Theorem bj-xpimasn 36585

Description: The image of a singleton, general case. [Change and relabel xpimasn 6191 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 6188	. 2 ⊢ ((𝐴 × 𝐵) “ {𝑋}) = if((𝐴 ∩ {𝑋}) = ∅, ∅, 𝐵)
2		disjsn 4717	. . 3 ⊢ ((𝐴 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ 𝐴)
3		eqid 2725	. . 3 ⊢ 𝐵 = 𝐵
4	2, 3	ifbieq2i 4555	. 2 ⊢ if((𝐴 ∩ {𝑋}) = ∅, ∅, 𝐵) = if(¬ 𝑋 ∈ 𝐴, ∅, 𝐵)
5		ifnot 4582	. 2 ⊢ if(¬ 𝑋 ∈ 𝐴, ∅, 𝐵) = if(𝑋 ∈ 𝐴, 𝐵, ∅)
6	1, 4, 5	3eqtri 2757	1 ⊢ ((𝐴 × 𝐵) “ {𝑋}) = if(𝑋 ∈ 𝐴, 𝐵, ∅)

Syntax hints:  ¬ wn 3   = wceq 1533   ∈ wcel 2098   ∩ cin 3943  ∅c0 4322  ifcif 4530  {csn 4630   × cxp 5676   “ cima 5681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-xp 5684  df-rel 5685  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691

This theorem is referenced by:  bj-xpima1sn  36586  bj-xpima2sn  36588