Theorem cardid2 8335

Description: Any numerable set is equinumerous to its cardinal number. Proposition 10.5 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.)

Assertion

Ref	Expression
cardid2	|- ( A e. dom card -> ( card ` A ) ~~ A )

Proof of Theorem cardid2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cardval3 8334	. . 3 |- ( A e. dom card -> ( card ` A ) = |^| { y e. On | y ~~ A } )
2		ssrab2 3585	. . . 4 |- { y e. On | y ~~ A } C_ On
3		fvex 5876	. . . . . 6 |- ( card ` A ) e. _V
4	1, 3	syl6eqelr 2564	. . . . 5 |- ( A e. dom card -> |^| { y e. On | y ~~ A } e. _V )
5		intex 4603	. . . . 5 |- ( { y e. On | y ~~ A } =/= (/) <-> |^| { y e. On | y ~~ A } e. _V )
6	4, 5	sylibr 212	. . . 4 |- ( A e. dom card -> { y e. On | y ~~ A } =/= (/) )
7		onint 6615	. . . 4 |- ( ( { y e. On | y ~~ A } C_ On /\ { y e. On | y ~~ A } =/= (/) ) -> |^| { y e. On | y ~~ A } e. { y e. On | y ~~ A } )
8	2, 6, 7	sylancr 663	. . 3 |- ( A e. dom card -> |^| { y e. On | y ~~ A } e. { y e. On | y ~~ A } )
9	1, 8	eqeltrd 2555	. 2 |- ( A e. dom card -> ( card ` A ) e. { y e. On | y ~~ A } )
10		breq1 4450	. . . 4 |- ( y = ( card ` A ) -> ( y ~~ A <-> ( card ` A ) ~~ A ) )
11	10	elrab 3261	. . 3 |- ( ( card ` A ) e. { y e. On | y ~~ A } <-> ( ( card ` A ) e. On /\ ( card ` A ) ~~ A ) )
12	11	simprbi 464	. 2 |- ( ( card ` A ) e. { y e. On | y ~~ A } -> ( card ` A ) ~~ A )
13	9, 12	syl 16	1 |- ( A e. dom card -> ( card ` A ) ~~ A )

Syntax hints:   -> wi 4   e. wcel 1767   =/= wne 2662   {crab 2818   _Vcvv 3113   C_ wss 3476   (/)c0 3785   |^|cint 4282   class class class wbr 4447   Oncon0 4878   dom cdm 4999   ` cfv 5588   ~~ cen 7514   cardccrd 8317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6577

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-en 7518  df-card 8321

This theorem is referenced by:  isnum3  8336  oncardid  8338  cardidm  8341  ficardom  8343  ficardid  8344  cardnn  8345  cardnueq0  8346  carden2a  8348  carden2b  8349  carddomi2  8352  sdomsdomcardi  8353  cardsdomelir  8355  cardsdomel  8356  infxpidm2  8395  dfac8b  8413  numdom  8420  alephnbtwn2  8454  alephsucdom  8461  infenaleph  8473  dfac12r  8527  cardacda  8579  pwsdompw  8585  cff1  8639  cfflb  8640  cflim2  8644  cfss  8646  cfslb  8647  domtriomlem  8823  cardid  8923  cardidg  8924  carden  8927  sdomsdomcard  8936  hargch  9052  gch2  9054  hashkf  12376