Theorem bj-xpima2sn 36431

Description: The image of a singleton by a direct product, nonempty case. [To replace xpimasn 6183.] (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 36428	. 2 ⊢ ((𝐴 × 𝐵) “ {𝑋}) = if(𝑋 ∈ 𝐴, 𝐵, ∅)
2		iftrue 4530	. 2 ⊢ (𝑋 ∈ 𝐴 → if(𝑋 ∈ 𝐴, 𝐵, ∅) = 𝐵)
3	1, 2	eqtrid 2780	1 ⊢ (𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099  ∅c0 4318  ifcif 4524  {csn 4624   × cxp 5670   “ cima 5675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-xp 5678  df-rel 5679  df-cnv 5680  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685

This theorem is referenced by:  bj-projval  36469