Theorem bj-xpimasn 31966

Description: The image of a singleton, general case. [Change and relabel xpimasn 5388 accordingly, maybe to xpima2sn.] (Contributed by BJ, 6-Apr-2019.)

Assertion

Ref	Expression
bj-xpimasn	⊢ ((𝐴 × 𝐵) “ {𝑋}) = if(𝑋 ∈ 𝐴, 𝐵, ∅)

Proof of Theorem bj-xpimasn

Step	Hyp	Ref	Expression
1		xpima 5385	. 2 ⊢ ((𝐴 × 𝐵) “ {𝑋}) = if((𝐴 ∩ {𝑋}) = ∅, ∅, 𝐵)
2		disjsn 4095	. . 3 ⊢ ((𝐴 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ 𝐴)
3		eqid 2514	. . 3 ⊢ 𝐵 = 𝐵
4	2, 3	ifbieq2i 3963	. 2 ⊢ if((𝐴 ∩ {𝑋}) = ∅, ∅, 𝐵) = if(¬ 𝑋 ∈ 𝐴, ∅, 𝐵)
5		ifnot 3986	. 2 ⊢ if(¬ 𝑋 ∈ 𝐴, ∅, 𝐵) = if(𝑋 ∈ 𝐴, 𝐵, ∅)
6	1, 4, 5	3eqtri 2540	1 ⊢ ((𝐴 × 𝐵) “ {𝑋}) = if(𝑋 ∈ 𝐴, 𝐵, ∅)

Syntax hints:  ¬ wn 3   = wceq 1474   ∈ wcel 1938   ∩ cin 3443  ∅c0 3777  ifcif 3939  {csn 4028   × cxp 4930   “ cima 4935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pr 4732

This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-br 4482  df-opab 4542  df-xp 4938  df-rel 4939  df-cnv 4940  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945

This theorem is referenced by:  bj-xpima1sn  31967  bj-xpima2sn  31969