Theorem xpimasn 6185

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	⊢ (𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)

Proof of Theorem xpimasn

Step	Hyp	Ref	Expression
1		disjsn 4716	. . . 4 ⊢ ((𝐴 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ 𝐴)
2	1	necon3abii 2988	. . 3 ⊢ ((𝐴 ∩ {𝑋}) ≠ ∅ ↔ ¬ ¬ 𝑋 ∈ 𝐴)
3		notnotb 315	. . 3 ⊢ (𝑋 ∈ 𝐴 ↔ ¬ ¬ 𝑋 ∈ 𝐴)
4	2, 3	bitr4i 278	. 2 ⊢ ((𝐴 ∩ {𝑋}) ≠ ∅ ↔ 𝑋 ∈ 𝐴)
5		xpima2 6184	. 2 ⊢ ((𝐴 ∩ {𝑋}) ≠ ∅ → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)
6	4, 5	sylbir 234	1 ⊢ (𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542   ∈ wcel 2107   ≠ wne 2941   ∩ cin 3948  ∅c0 4323  {csn 4629   × cxp 5675   “ cima 5680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690

This theorem is referenced by:  imasnopn  23194  imasncld  23195  imasncls  23196  restutopopn  23743  arearect  41964