Theorem cardval3ex 6365

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 6227	. . . 4 |- ( x ~~ A -> ( x e. _V /\ A e. _V ) )
2	1	simprd 107	. . 3 |- ( x ~~ A -> A e. _V )
3	2	rexlimivw 2429	. 2 |- ( E. x e. On x ~~ A -> A e. _V )
4		breq1 3767	. . . 4 |- ( y = x -> ( y ~~ A <-> x ~~ A ) )
5	4	cbvrexv 2534	. . 3 |- ( E. y e. On y ~~ A <-> E. x e. On x ~~ A )
6		intexrabim 3907	. . 3 |- ( E. y e. On y ~~ A -> |^| { y e. On | y ~~ A } e. _V )
7	5, 6	sylbir 125	. 2 |- ( E. x e. On x ~~ A -> |^| { y e. On | y ~~ A } e. _V )
8		breq2 3768	. . . . 5 |- ( z = A -> ( y ~~ z <-> y ~~ A ) )
9	8	rabbidv 2549	. . . 4 |- ( z = A -> { y e. On | y ~~ z } = { y e. On | y ~~ A } )
10	9	inteqd 3620	. . 3 |- ( z = A -> |^| { y e. On | y ~~ z } = |^| { y e. On | y ~~ A } )
11		df-card 6360	. . 3 |- card = ( z e. _V |-> |^| { y e. On | y ~~ z } )
12	10, 11	fvmptg 5248	. 2 |- ( ( A e. _V /\ |^| { y e. On | y ~~ A } e. _V ) -> ( card ` A ) = |^| { y e. On | y ~~ A } )
13	3, 7, 12	syl2anc 391	1 |- ( E. x e. On x ~~ A -> ( card ` A ) = |^| { y e. On | y ~~ A } )

Syntax hints:   -> wi 4   = wceq 1243   e. wcel 1393   E.wrex 2307   {crab 2310   _Vcvv 2557   |^|cint 3615   class class class wbr 3764   Oncon0 4100   ` cfv 4902   ~~ cen 6219   cardccrd 6359

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910  df-en 6222  df-card 6360

This theorem is referenced by:  oncardval  6366  carden2bex  6369