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Theorem xpimasn 6207
Description: Direct image of a singleton by a Cartesian product. (Contributed by Thierry Arnoux, 14-Jan-2018.) (Proof shortened by BJ, 6-Apr-2019.)
Assertion
Ref Expression
xpimasn (𝑋𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)

Proof of Theorem xpimasn
StepHypRef Expression
1 disjsn 4716 . . . 4 ((𝐴 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋𝐴)
21necon3abii 2985 . . 3 ((𝐴 ∩ {𝑋}) ≠ ∅ ↔ ¬ ¬ 𝑋𝐴)
3 notnotb 315 . . 3 (𝑋𝐴 ↔ ¬ ¬ 𝑋𝐴)
42, 3bitr4i 278 . 2 ((𝐴 ∩ {𝑋}) ≠ ∅ ↔ 𝑋𝐴)
5 xpima2 6206 . 2 ((𝐴 ∩ {𝑋}) ≠ ∅ → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)
64, 5sylbir 235 1 (𝑋𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1537  wcel 2106  wne 2938  cin 3962  c0 4339  {csn 4631   × cxp 5687  cima 5692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702
This theorem is referenced by:  imasnopn  23714  imasncld  23715  imasncls  23716  restutopopn  24263  arearect  43204
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