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Theorem xpima2 5393
Description: The image by a Cartesian product. (Contributed by Thierry Arnoux, 16-Dec-2017.)
Assertion
Ref Expression
xpima2  |-  ( ( A  i^i  C )  =/=  (/)  ->  ( ( A  X.  B ) " C )  =  B )

Proof of Theorem xpima2
StepHypRef Expression
1 xpima 5391 . 2  |-  ( ( A  X.  B )
" C )  =  if ( ( A  i^i  C )  =  (/) ,  (/) ,  B )
2 ifnefalse 3912 . 2  |-  ( ( A  i^i  C )  =/=  (/)  ->  if (
( A  i^i  C
)  =  (/) ,  (/) ,  B )  =  B )
31, 2syl5eq 2507 1  |-  ( ( A  i^i  C )  =/=  (/)  ->  ( ( A  X.  B ) " C )  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    =/= wne 2648    i^i cin 3438   (/)c0 3748   ifcif 3902    X. cxp 4949   "cima 4954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404  df-opab 4462  df-xp 4957  df-rel 4958  df-cnv 4959  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964
This theorem is referenced by:  xpimasn  5394  xpimasnOLD  5395  restutopopn  19955  ustuqtop1  19958  ustuqtop5  19962
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