Theorem xpimasn 4789

Description: The image of a singleton by a cross product. (Contributed by Thierry Arnoux, 14-Jan-2018.)

Assertion

Ref	Expression
xpimasn	|- ( X e. A -> ( ( A X. B ) " { X } ) = B )

Proof of Theorem xpimasn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		snmg 3508	. . 3 |- ( X e. A -> E. x x e. { X } )
2		snssi 3529	. . . . . 6 |- ( X e. A -> { X } C_ A )
3		dfss1 3170	. . . . . 6 |- ( { X } C_ A <-> ( A i^i { X } ) = { X } )
4	2, 3	sylib 120	. . . . 5 |- ( X e. A -> ( A i^i { X } ) = { X } )
5	4	eleq2d 2148	. . . 4 |- ( X e. A -> ( x e. ( A i^i { X } ) <-> x e. { X } ) )
6	5	exbidv 1746	. . 3 |- ( X e. A -> ( E. x x e. ( A i^i { X } ) <-> E. x x e. { X } ) )
7	1, 6	mpbird 165	. 2 |- ( X e. A -> E. x x e. ( A i^i { X } ) )
8		xpima2m 4788	. 2 |- ( E. x x e. ( A i^i { X } ) -> ( ( A X. B ) " { X } ) = B )
9	7, 8	syl 14	1 |- ( X e. A -> ( ( A X. B ) " { X } ) = B )

Syntax hints:   -> wi 4   = wceq 1284   E.wex 1421   e. wcel 1433   i^i cin 2972   C_ wss 2973   {csn 3398   X. cxp 4361   "cima 4366

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-cnv 4371  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376

This theorem is referenced by: (None)