Theorem bj-xpima2sn 36939

Description: The image of a singleton by a direct product, nonempty case. [To replace xpimasn 6146.] (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 36936	. 2 ⊢ ((𝐴 × 𝐵) “ {𝑋}) = if(𝑋 ∈ 𝐴, 𝐵, ∅)
2		iftrue 4490	. 2 ⊢ (𝑋 ∈ 𝐴 → if(𝑋 ∈ 𝐴, 𝐵, ∅) = 𝐵)
3	1, 2	eqtrid 2776	1 ⊢ (𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ∅c0 4292  ifcif 4484  {csn 4585   × cxp 5629   “ cima 5634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-xp 5637  df-rel 5638  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644

This theorem is referenced by:  bj-projval  36977