Theorem oncardval 7389

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 6936	. . 3 |- ( A e. On -> A ~~ A )
2		breq1 4091	. . . 4 |- ( y = A -> ( y ~~ A <-> A ~~ A ) )
3	2	rspcev 2910	. . 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 7388	. 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 1397   e. wcel 2202   E.wrex 2511   {crab 2514   |^|cint 3928   class class class wbr 4088   Oncon0 4460   ` cfv 5326   ~~ cen 6906   cardccrd 7380

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-en 6909  df-card 7382

This theorem is referenced by:  cardonle  7390