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Theorem bj-xpimasn 34303
Description: The image of a singleton, general case. [Change and relabel xpimasn 6029 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 6026 . 2 ((𝐴 × 𝐵) “ {𝑋}) = if((𝐴 ∩ {𝑋}) = ∅, ∅, 𝐵)
2 disjsn 4631 . . 3 ((𝐴 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋𝐴)
3 eqid 2824 . . 3 𝐵 = 𝐵
42, 3ifbieq2i 4473 . 2 if((𝐴 ∩ {𝑋}) = ∅, ∅, 𝐵) = if(¬ 𝑋𝐴, ∅, 𝐵)
5 ifnot 4499 . 2 if(¬ 𝑋𝐴, ∅, 𝐵) = if(𝑋𝐴, 𝐵, ∅)
61, 4, 53eqtri 2851 1 ((𝐴 × 𝐵) “ {𝑋}) = if(𝑋𝐴, 𝐵, ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1538  wcel 2115  cin 3918  c0 4275  ifcif 4449  {csn 4549   × cxp 5540  cima 5545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-sn 4550  df-pr 4552  df-op 4556  df-br 5053  df-opab 5115  df-xp 5548  df-rel 5549  df-cnv 5550  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555
This theorem is referenced by:  bj-xpima1sn  34304  bj-xpima2sn  34306
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