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Theorem xpimasn 6048
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 4627 . . . 4 ((𝐴 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋𝐴)
21necon3abii 2987 . . 3 ((𝐴 ∩ {𝑋}) ≠ ∅ ↔ ¬ ¬ 𝑋𝐴)
3 notnotb 318 . . 3 (𝑋𝐴 ↔ ¬ ¬ 𝑋𝐴)
42, 3bitr4i 281 . 2 ((𝐴 ∩ {𝑋}) ≠ ∅ ↔ 𝑋𝐴)
5 xpima2 6047 . 2 ((𝐴 ∩ {𝑋}) ≠ ∅ → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)
64, 5sylbir 238 1 (𝑋𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1543  wcel 2110  wne 2940  cin 3865  c0 4237  {csn 4541   × cxp 5549  cima 5554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-xp 5557  df-rel 5558  df-cnv 5559  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564
This theorem is referenced by:  imasnopn  22587  imasncld  22588  imasncls  22589  restutopopn  23136  arearect  40749
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