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Theorem bj-xpima2sn 36360
Description: The image of a singleton by a direct product, nonempty case. [To replace xpimasn 6183.] (Contributed by BJ, 6-Apr-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-xpima2sn (𝑋𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)

Proof of Theorem bj-xpima2sn
StepHypRef Expression
1 bj-xpimasn 36357 . 2 ((𝐴 × 𝐵) “ {𝑋}) = if(𝑋𝐴, 𝐵, ∅)
2 iftrue 4530 . 2 (𝑋𝐴 → if(𝑋𝐴, 𝐵, ∅) = 𝐵)
31, 2eqtrid 2779 1 (𝑋𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  c0 4318  ifcif 4524  {csn 4624   × cxp 5670  cima 5675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-xp 5678  df-rel 5679  df-cnv 5680  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685
This theorem is referenced by:  bj-projval  36398
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