Theorem xpcomen 6793

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 4719	. 2 |- ( A X. B ) e. _V
4	2, 1	xpex 4719	. 2 |- ( B X. A ) e. _V
5		eqid 2165	. . 3 |- ( x e. ( A X. B ) |-> U. `' { x } ) = ( x e. ( A X. B ) |-> U. `' { x } )
6	5	xpcomf1o 6791	. 2 |- ( x e. ( A X. B ) |-> U. `' { x } ) : ( A X. B ) -1-1-onto-> ( B X. A )
7		f1oen2g 6721	. 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 1327	1 |- ( A X. B ) ~~ ( B X. A )

Syntax hints:   e. wcel 2136   _Vcvv 2726   {csn 3576   U.cuni 3789   class class class wbr 3982   |-> cmpt 4043   X. cxp 4602   `'ccnv 4603   -1-1-onto->wf1o 5187   ~~ cen 6704

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-1st 6108  df-2nd 6109  df-en 6707

This theorem is referenced by:  xpcomeng  6794