Theorem xpcomeng 8052

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 5128	. . 3 |- ( x = A -> ( x X. y ) = ( A X. y ) )
2		xpeq2 5129	. . 3 |- ( x = A -> ( y X. x ) = ( y X. A ) )
3	1, 2	breq12d 4666	. 2 |- ( x = A -> ( ( x X. y ) ~~ ( y X. x ) <-> ( A X. y ) ~~ ( y X. A ) ) )
4		xpeq2 5129	. . 3 |- ( y = B -> ( A X. y ) = ( A X. B ) )
5		xpeq1 5128	. . 3 |- ( y = B -> ( y X. A ) = ( B X. A ) )
6	4, 5	breq12d 4666	. 2 |- ( y = B -> ( ( A X. y ) ~~ ( y X. A ) <-> ( A X. B ) ~~ ( B X. A ) ) )
7		vex 3203	. . 3 |- x e. _V
8		vex 3203	. . 3 |- y e. _V
9	7, 8	xpcomen 8051	. 2 |- ( x X. y ) ~~ ( y X. x )
10	3, 6, 9	vtocl2g 3270	1 |- ( ( A e. V /\ B e. W ) -> ( A X. B ) ~~ ( B X. A ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653   X. cxp 5112   ~~ cen 7952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-1st 7168  df-2nd 7169  df-en 7956

This theorem is referenced by:  xpsnen2g  8053  xpdom1g  8057  omxpen  8062  xpfir  8182  infxp  9037  infmap2  9040  enrelmap  38291  enrelmapr  38292