Theorem fmptsnxp 39349

Description: Maps-to notation and cross product for a singleton function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Assertion

Ref	Expression
fmptsnxp	|- ( ( A e. V /\ B e. W ) -> ( x e. { A } |-> B ) = ( { A } X. { B } ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hints:   V( x)   W( x)

Proof of Theorem fmptsnxp

Step	Hyp	Ref	Expression
1		xpsng 6406	. 2 |- ( ( A e. V /\ B e. W ) -> ( { A } X. { B } ) = { <. A , B >. } )
2		fmptsn 6433	. 2 |- ( ( A e. V /\ B e. W ) -> { <. A , B >. } = ( x e. { A } |-> B ) )
3	1, 2	eqtr2d 2657	1 |- ( ( A e. V /\ B e. W ) -> ( x e. { A } |-> B ) = ( { A } X. { B } ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {csn 4177   <.cop 4183   |-> cmpt 4729   X. cxp 5112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895

This theorem is referenced by:  cnfdmsn  40095