Theorem cardcf 6059

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 6054	. . . 4 |- (A e. On -> (cf` A) = |^|{x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))})
2		onint 3876	. . . . 5 |- (({x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))} C_ On /\ {x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))} =/= (/)) -> |^|{x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))} e. {x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))})
3		visset 2295	. . . . . . . . 9 |- v e. _V
4		eqeq1 1890	. . . . . . . . . . 11 |- (x = v -> (x = (card` y) <-> v = (card` y)))
5	4	anbi1d 679	. . . . . . . . . 10 |- (x = v -> ((x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w)) <-> (v = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))))
6	5	exbidv 1657	. . . . . . . . 9 |- (x = v -> (E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w)) <-> E.y(v = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))))
7	3, 6	elab 2403	. . . . . . . 8 |- (v e. {x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))} <-> E.y(v = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w)))
8		fveq2 4681	. . . . . . . . . . . 12 |- (v = (card` y) -> (card` v) = (card` (card` y)))
9		cardidm 6001	. . . . . . . . . . . 12 |- (card` (card` y)) = (card` y)
10	8, 9	syl6eq 1944	. . . . . . . . . . 11 |- (v = (card` y) -> (card` v) = (card` y))
11		eqeq2 1893	. . . . . . . . . . 11 |- (v = (card` y) -> ((card` v) = v <-> (card` v) = (card` y)))
12	10, 11	mpbird 213	. . . . . . . . . 10 |- (v = (card` y) -> (card` v) = v)
13	12	adantr 425	. . . . . . . . 9 |- ((v = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w)) -> (card` v) = v)
14	13	19.23aiv 1674	. . . . . . . 8 |- (E.y(v = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w)) -> (card` v) = v)
15	7, 14	sylbi 216	. . . . . . 7 |- (v e. {x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))} -> (card` v) = v)
16		cardon 5976	. . . . . . 7 |- (card` v) e. On
17	15, 16	syl6eqelr 1980	. . . . . 6 |- (v e. {x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))} -> v e. On)
18	17	ssriv 2621	. . . . 5 |- {x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))} C_ On
19		fvex 4689	. . . . . . 7 |- (cf` A) e. _V
20	1, 19	syl6eqelr 1980	. . . . . 6 |- (A e. On -> |^|{x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))} e. _V)
21		intex 3465	. . . . . 6 |- ({x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))} =/= (/) <-> |^|{x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))} e. _V)
22	20, 21	sylibr 217	. . . . 5 |- (A e. On -> {x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))} =/= (/))
23	2, 18, 22	sylancr 526	. . . 4 |- (A e. On -> |^|{x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))} e. {x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))})
24	1, 23	eqeltrd 1971	. . 3 |- (A e. On -> (cf` A) e. {x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))})
25		fveq2 4681	. . . . 5 |- (v = (cf` A) -> (card` v) = (card` (cf` A)))
26		id 73	. . . . 5 |- (v = (cf` A) -> v = (cf` A))
27	25, 26	eqeq12d 1899	. . . 4 |- (v = (cf` A) -> ((card` v) = v <-> (card` (cf` A)) = (cf` A)))
28	27, 15	vtoclga 2352	. . 3 |- ((cf` A) e. {x | E.y(x = (card` y) /\ (y C_ A /\ A.z e. A E.w e. y z C_ w))} -> (card` (cf` A)) = (cf` A))
29	24, 28	syl 12	. 2 |- (A e. On -> (card` (cf` A)) = (cf` A))
30		cffnon 6055	. . . . . . 7 |- cf Fn On
31		fndm 4512	. . . . . . 7 |- (cf Fn On -> dom cf = On)
32	30, 31	ax-mp 7	. . . . . 6 |- dom cf = On
33	32	eleq2i 1961	. . . . 5 |- (A e. dom cf <-> A e. On)
34	33	notbii 204	. . . 4 |- (-. A e. dom cf <-> -. A e. On)
35		ndmfv 4702	. . . 4 |- (-. A e. dom cf -> (cf` A) = (/))
36	34, 35	sylbir 218	. . 3 |- (-. A e. On -> (cf` A) = (/))
37		card0 5869	. . . 4 |- (card` (/)) = (/)
38		fveq2 4681	. . . 4 |- ((cf` A) = (/) -> (card` (cf` A)) = (card` (/)))
39		id 73	. . . 4 |- ((cf` A) = (/) -> (cf` A) = (/))
40	37, 38, 39	3eqtr4a 1954	. . 3 |- ((cf` A) = (/) -> (card` (cf` A)) = (cf` A))
41	36, 40	syl 12	. 2 |- (-. A e. On -> (card` (cf` A)) = (cf` A))
42	29, 41	pm2.61i 140	1 |- (card` (cf` A)) = (cf` A)

Syntax hints:  -. wn 2   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871   =/= wne 2017  A.wral 2105  E.wrex 2106  _Vcvv 2292   C_ wss 2593  (/)c0 2875  |^|cint 3214  Oncon0 3657  dom cdm 3986   Fn wfn 3993  ` cfv 3998  cardccrd 5859  cfccf 5861

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-er 5318  df-en 5427  df-card 5862  df-cf 5864

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