Theorem bj-xpima2sn 35148

Description: The image of a singleton by a direct product, nonempty case. [To replace xpimasn 6088.] (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 35145	. 2 ⊢ ((𝐴 × 𝐵) “ {𝑋}) = if(𝑋 ∈ 𝐴, 𝐵, ∅)
2		iftrue 4465	. 2 ⊢ (𝑋 ∈ 𝐴 → if(𝑋 ∈ 𝐴, 𝐵, ∅) = 𝐵)
3	1, 2	eqtrid 2790	1 ⊢ (𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  ∅c0 4256  ifcif 4459  {csn 4561   × cxp 5587   “ cima 5592

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602

This theorem is referenced by:  bj-projval  35186