Theorem cfval2 9082

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

Assertion

Ref	Expression
cfval2	|- ( A e. On -> ( cf ` A ) = |^|_ x e. { x e. ~P A | A. z e. A E. w e. x z C_ w } ( card ` x ) )

Distinct variable group:   w, A, x, z

Proof of Theorem cfval2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cfval 9069	. 2 |- ( 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 ) ) } )
2		fvex 6201	. . . 4 |- ( card ` x ) e. _V
3	2	dfiin2 4555	. . 3 |- |^|_ x e. { x e. ~P A | A. z e. A E. w e. x z C_ w } ( card ` x ) = |^| { y | E. x e. { x e. ~P A | A. z e. A E. w e. x z C_ w } y = ( card ` x ) }
4		df-rex 2918	. . . . . 6 |- ( E. x e. { x e. ~P A | A. z e. A E. w e. x z C_ w } y = ( card ` x ) <-> E. x ( x e. { x e. ~P A | A. z e. A E. w e. x z C_ w } /\ y = ( card ` x ) ) )
5		rabid 3116	. . . . . . . . 9 |- ( x e. { x e. ~P A | A. z e. A E. w e. x z C_ w } <-> ( x e. ~P A /\ A. z e. A E. w e. x z C_ w ) )
6		selpw 4165	. . . . . . . . . 10 |- ( x e. ~P A <-> x C_ A )
7	6	anbi1i 731	. . . . . . . . 9 |- ( ( x e. ~P A /\ A. z e. A E. w e. x z C_ w ) <-> ( x C_ A /\ A. z e. A E. w e. x z C_ w ) )
8	5, 7	bitri 264	. . . . . . . 8 |- ( x e. { x e. ~P A | A. z e. A E. w e. x z C_ w } <-> ( x C_ A /\ A. z e. A E. w e. x z C_ w ) )
9	8	anbi2ci 732	. . . . . . 7 |- ( ( x e. { x e. ~P A | A. z e. A E. w e. x z C_ w } /\ y = ( card ` x ) ) <-> ( y = ( card ` x ) /\ ( x C_ A /\ A. z e. A E. w e. x z C_ w ) ) )
10	9	exbii 1774	. . . . . 6 |- ( E. x ( x e. { x e. ~P A | A. z e. A E. w e. x z C_ w } /\ y = ( card ` x ) ) <-> E. x ( y = ( card ` x ) /\ ( x C_ A /\ A. z e. A E. w e. x z C_ w ) ) )
11	4, 10	bitri 264	. . . . 5 |- ( E. x e. { x e. ~P A | A. z e. A E. w e. x z C_ w } y = ( card ` x ) <-> E. x ( y = ( card ` x ) /\ ( x C_ A /\ A. z e. A E. w e. x z C_ w ) ) )
12	11	abbii 2739	. . . 4 |- { y | E. x e. { x e. ~P A | A. z e. A E. w e. x z C_ w } y = ( card ` x ) } = { y | E. x ( y = ( card ` x ) /\ ( x C_ A /\ A. z e. A E. w e. x z C_ w ) ) }
13	12	inteqi 4479	. . 3 |- |^| { y | E. x e. { x e. ~P A | A. z e. A E. w e. x z C_ w } y = ( card ` x ) } = |^| { y | E. x ( y = ( card ` x ) /\ ( x C_ A /\ A. z e. A E. w e. x z C_ w ) ) }
14	3, 13	eqtr2i 2645	. 2 |- |^| { 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 | A. z e. A E. w e. x z C_ w } ( card ` x )
15	1, 14	syl6eq 2672	1 |- ( A e. On -> ( cf ` A ) = |^|_ x e. { x e. ~P A | A. z e. A E. w e. x z C_ w } ( 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   C_ wss 3574   ~Pcpw 4158   |^|cint 4475   |^|_ciin 4521   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-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-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-cf 8767

This theorem is referenced by: (None)