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Theorem bj-xpima1sn 35534
Description: The image of a singleton by a direct product, empty case. [Change and relabel xpimasn 6157 accordingly, maybe to xpima2sn.] (Contributed by BJ, 6-Apr-2019.)
Assertion
Ref Expression
bj-xpima1sn 𝑋𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅)

Proof of Theorem bj-xpima1sn
StepHypRef Expression
1 bj-xpimasn 35533 . 2 ((𝐴 × 𝐵) “ {𝑋}) = if(𝑋𝐴, 𝐵, ∅)
2 iffalse 4515 . 2 𝑋𝐴 → if(𝑋𝐴, 𝐵, ∅) = ∅)
31, 2eqtrid 2783 1 𝑋𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1541  wcel 2106  c0 4302  ifcif 4506  {csn 4606   × cxp 5651  cima 5656
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5276  ax-nul 5283  ax-pr 5404
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rab 3419  df-v 3461  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-sn 4607  df-pr 4609  df-op 4613  df-br 5126  df-opab 5188  df-xp 5659  df-rel 5660  df-cnv 5661  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666
This theorem is referenced by:  bj-projval  35574
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