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Theorem bj-xpima2sn 34916
Description: The image of a singleton by a direct product, nonempty case. [To replace xpimasn 5437] (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 34913 . 2  |-  ( ( A  X.  B )
" { X }
)  =  if ( X  e.  A ,  B ,  (/) )
2 iftrue 3935 . 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 1398    e. wcel 1823   (/)c0 3783   ifcif 3929   {csn 4016    X. cxp 4986   "cima 4991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-cnv 4996  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001
This theorem is referenced by:  bj-projval  34955
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