Theorem xpcomen 6714

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 4649	. 2 |- ( A X. B ) e. _V
4	2, 1	xpex 4649	. 2 |- ( B X. A ) e. _V
5		eqid 2137	. . 3 |- ( x e. ( A X. B ) |-> U. `' { x } ) = ( x e. ( A X. B ) |-> U. `' { x } )
6	5	xpcomf1o 6712	. 2 |- ( x e. ( A X. B ) |-> U. `' { x } ) : ( A X. B ) -1-1-onto-> ( B X. A )
7		f1oen2g 6642	. 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 1315	1 |- ( A X. B ) ~~ ( B X. A )

Syntax hints:   e. wcel 1480   _Vcvv 2681   {csn 3522   U.cuni 3731   class class class wbr 3924   |-> cmpt 3984   X. cxp 4532   `'ccnv 4533   -1-1-onto->wf1o 5117   ~~ cen 6625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-1st 6031  df-2nd 6032  df-en 6628

This theorem is referenced by:  xpcomeng  6715