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Theorem xpimasn 5483
Description: The 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 4191 . . . 4 ((𝐴 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋𝐴)
21necon3abii 2827 . . 3 ((𝐴 ∩ {𝑋}) ≠ ∅ ↔ ¬ ¬ 𝑋𝐴)
3 notnotb 302 . . 3 (𝑋𝐴 ↔ ¬ ¬ 𝑋𝐴)
42, 3bitr4i 265 . 2 ((𝐴 ∩ {𝑋}) ≠ ∅ ↔ 𝑋𝐴)
5 xpima2 5482 . 2 ((𝐴 ∩ {𝑋}) ≠ ∅ → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)
64, 5sylbir 223 1 (𝑋𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1474  wcel 1976  wne 2779  cin 3538  c0 3873  {csn 4124   × cxp 5025  cima 5030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-xp 5033  df-rel 5034  df-cnv 5035  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040
This theorem is referenced by:  imasnopn  21250  imasncld  21251  imasncls  21252  restutopopn  21799  arearect  36603
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