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Theorem bj-xpima1sn 34861
Description: The image of a singleton by a direct product, empty case. [Change and relabel xpimasn 5362 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 34860 . 2  |-  ( ( A  X.  B )
" { X }
)  =  if ( X  e.  A ,  B ,  (/) )
2 df-nel 2580 . . 3  |-  ( X  e/  A  <->  -.  X  e.  A )
3 iffalse 3866 . . 3  |-  ( -.  X  e.  A  ->  if ( X  e.  A ,  B ,  (/) )  =  (/) )
42, 3sylbi 195 . 2  |-  ( X  e/  A  ->  if ( X  e.  A ,  B ,  (/) )  =  (/) )
51, 4syl5eq 2435 1  |-  ( X  e/  A  ->  (
( A  X.  B
) " { X } )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1399    e. wcel 1826    e/ wnel 2578   (/)c0 3711   ifcif 3857   {csn 3944    X. cxp 4911   "cima 4916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-br 4368  df-opab 4426  df-xp 4919  df-rel 4920  df-cnv 4921  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926
This theorem is referenced by:  bj-projval  34902
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