Theorem xpimasn 5185

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 3790	. . 3 |- ( X e. A -> E. x x e. { X } )
2		snssi 3817	. . . . . 6 |- ( X e. A -> { X } C_ A )
3		dfss1 3411	. . . . . 6 |- ( { X } C_ A <-> ( A i^i { X } ) = { X } )
4	2, 3	sylib 122	. . . . 5 |- ( X e. A -> ( A i^i { X } ) = { X } )
5	4	eleq2d 2301	. . . 4 |- ( X e. A -> ( x e. ( A i^i { X } ) <-> x e. { X } ) )
6	5	exbidv 1873	. . 3 |- ( X e. A -> ( E. x x e. ( A i^i { X } ) <-> E. x x e. { X } ) )
7	1, 6	mpbird 167	. 2 |- ( X e. A -> E. x x e. ( A i^i { X } ) )
8		xpima2m 5184	. 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 1397   E.wex 1540   e. wcel 2202   i^i cin 3199   C_ wss 3200   {csn 3669   X. cxp 4723   "cima 4728

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-cnv 4733  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738

This theorem is referenced by:  imasnopn  15022