Theorem bj-xpima1sn 32781

Description: The image of a singleton by a direct product, empty case. [Change and relabel xpimasn 5392 accordingly, maybe to xpima2sn.] (Contributed by BJ, 6-Apr-2019.)

Assertion

Ref	Expression
bj-xpima1sn	|- ( X e/ A -> ( ( A X. B ) " { X } ) = (/) )

Proof of Theorem bj-xpima1sn

Step	Hyp	Ref	Expression
1		bj-xpimasn 32780	. 2 |- ( ( A X. B ) " { X } ) = if ( X e. A , B , (/) )
2		df-nel 2651	. . 3 |- ( X e/ A <-> -. X e. A )
3		iffalse 3908	. . 3 |- ( -. X e. A -> if ( X e. A , B , (/) ) = (/) )
4	2, 3	sylbi 195	. 2 |- ( X e/ A -> if ( X e. A , B , (/) ) = (/) )
5	1, 4	syl5eq 2507	1 |- ( X e/ A -> ( ( A X. B ) " { X } ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1370   e. wcel 1758   e/ wnel 2649   (/)c0 3746   ifcif 3900   {csn 3986   X. cxp 4947   "cima 4952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-br 4402  df-opab 4460  df-xp 4955  df-rel 4956  df-cnv 4957  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962

This theorem is referenced by:  bj-projval  32822