Theorem xpsneng 6800

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 4625	. . 3 |- ( x = A -> ( x X. { y } ) = ( A X. { y } ) )
2		id 19	. . 3 |- ( x = A -> x = A )
3	1, 2	breq12d 4002	. 2 |- ( x = A -> ( ( x X. { y } ) ~~ x <-> ( A X. { y } ) ~~ A ) )
4		sneq 3594	. . . 4 |- ( y = B -> { y } = { B } )
5	4	xpeq2d 4635	. . 3 |- ( y = B -> ( A X. { y } ) = ( A X. { B } ) )
6	5	breq1d 3999	. 2 |- ( y = B -> ( ( A X. { y } ) ~~ A <-> ( A X. { B } ) ~~ A ) )
7		vex 2733	. . 3 |- x e. _V
8		vex 2733	. . 3 |- y e. _V
9	7, 8	xpsnen 6799	. 2 |- ( x X. { y } ) ~~ x
10	3, 6, 9	vtocl2g 2794	1 |- ( ( A e. V /\ B e. W ) -> ( A X. { B } ) ~~ A )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   {csn 3583   class class class wbr 3989   X. cxp 4609   ~~ cen 6716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-en 6719

This theorem is referenced by:  xp1en  6801  xpsnen2g  6807  xpdom3m  6812  hashxp  10761  pwf1oexmid  14032