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 4679	. . . 4 ⊢ ((𝐴 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ 𝐴)
2	1	necon3abii 3006	. . 3 ⊢ ((𝐴 ∩ {𝑋}) ≠ ∅ ↔ ¬ ¬ 𝑋 ∈ 𝐴)
3		notnotb 318	. . 3 ⊢ (𝑋 ∈ 𝐴 ↔ ¬ ¬ 𝑋 ∈ 𝐴)
4	2, 3	bitr4i 281	. 2 ⊢ ((𝐴 ∩ {𝑋}) ≠ ∅ ↔ 𝑋 ∈ 𝐴)
5		xpima2 6184	. 2 ⊢ ((𝐴 ∩ {𝑋}) ≠ ∅ → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)
6	4, 5	sylbir 238	1 ⊢ (𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2146   ≠ wne 2960   ∩ cin 3905  ∅c0 4286  {csn 4591   × cxp 5661   “ cima 5666

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-11 2195  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-rel 5670  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676

This theorem is used by:  imasnopn  23898  imasncld  23899  imasncls  23900  restutopopn  24446  arearect  44000