Theorem xpcomeng 7130

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 4790	. . 3 |- ( x = A -> ( x X. y ) = ( A X. y ) )
2		xpeq2 4791	. . 3 |- ( x = A -> ( y X. x ) = ( y X. A ) )
3	1, 2	breq12d 4144	. 2 |- ( x = A -> ( ( x X. y ) ~~ ( y X. x ) <-> ( A X. y ) ~~ ( y X. A ) ) )
4		xpeq2 4791	. . 3 |- ( y = B -> ( A X. y ) = ( A X. B ) )
5		xpeq1 4790	. . 3 |- ( y = B -> ( y X. A ) = ( B X. A ) )
6	4, 5	breq12d 4144	. 2 |- ( y = B -> ( ( A X. y ) ~~ ( y X. A ) <-> ( A X. B ) ~~ ( B X. A ) ) )
7		vex 2824	. . 3 |- x e. _V
8		vex 2824	. . 3 |- y e. _V
9	7, 8	xpcomen 7129	. 2 |- ( x X. y ) ~~ ( y X. x )
10	3, 6, 9	vtocl2g 2887	1 |- ( ( A e. V /\ B e. W ) -> ( A X. B ) ~~ ( B X. A ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   class class class wbr 4131   X. cxp 4774   ~~ cen 7024

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-1st 6378  df-2nd 6379  df-en 7027

This theorem is used by:  xpsnen2g  7131  xpdom1g  7135  hashxp  11289