Theorem oncardval 7354

Description: The value of the cardinal number function with an ordinal number as its argument. (Contributed by NM, 24-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.)

Assertion

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

Distinct variable group:   x, A

Proof of Theorem oncardval

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		enrefg 6913	. . 3 |- ( A e. On -> A ~~ A )
2		breq1 4085	. . . 4 |- ( y = A -> ( y ~~ A <-> A ~~ A ) )
3	2	rspcev 2907	. . 3 |- ( ( A e. On /\ A ~~ A ) -> E. y e. On y ~~ A )
4	1, 3	mpdan 421	. 2 |- ( A e. On -> E. y e. On y ~~ A )
5		cardval3ex 7353	. 2 |- ( E. y e. On y ~~ A -> ( card ` A ) = |^| { x e. On | x ~~ A } )
6	4, 5	syl 14	1 |- ( A e. On -> ( card ` A ) = |^| { x e. On | x ~~ A } )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   E.wrex 2509   {crab 2512   |^|cint 3922   class class class wbr 4082   Oncon0 4453   ` cfv 5317   ~~ cen 6883   cardccrd 7345

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-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523

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 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-en 6886  df-card 7347

This theorem is referenced by:  cardonle  7355