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Theorem bj-xpimasn 33267
Description: The image of a singleton, general case. [Change and relabel xpimasn 5738 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 5735 . 2 ((𝐴 × 𝐵) “ {𝑋}) = if((𝐴 ∩ {𝑋}) = ∅, ∅, 𝐵)
2 disjsn 4391 . . 3 ((𝐴 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋𝐴)
3 eqid 2761 . . 3 𝐵 = 𝐵
42, 3ifbieq2i 4255 . 2 if((𝐴 ∩ {𝑋}) = ∅, ∅, 𝐵) = if(¬ 𝑋𝐴, ∅, 𝐵)
5 ifnot 4278 . 2 if(¬ 𝑋𝐴, ∅, 𝐵) = if(𝑋𝐴, 𝐵, ∅)
61, 4, 53eqtri 2787 1 ((𝐴 × 𝐵) “ {𝑋}) = if(𝑋𝐴, 𝐵, ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1632  wcel 2140  cin 3715  c0 4059  ifcif 4231  {csn 4322   × cxp 5265  cima 5270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-br 4806  df-opab 4866  df-xp 5273  df-rel 5274  df-cnv 5275  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280
This theorem is referenced by:  bj-xpima1sn  33268  bj-xpima2sn  33270
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