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Theorem xpima2 6008
 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 6006 . 2 ((𝐴 × 𝐵) “ 𝐶) = if((𝐴𝐶) = ∅, ∅, 𝐵)
2 ifnefalse 4437 . 2 ((𝐴𝐶) ≠ ∅ → if((𝐴𝐶) = ∅, ∅, 𝐵) = 𝐵)
31, 2syl5eq 2845 1 ((𝐴𝐶) ≠ ∅ → ((𝐴 × 𝐵) “ 𝐶) = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ≠ wne 2987   ∩ cin 3880  ∅c0 4243  ifcif 4425   × cxp 5517   “ cima 5522 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295 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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532 This theorem is referenced by:  xpimasn  6009  ustuqtop1  22847  ustuqtop5  22851  brtrclfv2  40423  aacllem  45324
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