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Theorem xpima 5376
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 413 . . 3  |-  ( ( A  i^i  C )  =  (/)  \/  -.  ( A  i^i  C )  =  (/) )
2 df-ima 4943 . . . . . . . 8  |-  ( ( A  X.  B )
" C )  =  ran  ( ( A  X.  B )  |`  C )
3 df-res 4942 . . . . . . . . 9  |-  ( ( A  X.  B )  |`  C )  =  ( ( A  X.  B
)  i^i  ( C  X.  _V ) )
43rneqi 5159 . . . . . . . 8  |-  ran  (
( A  X.  B
)  |`  C )  =  ran  ( ( A  X.  B )  i^i  ( C  X.  _V ) )
52, 4eqtri 2425 . . . . . . 7  |-  ( ( A  X.  B )
" C )  =  ran  ( ( A  X.  B )  i^i  ( C  X.  _V ) )
6 inxp 5065 . . . . . . . 8  |-  ( ( A  X.  B )  i^i  ( C  X.  _V ) )  =  ( ( A  i^i  C
)  X.  ( B  i^i  _V ) )
76rneqi 5159 . . . . . . 7  |-  ran  (
( A  X.  B
)  i^i  ( C  X.  _V ) )  =  ran  ( ( A  i^i  C )  X.  ( B  i^i  _V ) )
8 inv1 3756 . . . . . . . . 9  |-  ( B  i^i  _V )  =  B
98xpeq2i 4951 . . . . . . . 8  |-  ( ( A  i^i  C )  X.  ( B  i^i  _V ) )  =  ( ( A  i^i  C
)  X.  B )
109rneqi 5159 . . . . . . 7  |-  ran  (
( A  i^i  C
)  X.  ( B  i^i  _V ) )  =  ran  ( ( A  i^i  C )  X.  B )
115, 7, 103eqtri 2429 . . . . . 6  |-  ( ( A  X.  B )
" C )  =  ran  ( ( A  i^i  C )  X.  B )
12 xpeq1 4944 . . . . . . . . 9  |-  ( ( A  i^i  C )  =  (/)  ->  ( ( A  i^i  C )  X.  B )  =  ( (/)  X.  B
) )
13 0xp 5011 . . . . . . . . 9  |-  ( (/)  X.  B )  =  (/)
1412, 13syl6eq 2453 . . . . . . . 8  |-  ( ( A  i^i  C )  =  (/)  ->  ( ( A  i^i  C )  X.  B )  =  (/) )
1514rneqd 5160 . . . . . . 7  |-  ( ( A  i^i  C )  =  (/)  ->  ran  (
( A  i^i  C
)  X.  B )  =  ran  (/) )
16 rn0 5184 . . . . . . 7  |-  ran  (/)  =  (/)
1715, 16syl6eq 2453 . . . . . 6  |-  ( ( A  i^i  C )  =  (/)  ->  ran  (
( A  i^i  C
)  X.  B )  =  (/) )
1811, 17syl5eq 2449 . . . . 5  |-  ( ( A  i^i  C )  =  (/)  ->  ( ( A  X.  B )
" C )  =  (/) )
1918ancli 549 . . . 4  |-  ( ( A  i^i  C )  =  (/)  ->  ( ( A  i^i  C )  =  (/)  /\  (
( A  X.  B
) " C )  =  (/) ) )
20 df-ne 2593 . . . . . . 7  |-  ( ( A  i^i  C )  =/=  (/)  <->  -.  ( A  i^i  C )  =  (/) )
21 rnxp 5364 . . . . . . 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 2449 . . . . 5  |-  ( -.  ( A  i^i  C
)  =  (/)  ->  (
( A  X.  B
) " C )  =  B )
2423ancli 549 . . . 4  |-  ( -.  ( A  i^i  C
)  =  (/)  ->  ( -.  ( A  i^i  C
)  =  (/)  /\  (
( A  X.  B
) " C )  =  B ) )
2519, 24orim12i 514 . . 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 3912 . 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 366    /\ wa 367    = wceq 1399    =/= wne 2591   _Vcvv 3051    i^i cin 3405   (/)c0 3728   ifcif 3874    X. cxp 4928   ran crn 4931    |` cres 4932   "cima 4933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-sep 4505  ax-nul 4513  ax-pr 4618
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-rab 2755  df-v 3053  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-sn 3962  df-pr 3964  df-op 3968  df-br 4385  df-opab 4443  df-xp 4936  df-rel 4937  df-cnv 4938  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943
This theorem is referenced by:  xpima1  5377  xpima2  5378  imadifxp  27625  bj-xpimasn  34899
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