Theorem cardval3ex 7141

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 6712	. . . 4 |- ( x ~~ A -> ( x e. _V /\ A e. _V ) )
2	1	simprd 113	. . 3 |- ( x ~~ A -> A e. _V )
3	2	rexlimivw 2579	. 2 |- ( E. x e. On x ~~ A -> A e. _V )
4		breq1 3985	. . . 4 |- ( y = x -> ( y ~~ A <-> x ~~ A ) )
5	4	cbvrexv 2693	. . 3 |- ( E. y e. On y ~~ A <-> E. x e. On x ~~ A )
6		intexrabim 4132	. . 3 |- ( E. y e. On y ~~ A -> |^| { y e. On | y ~~ A } e. _V )
7	5, 6	sylbir 134	. 2 |- ( E. x e. On x ~~ A -> |^| { y e. On | y ~~ A } e. _V )
8		breq2 3986	. . . . 5 |- ( z = A -> ( y ~~ z <-> y ~~ A ) )
9	8	rabbidv 2715	. . . 4 |- ( z = A -> { y e. On | y ~~ z } = { y e. On | y ~~ A } )
10	9	inteqd 3829	. . 3 |- ( z = A -> |^| { y e. On | y ~~ z } = |^| { y e. On | y ~~ A } )
11		df-card 7136	. . 3 |- card = ( z e. _V |-> |^| { y e. On | y ~~ z } )
12	10, 11	fvmptg 5562	. 2 |- ( ( A e. _V /\ |^| { y e. On | y ~~ A } e. _V ) -> ( card ` A ) = |^| { y e. On | y ~~ A } )
13	3, 7, 12	syl2anc 409	1 |- ( E. x e. On x ~~ A -> ( card ` A ) = |^| { y e. On | y ~~ A } )

Syntax hints:   -> wi 4   = wceq 1343   e. wcel 2136   E.wrex 2445   {crab 2448   _Vcvv 2726   |^|cint 3824   class class class wbr 3982   Oncon0 4341   ` cfv 5188   ~~ cen 6704   cardccrd 7135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-en 6707  df-card 7136

This theorem is referenced by:  oncardval  7142  carden2bex  7145