MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpima Structured version   Unicode version

Theorem xpima 5435
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 4998 . . . . . . . 8  |-  ( ( A  X.  B )
" C )  =  ran  ( ( A  X.  B )  |`  C )
3 df-res 4997 . . . . . . . . 9  |-  ( ( A  X.  B )  |`  C )  =  ( ( A  X.  B
)  i^i  ( C  X.  _V ) )
43rneqi 5215 . . . . . . . 8  |-  ran  (
( A  X.  B
)  |`  C )  =  ran  ( ( A  X.  B )  i^i  ( C  X.  _V ) )
52, 4eqtri 2470 . . . . . . 7  |-  ( ( A  X.  B )
" C )  =  ran  ( ( A  X.  B )  i^i  ( C  X.  _V ) )
6 inxp 5121 . . . . . . . 8  |-  ( ( A  X.  B )  i^i  ( C  X.  _V ) )  =  ( ( A  i^i  C
)  X.  ( B  i^i  _V ) )
76rneqi 5215 . . . . . . 7  |-  ran  (
( A  X.  B
)  i^i  ( C  X.  _V ) )  =  ran  ( ( A  i^i  C )  X.  ( B  i^i  _V ) )
8 inv1 3794 . . . . . . . . 9  |-  ( B  i^i  _V )  =  B
98xpeq2i 5006 . . . . . . . 8  |-  ( ( A  i^i  C )  X.  ( B  i^i  _V ) )  =  ( ( A  i^i  C
)  X.  B )
109rneqi 5215 . . . . . . 7  |-  ran  (
( A  i^i  C
)  X.  ( B  i^i  _V ) )  =  ran  ( ( A  i^i  C )  X.  B )
115, 7, 103eqtri 2474 . . . . . 6  |-  ( ( A  X.  B )
" C )  =  ran  ( ( A  i^i  C )  X.  B )
12 xpeq1 4999 . . . . . . . . 9  |-  ( ( A  i^i  C )  =  (/)  ->  ( ( A  i^i  C )  X.  B )  =  ( (/)  X.  B
) )
13 0xp 5066 . . . . . . . . 9  |-  ( (/)  X.  B )  =  (/)
1412, 13syl6eq 2498 . . . . . . . 8  |-  ( ( A  i^i  C )  =  (/)  ->  ( ( A  i^i  C )  X.  B )  =  (/) )
1514rneqd 5216 . . . . . . 7  |-  ( ( A  i^i  C )  =  (/)  ->  ran  (
( A  i^i  C
)  X.  B )  =  ran  (/) )
16 rn0 5240 . . . . . . 7  |-  ran  (/)  =  (/)
1715, 16syl6eq 2498 . . . . . 6  |-  ( ( A  i^i  C )  =  (/)  ->  ran  (
( A  i^i  C
)  X.  B )  =  (/) )
1811, 17syl5eq 2494 . . . . 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 2638 . . . . . . 7  |-  ( ( A  i^i  C )  =/=  (/)  <->  -.  ( A  i^i  C )  =  (/) )
21 rnxp 5423 . . . . . . 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 2494 . . . . 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 3960 . 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 1381    =/= wne 2636   _Vcvv 3093    i^i cin 3457   (/)c0 3767   ifcif 3922    X. cxp 4983   ran crn 4986    |` cres 4987   "cima 4988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pr 4672
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-br 4434  df-opab 4492  df-xp 4991  df-rel 4992  df-cnv 4993  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998
This theorem is referenced by:  xpima1  5436  xpima2  5437  imadifxp  27323  bj-xpimasn  34224
  Copyright terms: Public domain W3C validator