Theorem cardcf 9074

Description: Cofinality is a cardinal number. Proposition 11.11 of [TakeutiZaring] p. 103. (Contributed by NM, 24-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)

Assertion

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

Proof of Theorem cardcf

Dummy variables x y z w v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cfval 9069	. . . 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		vex 3203	. . . . . . . . 9 |- v e. _V
3		eqeq1 2626	. . . . . . . . . . 11 |- ( x = v -> ( x = ( card ` y ) <-> v = ( card ` y ) ) )
4	3	anbi1d 741	. . . . . . . . . 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 ) ) ) )
5	4	exbidv 1850	. . . . . . . . 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 ) ) ) )
6	2, 5	elab 3350	. . . . . . . 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 ) ) )
7		fveq2 6191	. . . . . . . . . . . 12 |- ( v = ( card ` y ) -> ( card ` v ) = ( card ` ( card ` y ) ) )
8		cardidm 8785	. . . . . . . . . . . 12 |- ( card ` ( card ` y ) ) = ( card ` y )
9	7, 8	syl6eq 2672	. . . . . . . . . . 11 |- ( v = ( card ` y ) -> ( card ` v ) = ( card ` y ) )
10		eqeq2 2633	. . . . . . . . . . 11 |- ( v = ( card ` y ) -> ( ( card ` v ) = v <-> ( card ` v ) = ( card ` y ) ) )
11	9, 10	mpbird 247	. . . . . . . . . 10 |- ( v = ( card ` y ) -> ( card ` v ) = v )
12	11	adantr 481	. . . . . . . . 9 |- ( ( v = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) -> ( card ` v ) = v )
13	12	exlimiv 1858	. . . . . . . 8 |- ( E. y ( v = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) -> ( card ` v ) = v )
14	6, 13	sylbi 207	. . . . . . 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 )
15		cardon 8770	. . . . . . 7 |- ( card ` v ) e. On
16	14, 15	syl6eqelr 2710	. . . . . 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 )
17	16	ssriv 3607	. . . . 5 |- { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) } C_ On
18		fvex 6201	. . . . . . 7 |- ( cf ` A ) e. _V
19	1, 18	syl6eqelr 2710	. . . . . 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 )
20		intex 4820	. . . . . 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 )
21	19, 20	sylibr 224	. . . . 5 |- ( A e. On -> { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) } =/= (/) )
22		onint 6995	. . . . 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 ) ) } )
23	17, 21, 22	sylancr 695	. . . 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 2701	. . 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 6191	. . . . 5 |- ( v = ( cf ` A ) -> ( card ` v ) = ( card ` ( cf ` A ) ) )
26		id 22	. . . . 5 |- ( v = ( cf ` A ) -> v = ( cf ` A ) )
27	25, 26	eqeq12d 2637	. . . 4 |- ( v = ( cf ` A ) -> ( ( card ` v ) = v <-> ( card ` ( cf ` A ) ) = ( cf ` A ) ) )
28	27, 14	vtoclga 3272	. . 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 17	. 2 |- ( A e. On -> ( card ` ( cf ` A ) ) = ( cf ` A ) )
30		cff 9070	. . . . . 6 |- cf : On --> On
31	30	fdmi 6052	. . . . 5 |- dom cf = On
32	31	eleq2i 2693	. . . 4 |- ( A e. dom cf <-> A e. On )
33		ndmfv 6218	. . . 4 |- ( -. A e. dom cf -> ( cf ` A ) = (/) )
34	32, 33	sylnbir 321	. . 3 |- ( -. A e. On -> ( cf ` A ) = (/) )
35		card0 8784	. . . 4 |- ( card ` (/) ) = (/)
36		fveq2 6191	. . . 4 |- ( ( cf ` A ) = (/) -> ( card ` ( cf ` A ) ) = ( card ` (/) ) )
37		id 22	. . . 4 |- ( ( cf ` A ) = (/) -> ( cf ` A ) = (/) )
38	35, 36, 37	3eqtr4a 2682	. . 3 |- ( ( cf ` A ) = (/) -> ( card ` ( cf ` A ) ) = ( cf ` A ) )
39	34, 38	syl 17	. 2 |- ( -. A e. On -> ( card ` ( cf ` A ) ) = ( cf ` A ) )
40	29, 39	pm2.61i 176	1 |- ( card ` ( cf ` A ) ) = ( cf ` A )

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   (/)c0 3915   |^|cint 4475   dom cdm 5114   Oncon0 5723   ` cfv 5888   cardccrd 8761   cfccf 8763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-er 7742  df-en 7956  df-card 8765  df-cf 8767

This theorem is referenced by:  cfon  9077  winacard  9514