Theorem xpimasn 5177

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 3785	. . 3 |- ( X e. A -> E. x x e. { X } )
2		snssi 3812	. . . . . 6 |- ( X e. A -> { X } C_ A )
3		dfss1 3408	. . . . . 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 2299	. . . 4 |- ( X e. A -> ( x e. ( A i^i { X } ) <-> x e. { X } ) )
6	5	exbidv 1871	. . 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 5176	. 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 1395   E.wex 1538   e. wcel 2200   i^i cin 3196   C_ wss 3197   {csn 3666   X. cxp 4717   "cima 4722

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4725  df-rel 4726  df-cnv 4727  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732

This theorem is referenced by:  imasnopn  14973