Theorem xpsneng 6917

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 4689	. . 3 |- ( x = A -> ( x X. { y } ) = ( A X. { y } ) )
2		id 19	. . 3 |- ( x = A -> x = A )
3	1, 2	breq12d 4057	. 2 |- ( x = A -> ( ( x X. { y } ) ~~ x <-> ( A X. { y } ) ~~ A ) )
4		sneq 3644	. . . 4 |- ( y = B -> { y } = { B } )
5	4	xpeq2d 4699	. . 3 |- ( y = B -> ( A X. { y } ) = ( A X. { B } ) )
6	5	breq1d 4054	. 2 |- ( y = B -> ( ( A X. { y } ) ~~ A <-> ( A X. { B } ) ~~ A ) )
7		vex 2775	. . 3 |- x e. _V
8		vex 2775	. . 3 |- y e. _V
9	7, 8	xpsnen 6916	. 2 |- ( x X. { y } ) ~~ x
10	3, 6, 9	vtocl2g 2837	1 |- ( ( A e. V /\ B e. W ) -> ( A X. { B } ) ~~ A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   {csn 3633   class class class wbr 4044   X. cxp 4673   ~~ cen 6825

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-en 6828

This theorem is referenced by:  xp1en  6918  xpsnen2g  6924  xpdom3m  6929  hashxp  10971  pwf1oexmid  15936