Theorem xpimasn 5150

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 3761	. . 3 |- ( X e. A -> E. x x e. { X } )
2		snssi 3788	. . . . . 6 |- ( X e. A -> { X } C_ A )
3		dfss1 3385	. . . . . 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 2277	. . . 4 |- ( X e. A -> ( x e. ( A i^i { X } ) <-> x e. { X } ) )
6	5	exbidv 1849	. . 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 5149	. 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 1373   E.wex 1516   e. wcel 2178   i^i cin 3173   C_ wss 3174   {csn 3643   X. cxp 4691   "cima 4696

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-xp 4699  df-rel 4700  df-cnv 4701  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706

This theorem is referenced by:  imasnopn  14886