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Theorem bj-xpimasn 34908
Description: The image of a singleton, general case. [Change and relabel xpimasn 6062 accordingly, maybe to xpima2sn.] (Contributed by BJ, 6-Apr-2019.)
Assertion
Ref Expression
bj-xpimasn ((𝐴 × 𝐵) “ {𝑋}) = if(𝑋𝐴, 𝐵, ∅)

Proof of Theorem bj-xpimasn
StepHypRef Expression
1 xpima 6059 . 2 ((𝐴 × 𝐵) “ {𝑋}) = if((𝐴 ∩ {𝑋}) = ∅, ∅, 𝐵)
2 disjsn 4641 . . 3 ((𝐴 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋𝐴)
3 eqid 2738 . . 3 𝐵 = 𝐵
42, 3ifbieq2i 4478 . 2 if((𝐴 ∩ {𝑋}) = ∅, ∅, 𝐵) = if(¬ 𝑋𝐴, ∅, 𝐵)
5 ifnot 4505 . 2 if(¬ 𝑋𝐴, ∅, 𝐵) = if(𝑋𝐴, 𝐵, ∅)
61, 4, 53eqtri 2770 1 ((𝐴 × 𝐵) “ {𝑋}) = if(𝑋𝐴, 𝐵, ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1543  wcel 2111  cin 3879  c0 4251  ifcif 4453  {csn 4555   × cxp 5563  cima 5568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5206  ax-nul 5213  ax-pr 5336
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-ral 3067  df-rab 3071  df-v 3422  df-dif 3883  df-un 3885  df-in 3887  df-ss 3897  df-nul 4252  df-if 4454  df-sn 4556  df-pr 4558  df-op 4562  df-br 5068  df-opab 5130  df-xp 5571  df-rel 5572  df-cnv 5573  df-dm 5575  df-rn 5576  df-res 5577  df-ima 5578
This theorem is referenced by:  bj-xpima1sn  34909  bj-xpima2sn  34911
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