Theorem xpimasn 4769

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 3486	. . 3 |- ( X e. A -> E. x x e. { X } )
2		snssi 3508	. . . . . 6 |- ( X e. A -> { X } C_ A )
3		dfss1 3141	. . . . . 6 |- ( { X } C_ A <-> ( A i^i { X } ) = { X } )
4	2, 3	sylib 127	. . . . 5 |- ( X e. A -> ( A i^i { X } ) = { X } )
5	4	eleq2d 2107	. . . 4 |- ( X e. A -> ( x e. ( A i^i { X } ) <-> x e. { X } ) )
6	5	exbidv 1706	. . 3 |- ( X e. A -> ( E. x x e. ( A i^i { X } ) <-> E. x x e. { X } ) )
7	1, 6	mpbird 156	. 2 |- ( X e. A -> E. x x e. ( A i^i { X } ) )
8		xpima2m 4768	. 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 1243   E.wex 1381   e. wcel 1393   i^i cin 2916   C_ wss 2917   {csn 3375   X. cxp 4343   "cima 4348

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358

This theorem is referenced by: (None)