Theorem xpsneng 7110

Description: A set is equinumerous to its Cartesian product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM, 22-Oct-2004.)

Assertion

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

Proof of Theorem xpsneng

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

Step	Hyp	Ref	Expression
1		xpeq1 4783	. . 3 |- ( x = A -> ( x X. { y } ) = ( A X. { y } ) )
2		id 19	. . 3 |- ( x = A -> x = A )
3	1, 2	breq12d 4138	. 2 |- ( x = A -> ( ( x X. { y } ) ~~ x <-> ( A X. { y } ) ~~ A ) )
4		sneq 3716	. . . 4 |- ( y = B -> { y } = { B } )
5	4	xpeq2d 4793	. . 3 |- ( y = B -> ( A X. { y } ) = ( A X. { B } ) )
6	5	breq1d 4135	. 2 |- ( y = B -> ( ( A X. { y } ) ~~ A <-> ( A X. { B } ) ~~ A ) )
7		vex 2824	. . 3 |- x e. _V
8		vex 2824	. . 3 |- y e. _V
9	7, 8	xpsnen 7109	. 2 |- ( x X. { y } ) ~~ x
10	3, 6, 9	vtocl2g 2887	1 |- ( ( A e. V /\ B e. W ) -> ( A X. { B } ) ~~ A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   {csn 3705   class class class wbr 4125   X. cxp 4767   ~~ cen 7010

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 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 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem 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-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-en 7013

This theorem is referenced by:  xp1en  7111  xpsnen2g  7117  xpdom3m  7122  hashxp  11245  pwf1oexmid  16943