Theorem xpsneng 7049

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 4745	. . 3 |- ( x = A -> ( x X. { y } ) = ( A X. { y } ) )
2		id 19	. . 3 |- ( x = A -> x = A )
3	1, 2	breq12d 4106	. 2 |- ( x = A -> ( ( x X. { y } ) ~~ x <-> ( A X. { y } ) ~~ A ) )
4		sneq 3684	. . . 4 |- ( y = B -> { y } = { B } )
5	4	xpeq2d 4755	. . 3 |- ( y = B -> ( A X. { y } ) = ( A X. { B } ) )
6	5	breq1d 4103	. 2 |- ( y = B -> ( ( A X. { y } ) ~~ A <-> ( A X. { B } ) ~~ A ) )
7		vex 2806	. . 3 |- x e. _V
8		vex 2806	. . 3 |- y e. _V
9	7, 8	xpsnen 7048	. 2 |- ( x X. { y } ) ~~ x
10	3, 6, 9	vtocl2g 2869	1 |- ( ( A e. V /\ B e. W ) -> ( A X. { B } ) ~~ A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   {csn 3673   class class class wbr 4093   X. cxp 4729   ~~ cen 6950

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-en 6953

This theorem is referenced by:  xp1en  7050  xpsnen2g  7056  xpdom3m  7061  hashxp  11136  pwf1oexmid  16704