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Theorem xpimasn 6194
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 4720 . . . 4 ((𝐴 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋𝐴)
21necon3abii 2984 . . 3 ((𝐴 ∩ {𝑋}) ≠ ∅ ↔ ¬ ¬ 𝑋𝐴)
3 notnotb 314 . . 3 (𝑋𝐴 ↔ ¬ ¬ 𝑋𝐴)
42, 3bitr4i 277 . 2 ((𝐴 ∩ {𝑋}) ≠ ∅ ↔ 𝑋𝐴)
5 xpima2 6193 . 2 ((𝐴 ∩ {𝑋}) ≠ ∅ → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)
64, 5sylbir 234 1 (𝑋𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1533  wcel 2098  wne 2937  cin 3948  c0 4326  {csn 4632   × cxp 5680  cima 5685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-xp 5688  df-rel 5689  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695
This theorem is referenced by:  imasnopn  23614  imasncld  23615  imasncls  23616  restutopopn  24163  arearect  42674
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