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

Theorem xpima 5389
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 4962 . . . . . . . 8  |-  ( ( A  X.  B )
" C )  =  ran  ( ( A  X.  B )  |`  C )
3 df-res 4961 . . . . . . . . 9  |-  ( ( A  X.  B )  |`  C )  =  ( ( A  X.  B
)  i^i  ( C  X.  _V ) )
43rneqi 5175 . . . . . . . 8  |-  ran  (
( A  X.  B
)  |`  C )  =  ran  ( ( A  X.  B )  i^i  ( C  X.  _V ) )
52, 4eqtri 2483 . . . . . . 7  |-  ( ( A  X.  B )
" C )  =  ran  ( ( A  X.  B )  i^i  ( C  X.  _V ) )
6 inxp 5081 . . . . . . . 8  |-  ( ( A  X.  B )  i^i  ( C  X.  _V ) )  =  ( ( A  i^i  C
)  X.  ( B  i^i  _V ) )
76rneqi 5175 . . . . . . 7  |-  ran  (
( A  X.  B
)  i^i  ( C  X.  _V ) )  =  ran  ( ( A  i^i  C )  X.  ( B  i^i  _V ) )
8 inv1 3773 . . . . . . . . 9  |-  ( B  i^i  _V )  =  B
98xpeq2i 4970 . . . . . . . 8  |-  ( ( A  i^i  C )  X.  ( B  i^i  _V ) )  =  ( ( A  i^i  C
)  X.  B )
109rneqi 5175 . . . . . . 7  |-  ran  (
( A  i^i  C
)  X.  ( B  i^i  _V ) )  =  ran  ( ( A  i^i  C )  X.  B )
115, 7, 103eqtri 2487 . . . . . 6  |-  ( ( A  X.  B )
" C )  =  ran  ( ( A  i^i  C )  X.  B )
12 xpeq1 4963 . . . . . . . . 9  |-  ( ( A  i^i  C )  =  (/)  ->  ( ( A  i^i  C )  X.  B )  =  ( (/)  X.  B
) )
13 0xp 5026 . . . . . . . . 9  |-  ( (/)  X.  B )  =  (/)
1412, 13syl6eq 2511 . . . . . . . 8  |-  ( ( A  i^i  C )  =  (/)  ->  ( ( A  i^i  C )  X.  B )  =  (/) )
1514rneqd 5176 . . . . . . 7  |-  ( ( A  i^i  C )  =  (/)  ->  ran  (
( A  i^i  C
)  X.  B )  =  ran  (/) )
16 rn0 5200 . . . . . . 7  |-  ran  (/)  =  (/)
1715, 16syl6eq 2511 . . . . . 6  |-  ( ( A  i^i  C )  =  (/)  ->  ran  (
( A  i^i  C
)  X.  B )  =  (/) )
1811, 17syl5eq 2507 . . . . 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 2650 . . . . . . 7  |-  ( ( A  i^i  C )  =/=  (/)  <->  -.  ( A  i^i  C )  =  (/) )
21 rnxp 5377 . . . . . . 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 2507 . . . . 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 3936 . 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 1370    =/= wne 2648   _Vcvv 3078    i^i cin 3436   (/)c0 3746   ifcif 3900    X. cxp 4947   ran crn 4950    |` cres 4951   "cima 4952
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 4522  ax-nul 4530  ax-pr 4640
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 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-br 4402  df-opab 4460  df-xp 4955  df-rel 4956  df-cnv 4957  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962
This theorem is referenced by:  xpima1  5390  xpima2  5391  imadifxp  26091  bj-xpimasn  32780
  Copyright terms: Public domain W3C validator