Theorem cardcf 4899

Description: Cofinality is a cardinal number. Proposition 11.11 of [TakeutiZaring] p. 103.

Assertion

Ref	Expression
cardcf	|- (card` (cf` A)) = (cf` A)

Proof of Theorem cardcf

Step	Hyp	Ref	Expression
1		cfval 4894	. . . 4 |- (A e. On -> (cf` A) = |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))})
2		fvex 3730	. . . . . . 7 |- (cf` A) e. V
3	1, 2	syl6eqelr 1556	. . . . . 6 |- (A e. On -> |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} e. V)
4		intex 2727	. . . . . 6 |- ({x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} =/= (/) <-> |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} e. V)
5	3, 4	sylibr 200	. . . . 5 |- (A e. On -> {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} =/= (/))
6		visset 1811	. . . . . . . . . 10 |- v e. V
7		eqeq1 1480	. . . . . . . . . . . 12 |- (x = v -> (x = (card` y) <-> v = (card` y)))
8	7	anbi1d 617	. . . . . . . . . . 11 |- (x = v -> ((x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)) <-> (v = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))))
9	8	exbidv 1279	. . . . . . . . . 10 |- (x = v -> (E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)) <-> E.y(v = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))))
10	6, 9	elab 1895	. . . . . . . . 9 |- (v e. {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} <-> E.y(v = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)))
11		fveq2 3722	. . . . . . . . . . . . 13 |- (v = (card` y) -> (card` v) = (card` (card` y)))
12		cardidm 4837	. . . . . . . . . . . . 13 |- (card` (card` y)) = (card` y)
13	11, 12	syl6eq 1522	. . . . . . . . . . . 12 |- (v = (card` y) -> (card` v) = (card` y))
14		eqeq2 1483	. . . . . . . . . . . 12 |- (v = (card` y) -> ((card` v) = v <-> (card` v) = (card` y)))
15	13, 14	mpbird 196	. . . . . . . . . . 11 |- (v = (card` y) -> (card` v) = v)
16	15	adantr 389	. . . . . . . . . 10 |- ((v = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)) -> (card` v) = v)
17	16	19.23aiv 1295	. . . . . . . . 9 |- (E.y(v = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)) -> (card` v) = v)
18	10, 17	sylbi 199	. . . . . . . 8 |- (v e. {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} -> (card` v) = v)
19		cardon 4815	. . . . . . . 8 |- (card` v) e. On
20	18, 19	syl6eqelr 1556	. . . . . . 7 |- (v e. {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} -> v e. On)
21	20	ssriv 2067	. . . . . 6 |- {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} (_ On
22		onint 3004	. . . . . 6 |- (({x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} (_ On /\ {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} =/= (/)) -> |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} e. {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))})
23	21, 22	mpan 695	. . . . 5 |- ({x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} =/= (/) -> |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} e. {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))})
24	5, 23	syl 10	. . . 4 |- (A e. On -> |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} e. {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))})
25	1, 24	eqeltrd 1547	. . 3 |- (A e. On -> (cf` A) e. {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))})
26		fveq2 3722	. . . . 5 |- (v = (cf` A) -> (card` v) = (card` (cf` A)))
27		id 59	. . . . 5 |- (v = (cf` A) -> v = (cf` A))
28	26, 27	eqeq12d 1488	. . . 4 |- (v = (cf` A) -> ((card` v) = v <-> (card` (cf` A)) = (cf` A)))
29	28, 18	vtoclga 1850	. . 3 |- ((cf` A) e. {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} -> (card` (cf` A)) = (cf` A))
30	25, 29	syl 10	. 2 |- (A e. On -> (card` (cf` A)) = (cf` A))
31		cffnon 4895	. . . . . . 7 |- cf Fn On
32		fndm 3585	. . . . . . 7 |- (cf Fn On -> dom cf = On)
33	31, 32	ax-mp 7	. . . . . 6 |- dom cf = On
34	33	eleq2i 1537	. . . . 5 |- (A e. dom cf <-> A e. On)
35	34	negbii 187	. . . 4 |- (-. A e. dom cf <-> -. A e. On)
36		ndmfv 3743	. . . 4 |- (-. A e. dom cf -> (cf` A) = (/))
37	35, 36	sylbir 201	. . 3 |- (-. A e. On -> (cf` A) = (/))
38		card0 4811	. . . 4 |- (card` (/)) = (/)
39		fveq2 3722	. . . 4 |- ((cf` A) = (/) -> (card` (cf` A)) = (card` (/)))
40		id 59	. . . 4 |- ((cf` A) = (/) -> (cf` A) = (/))
41	38, 39, 40	3eqtr4a 1531	. . 3 |- ((cf` A) = (/) -> (card` (cf` A)) = (cf` A))
42	37, 41	syl 10	. 2 |- (-. A e. On -> (card` (cf` A)) = (cf` A))
43	30, 42	pm2.61i 126	1 |- (card` (cf` A)) = (cf` A)

Syntax hints:  -. wn 2   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  {cab 1463   =/= wne 1584  A.wral 1644  E.wrex 1645  Vcvv 1809   (_ wss 2045  (/)c0 2278  |^|cint 2531  Oncon0 2946  dom cdm 3168   Fn wfn 3175  ` cfv 3180  cardccrd 4801  cfccf 4803

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  ax-ac 4732

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-reu 1650  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-pw 2400  df-sn 2410  df-pr 2411  df-tp 2413  df-op 2414  df-uni 2502  df-int 2532  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-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-f 3192  df-f1 3193  df-fo 3194  df-f1o 3195  df-fv 3196  df-er 4259  df-en 4365  df-card 4804  df-cf 4806

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