Theorem cflim3 9084

Description: Another expression for the cofinality function. (Contributed by Mario Carneiro, 28-Feb-2013.)

Hypothesis

Ref	Expression
cflim3.1	|- A e. _V

Assertion

Ref	Expression
cflim3	|- ( Lim A -> ( cf ` A ) = |^|_ x e. { x e. ~P A | U. x = A } ( card ` x ) )

Distinct variable group:   x, A

Proof of Theorem cflim3

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

Step	Hyp	Ref	Expression
1		limord 5784	. . . 4 |- ( Lim A -> Ord A )
2		cflim3.1	. . . . 5 |- A e. _V
3	2	elon 5732	. . . 4 |- ( A e. On <-> Ord A )
4	1, 3	sylibr 224	. . 3 |- ( Lim A -> A e. On )
5		cfval 9069	. . 3 |- ( A e. On -> ( cf ` A ) = |^| { y | E. x ( y = ( card ` x ) /\ ( x C_ A /\ A. z e. A E. w e. x z C_ w ) ) } )
6	4, 5	syl 17	. 2 |- ( Lim A -> ( cf ` A ) = |^| { y | E. x ( y = ( card ` x ) /\ ( x C_ A /\ A. z e. A E. w e. x z C_ w ) ) } )
7		fvex 6201	. . . 4 |- ( card ` x ) e. _V
8	7	dfiin2 4555	. . 3 |- |^|_ x e. { x e. ~P A | U. x = A } ( card ` x ) = |^| { y | E. x e. { x e. ~P A | U. x = A } y = ( card ` x ) }
9		df-rex 2918	. . . . . 6 |- ( E. x e. { x e. ~P A | U. x = A } y = ( card ` x ) <-> E. x ( x e. { x e. ~P A | U. x = A } /\ y = ( card ` x ) ) )
10		ancom 466	. . . . . . . 8 |- ( ( x e. { x e. ~P A | U. x = A } /\ y = ( card ` x ) ) <-> ( y = ( card ` x ) /\ x e. { x e. ~P A | U. x = A } ) )
11		rabid 3116	. . . . . . . . . 10 |- ( x e. { x e. ~P A | U. x = A } <-> ( x e. ~P A /\ U. x = A ) )
12		selpw 4165	. . . . . . . . . . . 12 |- ( x e. ~P A <-> x C_ A )
13	12	anbi1i 731	. . . . . . . . . . 11 |- ( ( x e. ~P A /\ U. x = A ) <-> ( x C_ A /\ U. x = A ) )
14		coflim 9083	. . . . . . . . . . . 12 |- ( ( Lim A /\ x C_ A ) -> ( U. x = A <-> A. z e. A E. w e. x z C_ w ) )
15	14	pm5.32da 673	. . . . . . . . . . 11 |- ( Lim A -> ( ( x C_ A /\ U. x = A ) <-> ( x C_ A /\ A. z e. A E. w e. x z C_ w ) ) )
16	13, 15	syl5bb 272	. . . . . . . . . 10 |- ( Lim A -> ( ( x e. ~P A /\ U. x = A ) <-> ( x C_ A /\ A. z e. A E. w e. x z C_ w ) ) )
17	11, 16	syl5bb 272	. . . . . . . . 9 |- ( Lim A -> ( x e. { x e. ~P A | U. x = A } <-> ( x C_ A /\ A. z e. A E. w e. x z C_ w ) ) )
18	17	anbi2d 740	. . . . . . . 8 |- ( Lim A -> ( ( y = ( card ` x ) /\ x e. { x e. ~P A | U. x = A } ) <-> ( y = ( card ` x ) /\ ( x C_ A /\ A. z e. A E. w e. x z C_ w ) ) ) )
19	10, 18	syl5bb 272	. . . . . . 7 |- ( Lim A -> ( ( x e. { x e. ~P A | U. x = A } /\ y = ( card ` x ) ) <-> ( y = ( card ` x ) /\ ( x C_ A /\ A. z e. A E. w e. x z C_ w ) ) ) )
20	19	exbidv 1850	. . . . . 6 |- ( Lim A -> ( E. x ( x e. { x e. ~P A | U. x = A } /\ y = ( card ` x ) ) <-> E. x ( y = ( card ` x ) /\ ( x C_ A /\ A. z e. A E. w e. x z C_ w ) ) ) )
21	9, 20	syl5bb 272	. . . . 5 |- ( Lim A -> ( E. x e. { x e. ~P A | U. x = A } y = ( card ` x ) <-> E. x ( y = ( card ` x ) /\ ( x C_ A /\ A. z e. A E. w e. x z C_ w ) ) ) )
22	21	abbidv 2741	. . . 4 |- ( Lim A -> { y | E. x e. { x e. ~P A | U. x = A } y = ( card ` x ) } = { y | E. x ( y = ( card ` x ) /\ ( x C_ A /\ A. z e. A E. w e. x z C_ w ) ) } )
23	22	inteqd 4480	. . 3 |- ( Lim A -> |^| { y | E. x e. { x e. ~P A | U. x = A } y = ( card ` x ) } = |^| { y | E. x ( y = ( card ` x ) /\ ( x C_ A /\ A. z e. A E. w e. x z C_ w ) ) } )
24	8, 23	syl5req 2669	. 2 |- ( Lim A -> |^| { y | E. x ( y = ( card ` x ) /\ ( x C_ A /\ A. z e. A E. w e. x z C_ w ) ) } = |^|_ x e. { x e. ~P A | U. x = A } ( card ` x ) )
25	6, 24	eqtrd 2656	1 |- ( Lim A -> ( cf ` A ) = |^|_ x e. { x e. ~P A | U. x = A } ( card ` x ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   |^|cint 4475   |^|_ciin 4521   Ord word 5722   Oncon0 5723   Lim wlim 5724   ` 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-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iin 4523  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-ord 5726  df-on 5727  df-lim 5728  df-iota 5851  df-fun 5890  df-fv 5896  df-cf 8767

This theorem is referenced by:  cflim2  9085  cfss  9087  cfslb  9088