Theorem xpcomeng 6983

Description: Commutative law for equinumerosity of Cartesian product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 27-Mar-2006.)

Assertion

Ref	Expression
xpcomeng	|- ( ( A e. V /\ B e. W ) -> ( A X. B ) ~~ ( B X. A ) )

Proof of Theorem xpcomeng

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpeq1 4732	. . 3 |- ( x = A -> ( x X. y ) = ( A X. y ) )
2		xpeq2 4733	. . 3 |- ( x = A -> ( y X. x ) = ( y X. A ) )
3	1, 2	breq12d 4095	. 2 |- ( x = A -> ( ( x X. y ) ~~ ( y X. x ) <-> ( A X. y ) ~~ ( y X. A ) ) )
4		xpeq2 4733	. . 3 |- ( y = B -> ( A X. y ) = ( A X. B ) )
5		xpeq1 4732	. . 3 |- ( y = B -> ( y X. A ) = ( B X. A ) )
6	4, 5	breq12d 4095	. 2 |- ( y = B -> ( ( A X. y ) ~~ ( y X. A ) <-> ( A X. B ) ~~ ( B X. A ) ) )
7		vex 2802	. . 3 |- x e. _V
8		vex 2802	. . 3 |- y e. _V
9	7, 8	xpcomen 6982	. 2 |- ( x X. y ) ~~ ( y X. x )
10	3, 6, 9	vtocl2g 2865	1 |- ( ( A e. V /\ B e. W ) -> ( A X. B ) ~~ ( B X. A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   class class class wbr 4082   X. cxp 4716   ~~ cen 6883

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-1st 6284  df-2nd 6285  df-en 6886

This theorem is referenced by:  xpsnen2g  6984  xpdom1g  6988  hashxp  11043