Theorem xpcomen 6947

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 4808	. 2 |- ( A X. B ) e. _V
4	2, 1	xpex 4808	. 2 |- ( B X. A ) e. _V
5		eqid 2207	. . 3 |- ( x e. ( A X. B ) |-> U. `' { x } ) = ( x e. ( A X. B ) |-> U. `' { x } )
6	5	xpcomf1o 6945	. 2 |- ( x e. ( A X. B ) |-> U. `' { x } ) : ( A X. B ) -1-1-onto-> ( B X. A )
7		f1oen2g 6869	. 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 1350	1 |- ( A X. B ) ~~ ( B X. A )

Syntax hints:   e. wcel 2178   _Vcvv 2776   {csn 3643   U.cuni 3864   class class class wbr 4059   |-> cmpt 4121   X. cxp 4691   `'ccnv 4692   -1-1-onto->wf1o 5289   ~~ cen 6848

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-1st 6249  df-2nd 6250  df-en 6851

This theorem is referenced by:  xpcomeng  6948