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Theorem xpimasn 5614
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 4278 . . . 4 ((𝐴 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋𝐴)
21necon3abii 2869 . . 3 ((𝐴 ∩ {𝑋}) ≠ ∅ ↔ ¬ ¬ 𝑋𝐴)
3 notnotb 304 . . 3 (𝑋𝐴 ↔ ¬ ¬ 𝑋𝐴)
42, 3bitr4i 267 . 2 ((𝐴 ∩ {𝑋}) ≠ ∅ ↔ 𝑋𝐴)
5 xpima2 5613 . 2 ((𝐴 ∩ {𝑋}) ≠ ∅ → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)
64, 5sylbir 225 1 (𝑋𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1523  wcel 2030  wne 2823  cin 3606  c0 3948  {csn 4210   × cxp 5141  cima 5146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156
This theorem is referenced by:  imasnopn  21541  imasncld  21542  imasncls  21543  restutopopn  22089  arearect  38118
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