Theorem xpcomen 7091

Description: Commutative law for equinumerosity of Cartesian product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 5-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)

Hypotheses

Ref	Expression
xpcomen.1	|- A e. _V
xpcomen.2	|- B e. _V

Assertion

Ref	Expression
xpcomen	|- ( A X. B ) ~~ ( B X. A )

Proof of Theorem xpcomen

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpcomen.1	. . 3 |- A e. _V
2		xpcomen.2	. . 3 |- B e. _V
3	1, 2	xpex 4871	. 2 |- ( A X. B ) e. _V
4	2, 1	xpex 4871	. 2 |- ( B X. A ) e. _V
5		eqid 2234	. . 3 |- ( x e. ( A X. B ) |-> U. `' { x } ) = ( x e. ( A X. B ) |-> U. `' { x } )
6	5	xpcomf1o 7089	. 2 |- ( x e. ( A X. B ) |-> U. `' { x } ) : ( A X. B ) -1-1-onto-> ( B X. A )
7		f1oen2g 7007	. 2 |- ( ( ( A X. B ) e. _V /\ ( B X. A ) e. _V /\ ( x e. ( A X. B ) |-> U. `' { x } ) : ( A X. B ) -1-1-onto-> ( B X. A ) ) -> ( A X. B ) ~~ ( B X. A ) )
8	3, 4, 6, 7	mp3an 1374	1 |- ( A X. B ) ~~ ( B X. A )

Syntax hints:   e. wcel 2205   _Vcvv 2815   {csn 3694   U.cuni 3919   class class class wbr 4114   |-> cmpt 4176   X. cxp 4752   `'ccnv 4753   -1-1-onto->wf1o 5356   ~~ cen 6986

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-1st 6347  df-2nd 6348  df-en 6989

This theorem is referenced by:  xpcomeng  7092