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

Theorem inimass 5332
Description: The image of an intersection (Contributed by Thierry Arnoux, 16-Dec-2017.)
Assertion
Ref Expression
inimass  |-  ( ( A  i^i  B )
" C )  C_  ( ( A " C )  i^i  ( B " C ) )

Proof of Theorem inimass
StepHypRef Expression
1 rnin 5325 . 2  |-  ran  (
( A  |`  C )  i^i  ( B  |`  C ) )  C_  ( ran  ( A  |`  C )  i^i  ran  ( B  |`  C ) )
2 df-ima 4926 . . 3  |-  ( ( A  i^i  B )
" C )  =  ran  ( ( A  i^i  B )  |`  C )
3 resindir 5202 . . . 4  |-  ( ( A  i^i  B )  |`  C )  =  ( ( A  |`  C )  i^i  ( B  |`  C ) )
43rneqi 5142 . . 3  |-  ran  (
( A  i^i  B
)  |`  C )  =  ran  ( ( A  |`  C )  i^i  ( B  |`  C ) )
52, 4eqtri 2411 . 2  |-  ( ( A  i^i  B )
" C )  =  ran  ( ( A  |`  C )  i^i  ( B  |`  C ) )
6 df-ima 4926 . . 3  |-  ( A
" C )  =  ran  ( A  |`  C )
7 df-ima 4926 . . 3  |-  ( B
" C )  =  ran  ( B  |`  C )
86, 7ineq12i 3612 . 2  |-  ( ( A " C )  i^i  ( B " C ) )  =  ( ran  ( A  |`  C )  i^i  ran  ( B  |`  C ) )
91, 5, 83sstr4i 3456 1  |-  ( ( A  i^i  B )
" C )  C_  ( ( A " C )  i^i  ( B " C ) )
Colors of variables: wff setvar class
Syntax hints:    i^i cin 3388    C_ wss 3389   ran crn 4914    |` cres 4915   "cima 4916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-br 4368  df-opab 4426  df-xp 4919  df-rel 4920  df-cnv 4921  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926
This theorem is referenced by:  restutopopn  20826
  Copyright terms: Public domain W3C validator