Theorem cardval3ex 7186

Description: The value of ( card ` A ). (Contributed by Jim Kingdon, 30-Aug-2021.)

Assertion

Ref	Expression
cardval3ex	|- ( E. x e. On x ~~ A -> ( card ` A ) = |^| { y e. On | y ~~ A } )

Distinct variable group:   x, A, y

Proof of Theorem cardval3ex

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		encv 6748	. . . 4 |- ( x ~~ A -> ( x e. _V /\ A e. _V ) )
2	1	simprd 114	. . 3 |- ( x ~~ A -> A e. _V )
3	2	rexlimivw 2590	. 2 |- ( E. x e. On x ~~ A -> A e. _V )
4		breq1 4008	. . . 4 |- ( y = x -> ( y ~~ A <-> x ~~ A ) )
5	4	cbvrexv 2706	. . 3 |- ( E. y e. On y ~~ A <-> E. x e. On x ~~ A )
6		intexrabim 4155	. . 3 |- ( E. y e. On y ~~ A -> |^| { y e. On | y ~~ A } e. _V )
7	5, 6	sylbir 135	. 2 |- ( E. x e. On x ~~ A -> |^| { y e. On | y ~~ A } e. _V )
8		breq2 4009	. . . . 5 |- ( z = A -> ( y ~~ z <-> y ~~ A ) )
9	8	rabbidv 2728	. . . 4 |- ( z = A -> { y e. On | y ~~ z } = { y e. On | y ~~ A } )
10	9	inteqd 3851	. . 3 |- ( z = A -> |^| { y e. On | y ~~ z } = |^| { y e. On | y ~~ A } )
11		df-card 7181	. . 3 |- card = ( z e. _V |-> |^| { y e. On | y ~~ z } )
12	10, 11	fvmptg 5594	. 2 |- ( ( A e. _V /\ |^| { y e. On | y ~~ A } e. _V ) -> ( card ` A ) = |^| { y e. On | y ~~ A } )
13	3, 7, 12	syl2anc 411	1 |- ( E. x e. On x ~~ A -> ( card ` A ) = |^| { y e. On | y ~~ A } )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   E.wrex 2456   {crab 2459   _Vcvv 2739   |^|cint 3846   class class class wbr 4005   Oncon0 4365   ` cfv 5218   ~~ cen 6740   cardccrd 7180

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-en 6743  df-card 7181

This theorem is referenced by:  oncardval  7187  carden2bex  7190