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Theorem xpima2 6174
Description: Direct image by a Cartesian product (case of nonempty intersection with the domain). (Contributed by Thierry Arnoux, 16-Dec-2017.)
Assertion
Ref Expression
xpima2 ((𝐴𝐶) ≠ ∅ → ((𝐴 × 𝐵) “ 𝐶) = 𝐵)

Proof of Theorem xpima2
StepHypRef Expression
1 xpima 6172 . 2 ((𝐴 × 𝐵) “ 𝐶) = if((𝐴𝐶) = ∅, ∅, 𝐵)
2 ifnefalse 4533 . 2 ((𝐴𝐶) ≠ ∅ → if((𝐴𝐶) = ∅, ∅, 𝐵) = 𝐵)
31, 2eqtrid 2776 1 ((𝐴𝐶) ≠ ∅ → ((𝐴 × 𝐵) “ 𝐶) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wne 2932  cin 3940  c0 4315  ifcif 4521   × cxp 5665  cima 5670
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 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-xp 5673  df-rel 5674  df-cnv 5675  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680
This theorem is referenced by:  xpimasn  6175  ustuqtop1  24090  ustuqtop5  24094  prjcrv0  41927  brtrclfv2  43028  aacllem  48096
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