Theorem xpsneng 6916

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 4688	. . 3 |- ( x = A -> ( x X. { y } ) = ( A X. { y } ) )
2		id 19	. . 3 |- ( x = A -> x = A )
3	1, 2	breq12d 4056	. 2 |- ( x = A -> ( ( x X. { y } ) ~~ x <-> ( A X. { y } ) ~~ A ) )
4		sneq 3643	. . . 4 |- ( y = B -> { y } = { B } )
5	4	xpeq2d 4698	. . 3 |- ( y = B -> ( A X. { y } ) = ( A X. { B } ) )
6	5	breq1d 4053	. 2 |- ( y = B -> ( ( A X. { y } ) ~~ A <-> ( A X. { B } ) ~~ A ) )
7		vex 2774	. . 3 |- x e. _V
8		vex 2774	. . 3 |- y e. _V
9	7, 8	xpsnen 6915	. 2 |- ( x X. { y } ) ~~ x
10	3, 6, 9	vtocl2g 2836	1 |- ( ( A e. V /\ B e. W ) -> ( A X. { B } ) ~~ A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1372   e. wcel 2175   {csn 3632   class class class wbr 4043   X. cxp 4672   ~~ cen 6824

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-en 6827

This theorem is referenced by:  xp1en  6917  xpsnen2g  6923  xpdom3m  6928  hashxp  10969  pwf1oexmid  15898