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

Theorem inimass 4913
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 4906 . 2  |-  ran  (
( A  |`  C )  i^i  ( B  |`  C ) )  C_  ( ran  ( A  |`  C )  i^i  ran  ( B  |`  C ) )
2 df-ima 4512 . . 3  |-  ( ( A  i^i  B )
" C )  =  ran  ( ( A  i^i  B )  |`  C )
3 resindir 4793 . . . 4  |-  ( ( A  i^i  B )  |`  C )  =  ( ( A  |`  C )  i^i  ( B  |`  C ) )
43rneqi 4727 . . 3  |-  ran  (
( A  i^i  B
)  |`  C )  =  ran  ( ( A  |`  C )  i^i  ( B  |`  C ) )
52, 4eqtri 2135 . 2  |-  ( ( A  i^i  B )
" C )  =  ran  ( ( A  |`  C )  i^i  ( B  |`  C ) )
6 df-ima 4512 . . 3  |-  ( A
" C )  =  ran  ( A  |`  C )
7 df-ima 4512 . . 3  |-  ( B
" C )  =  ran  ( B  |`  C )
86, 7ineq12i 3241 . 2  |-  ( ( A " C )  i^i  ( B " C ) )  =  ( ran  ( A  |`  C )  i^i  ran  ( B  |`  C ) )
91, 5, 83sstr4i 3104 1  |-  ( ( A  i^i  B )
" C )  C_  ( ( A " C )  i^i  ( B " C ) )
Colors of variables: wff set class
Syntax hints:    i^i cin 3036    C_ wss 3037   ran crn 4500    |` cres 4501   "cima 4502
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-br 3896  df-opab 3950  df-xp 4505  df-rel 4506  df-cnv 4507  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator