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Theorem xpima 5440
Description: The image by a constant function (or other Cartesian product). (Contributed by Thierry Arnoux, 4-Feb-2017.)
Assertion
Ref Expression
xpima  |-  ( ( A  X.  B )
" C )  =  if ( ( A  i^i  C )  =  (/) ,  (/) ,  B )

Proof of Theorem xpima
StepHypRef Expression
1 exmid 415 . . 3  |-  ( ( A  i^i  C )  =  (/)  \/  -.  ( A  i^i  C )  =  (/) )
2 df-ima 5005 . . . . . . . 8  |-  ( ( A  X.  B )
" C )  =  ran  ( ( A  X.  B )  |`  C )
3 df-res 5004 . . . . . . . . 9  |-  ( ( A  X.  B )  |`  C )  =  ( ( A  X.  B
)  i^i  ( C  X.  _V ) )
43rneqi 5220 . . . . . . . 8  |-  ran  (
( A  X.  B
)  |`  C )  =  ran  ( ( A  X.  B )  i^i  ( C  X.  _V ) )
52, 4eqtri 2489 . . . . . . 7  |-  ( ( A  X.  B )
" C )  =  ran  ( ( A  X.  B )  i^i  ( C  X.  _V ) )
6 inxp 5126 . . . . . . . 8  |-  ( ( A  X.  B )  i^i  ( C  X.  _V ) )  =  ( ( A  i^i  C
)  X.  ( B  i^i  _V ) )
76rneqi 5220 . . . . . . 7  |-  ran  (
( A  X.  B
)  i^i  ( C  X.  _V ) )  =  ran  ( ( A  i^i  C )  X.  ( B  i^i  _V ) )
8 inv1 3805 . . . . . . . . 9  |-  ( B  i^i  _V )  =  B
98xpeq2i 5013 . . . . . . . 8  |-  ( ( A  i^i  C )  X.  ( B  i^i  _V ) )  =  ( ( A  i^i  C
)  X.  B )
109rneqi 5220 . . . . . . 7  |-  ran  (
( A  i^i  C
)  X.  ( B  i^i  _V ) )  =  ran  ( ( A  i^i  C )  X.  B )
115, 7, 103eqtri 2493 . . . . . 6  |-  ( ( A  X.  B )
" C )  =  ran  ( ( A  i^i  C )  X.  B )
12 xpeq1 5006 . . . . . . . . 9  |-  ( ( A  i^i  C )  =  (/)  ->  ( ( A  i^i  C )  X.  B )  =  ( (/)  X.  B
) )
13 0xp 5071 . . . . . . . . 9  |-  ( (/)  X.  B )  =  (/)
1412, 13syl6eq 2517 . . . . . . . 8  |-  ( ( A  i^i  C )  =  (/)  ->  ( ( A  i^i  C )  X.  B )  =  (/) )
1514rneqd 5221 . . . . . . 7  |-  ( ( A  i^i  C )  =  (/)  ->  ran  (
( A  i^i  C
)  X.  B )  =  ran  (/) )
16 rn0 5245 . . . . . . 7  |-  ran  (/)  =  (/)
1715, 16syl6eq 2517 . . . . . 6  |-  ( ( A  i^i  C )  =  (/)  ->  ran  (
( A  i^i  C
)  X.  B )  =  (/) )
1811, 17syl5eq 2513 . . . . 5  |-  ( ( A  i^i  C )  =  (/)  ->  ( ( A  X.  B )
" C )  =  (/) )
1918ancli 551 . . . 4  |-  ( ( A  i^i  C )  =  (/)  ->  ( ( A  i^i  C )  =  (/)  /\  (
( A  X.  B
) " C )  =  (/) ) )
20 df-ne 2657 . . . . . . 7  |-  ( ( A  i^i  C )  =/=  (/)  <->  -.  ( A  i^i  C )  =  (/) )
21 rnxp 5428 . . . . . . 7  |-  ( ( A  i^i  C )  =/=  (/)  ->  ran  ( ( A  i^i  C )  X.  B )  =  B )
2220, 21sylbir 213 . . . . . 6  |-  ( -.  ( A  i^i  C
)  =  (/)  ->  ran  ( ( A  i^i  C )  X.  B )  =  B )
2311, 22syl5eq 2513 . . . . 5  |-  ( -.  ( A  i^i  C
)  =  (/)  ->  (
( A  X.  B
) " C )  =  B )
2423ancli 551 . . . 4  |-  ( -.  ( A  i^i  C
)  =  (/)  ->  ( -.  ( A  i^i  C
)  =  (/)  /\  (
( A  X.  B
) " C )  =  B ) )
2519, 24orim12i 516 . . 3  |-  ( ( ( A  i^i  C
)  =  (/)  \/  -.  ( A  i^i  C )  =  (/) )  ->  (
( ( A  i^i  C )  =  (/)  /\  (
( A  X.  B
) " C )  =  (/) )  \/  ( -.  ( A  i^i  C
)  =  (/)  /\  (
( A  X.  B
) " C )  =  B ) ) )
261, 25ax-mp 5 . 2  |-  ( ( ( A  i^i  C
)  =  (/)  /\  (
( A  X.  B
) " C )  =  (/) )  \/  ( -.  ( A  i^i  C
)  =  (/)  /\  (
( A  X.  B
) " C )  =  B ) )
27 eqif 3970 . 2  |-  ( ( ( A  X.  B
) " C )  =  if ( ( A  i^i  C )  =  (/) ,  (/) ,  B
)  <->  ( ( ( A  i^i  C )  =  (/)  /\  (
( A  X.  B
) " C )  =  (/) )  \/  ( -.  ( A  i^i  C
)  =  (/)  /\  (
( A  X.  B
) " C )  =  B ) ) )
2826, 27mpbir 209 1  |-  ( ( A  X.  B )
" C )  =  if ( ( A  i^i  C )  =  (/) ,  (/) ,  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 368    /\ wa 369    = wceq 1374    =/= wne 2655   _Vcvv 3106    i^i cin 3468   (/)c0 3778   ifcif 3932    X. cxp 4990   ran crn 4993    |` cres 4994   "cima 4995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441  df-opab 4499  df-xp 4998  df-rel 4999  df-cnv 5000  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005
This theorem is referenced by:  xpima1  5441  xpima2  5442  imadifxp  27117  bj-xpimasn  33468
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