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Theorem xpima2 5457
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 5455 . 2  |-  ( ( A  X.  B )
" C )  =  if ( ( A  i^i  C )  =  (/) ,  (/) ,  B )
2 ifnefalse 3957 . 2  |-  ( ( A  i^i  C )  =/=  (/)  ->  if (
( A  i^i  C
)  =  (/) ,  (/) ,  B )  =  B )
31, 2syl5eq 2520 1  |-  ( ( A  i^i  C )  =/=  (/)  ->  ( ( A  X.  B ) " C )  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    =/= wne 2662    i^i cin 3480   (/)c0 3790   ifcif 3945    X. cxp 5003   "cima 5008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-xp 5011  df-rel 5012  df-cnv 5013  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018
This theorem is referenced by:  xpimasn  5458  xpimasnOLD  5459  restutopopn  20609  ustuqtop1  20612  ustuqtop5  20616
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