Theorem cardval3ex 7530

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 7028	. . . 4 |- ( x ~~ A -> ( x e. _V /\ A e. _V ) )
2	1	simprd 114	. . 3 |- ( x ~~ A -> A e. _V )
3	2	rexlimivw 2664	. 2 |- ( E. x e. On x ~~ A -> A e. _V )
4		breq1 4133	. . . 4 |- ( y = x -> ( y ~~ A <-> x ~~ A ) )
5	4	cbvrexv 2787	. . 3 |- ( E. y e. On y ~~ A <-> E. x e. On x ~~ A )
6		intexrabim 4289	. . 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 4134	. . . . 5 |- ( z = A -> ( y ~~ z <-> y ~~ A ) )
9	8	rabbidv 2810	. . . 4 |- ( z = A -> { y e. On | y ~~ z } = { y e. On | y ~~ A } )
10	9	inteqd 3975	. . 3 |- ( z = A -> |^| { y e. On | y ~~ z } = |^| { y e. On | y ~~ A } )
11		df-card 7524	. . 3 |- card = ( z e. _V |-> |^| { y e. On | y ~~ z } )
12	10, 11	fvmptg 5781	. 2 |- ( ( A e. _V /\ |^| { y e. On | y ~~ A } e. _V ) -> ( card ` A ) = |^| { y e. On | y ~~ A } )
13	3, 7, 12	syl2anc 415	1 |- ( E. x e. On x ~~ A -> ( card ` A ) = |^| { y e. On | y ~~ A } )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   E.wrex 2529   {crab 2532   _Vcvv 2821   |^|cint 3970   class class class wbr 4130   Oncon0 4508   ` cfv 5377   ~~ cen 7020   cardccrd 7522

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-en 7023  df-card 7524

This theorem is used by:  oncardval  7531  carden2bex  7535