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Theorem bj-xpima2sn 32767
Description: The image of a singleton by a direct product, nonempty case. [To replace xpimasn 5390] (Contributed by BJ, 6-Apr-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-xpima2sn  |-  ( X  e.  A  ->  (
( A  X.  B
) " { X } )  =  B )

Proof of Theorem bj-xpima2sn
StepHypRef Expression
1 bj-xpimasn 32764 . 2  |-  ( ( A  X.  B )
" { X }
)  =  if ( X  e.  A ,  B ,  (/) )
2 iftrue 3904 . 2  |-  ( X  e.  A  ->  if ( X  e.  A ,  B ,  (/) )  =  B )
31, 2syl5eq 2507 1  |-  ( X  e.  A  ->  (
( A  X.  B
) " { X } )  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   (/)c0 3744   ifcif 3898   {csn 3984    X. cxp 4945   "cima 4950
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 4520  ax-nul 4528  ax-pr 4638
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 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-br 4400  df-opab 4458  df-xp 4953  df-rel 4954  df-cnv 4955  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960
This theorem is referenced by:  bj-projval  32806
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