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Theorem xpima2 5482
Description: The image by a Cartesian product. (Contributed by Thierry Arnoux, 16-Dec-2017.)
Assertion
Ref Expression
xpima2 ((𝐴𝐶) ≠ ∅ → ((𝐴 × 𝐵) “ 𝐶) = 𝐵)

Proof of Theorem xpima2
StepHypRef Expression
1 xpima 5480 . 2 ((𝐴 × 𝐵) “ 𝐶) = if((𝐴𝐶) = ∅, ∅, 𝐵)
2 ifnefalse 4047 . 2 ((𝐴𝐶) ≠ ∅ → if((𝐴𝐶) = ∅, ∅, 𝐵) = 𝐵)
31, 2syl5eq 2655 1 ((𝐴𝐶) ≠ ∅ → ((𝐴 × 𝐵) “ 𝐶) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wne 2779  cin 3538  c0 3873  ifcif 4035   × cxp 5025  cima 5030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-xp 5033  df-rel 5034  df-cnv 5035  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040
This theorem is referenced by:  xpimasn  5483  ustuqtop1  21802  ustuqtop5  21806  brtrclfv2  36821  aacllem  42298
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