Theorem oncardval 7531

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 7050	. . 3 |- ( A e. On -> A ~~ A )
2		breq1 4133	. . . 4 |- ( y = A -> ( y ~~ A <-> A ~~ A ) )
3	2	rspcev 2929	. . 3 |- ( ( A e. On /\ A ~~ A ) -> E. y e. On y ~~ A )
4	1, 3	mpdan 425	. 2 |- ( A e. On -> E. y e. On y ~~ A )
5		cardval3ex 7530	. 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 } )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   E.wrex 2529   {crab 2532   |^|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  ax-un 4578

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-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-en 7023  df-card 7524

This theorem is used by:  cardonle  7532