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Theorem xpima2 5834
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 5832 . 2 ((𝐴 × 𝐵) “ 𝐶) = if((𝐴𝐶) = ∅, ∅, 𝐵)
2 ifnefalse 4319 . 2 ((𝐴𝐶) ≠ ∅ → if((𝐴𝐶) = ∅, ∅, 𝐵) = 𝐵)
31, 2syl5eq 2826 1 ((𝐴𝐶) ≠ ∅ → ((𝐴 × 𝐵) “ 𝐶) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wne 2969  cin 3791  c0 4141  ifcif 4307   × cxp 5355  cima 5360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4889  df-opab 4951  df-xp 5363  df-rel 5364  df-cnv 5365  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370
This theorem is referenced by:  xpimasn  5835  ustuqtop1  22464  ustuqtop5  22468  brtrclfv2  38990  aacllem  43667
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