Theorem bj-xpima2sn 37622

Description: The image of a singleton by a direct product, nonempty case. [To replace xpimasn 6182.] (Contributed by BJ, 6-Apr-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-xpima2sn	⊢ (𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)

Proof of Theorem bj-xpima2sn

Step	Hyp	Ref	Expression
1		bj-xpimasn 37619	. 2 ⊢ ((𝐴 × 𝐵) “ {𝑋}) = if(𝑋 ∈ 𝐴, 𝐵, ∅)
2		iftrue 4492	. 2 ⊢ (𝑋 ∈ 𝐴 → if(𝑋 ∈ 𝐴, 𝐵, ∅) = 𝐵)
3	1, 2	eqtrid 2809	1 ⊢ (𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2142  ∅c0 4285  ifcif 4486  {csn 4588   × cxp 5658   “ cima 5663

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-11 2191  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-xp 5666  df-rel 5667  df-cnv 5668  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673

This theorem is used by:  bj-projval  37660