Theorem cardval3ex 7380

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 6910	. . . 4 |- ( x ~~ A -> ( x e. _V /\ A e. _V ) )
2	1	simprd 114	. . 3 |- ( x ~~ A -> A e. _V )
3	2	rexlimivw 2644	. 2 |- ( E. x e. On x ~~ A -> A e. _V )
4		breq1 4089	. . . 4 |- ( y = x -> ( y ~~ A <-> x ~~ A ) )
5	4	cbvrexv 2766	. . 3 |- ( E. y e. On y ~~ A <-> E. x e. On x ~~ A )
6		intexrabim 4241	. . 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 4090	. . . . 5 |- ( z = A -> ( y ~~ z <-> y ~~ A ) )
9	8	rabbidv 2789	. . . 4 |- ( z = A -> { y e. On | y ~~ z } = { y e. On | y ~~ A } )
10	9	inteqd 3931	. . 3 |- ( z = A -> |^| { y e. On | y ~~ z } = |^| { y e. On | y ~~ A } )
11		df-card 7374	. . 3 |- card = ( z e. _V |-> |^| { y e. On | y ~~ z } )
12	10, 11	fvmptg 5718	. 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 1395   e. wcel 2200   E.wrex 2509   {crab 2512   _Vcvv 2800   |^|cint 3926   class class class wbr 4086   Oncon0 4458   ` cfv 5324   ~~ cen 6902   cardccrd 7372

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-iota 5284  df-fun 5326  df-fv 5332  df-en 6905  df-card 7374

This theorem is referenced by:  oncardval  7381  carden2bex  7385