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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
StepHypRef 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 ,  (/) )  =  (/) )
42, 3sylbi 195 . 2  |-  ( X  e/  A  ->  if ( X  e.  A ,  B ,  (/) )  =  (/) )
51, 4syl5eq 2507 1  |-  ( X  e/  A  ->  (
( A  X.  B
) " { X } )  =  (/) )
Colors of variables: wff setvar class
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
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