ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  inimass Unicode version

Theorem inimass 5057
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 5050 . 2  |-  ran  (
( A  |`  C )  i^i  ( B  |`  C ) )  C_  ( ran  ( A  |`  C )  i^i  ran  ( B  |`  C ) )
2 df-ima 4651 . . 3  |-  ( ( A  i^i  B )
" C )  =  ran  ( ( A  i^i  B )  |`  C )
3 resindir 4935 . . . 4  |-  ( ( A  i^i  B )  |`  C )  =  ( ( A  |`  C )  i^i  ( B  |`  C ) )
43rneqi 4867 . . 3  |-  ran  (
( A  i^i  B
)  |`  C )  =  ran  ( ( A  |`  C )  i^i  ( B  |`  C ) )
52, 4eqtri 2208 . 2  |-  ( ( A  i^i  B )
" C )  =  ran  ( ( A  |`  C )  i^i  ( B  |`  C ) )
6 df-ima 4651 . . 3  |-  ( A
" C )  =  ran  ( A  |`  C )
7 df-ima 4651 . . 3  |-  ( B
" C )  =  ran  ( B  |`  C )
86, 7ineq12i 3346 . 2  |-  ( ( A " C )  i^i  ( B " C ) )  =  ( ran  ( A  |`  C )  i^i  ran  ( B  |`  C ) )
91, 5, 83sstr4i 3208 1  |-  ( ( A  i^i  B )
" C )  C_  ( ( A " C )  i^i  ( B " C ) )
Colors of variables: wff set class
Syntax hints:    i^i cin 3140    C_ wss 3141   ran crn 4639    |` cres 4640   "cima 4641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-xp 4644  df-rel 4645  df-cnv 4646  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator