Theorem bj-xpima2sn 37278

Description: The image of a singleton by a direct product, nonempty case. [To replace xpimasn 6141.] (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 37275	. 2 ⊢ ((𝐴 × 𝐵) “ {𝑋}) = if(𝑋 ∈ 𝐴, 𝐵, ∅)
2		iftrue 4473	. 2 ⊢ (𝑋 ∈ 𝐴 → if(𝑋 ∈ 𝐴, 𝐵, ∅) = 𝐵)
3	1, 2	eqtrid 2784	1 ⊢ (𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∅c0 4274  ifcif 4467  {csn 4568   × cxp 5620   “ cima 5625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2709  ax-sep 5231  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5628  df-rel 5629  df-cnv 5630  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635

This theorem is referenced by:  bj-projval  37316