab 2665  df-tr 2679  df-eprel 2830  df-id 2833  df-po 2838  df-so 2848  df-fr 2915  df-we 2932  df-ord 2949  df-on 2950  df-lim 2951  df-suc 2952  df-xp 3182  df-rel 3183  df-cnv 3184  df-co 3185  df-dm 3186  df-rn 3187  df-res 3188  df-ima 3189  df-fun 3190  df-fn 3191  df-fv 3196  df-rdg 3930

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