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Theorem inimass 4925
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 4918 . 2  |-  ran  (
( A  |`  C )  i^i  ( B  |`  C ) )  C_  ( ran  ( A  |`  C )  i^i  ran  ( B  |`  C ) )
2 df-ima 4522 . . 3  |-  ( ( A  i^i  B )
" C )  =  ran  ( ( A  i^i  B )  |`  C )
3 resindir 4805 . . . 4  |-  ( ( A  i^i  B )  |`  C )  =  ( ( A  |`  C )  i^i  ( B  |`  C ) )
43rneqi 4737 . . 3  |-  ran  (
( A  i^i  B
)  |`  C )  =  ran  ( ( A  |`  C )  i^i  ( B  |`  C ) )
52, 4eqtri 2138 . 2  |-  ( ( A  i^i  B )
" C )  =  ran  ( ( A  |`  C )  i^i  ( B  |`  C ) )
6 df-ima 4522 . . 3  |-  ( A
" C )  =  ran  ( A  |`  C )
7 df-ima 4522 . . 3  |-  ( B
" C )  =  ran  ( B  |`  C )
86, 7ineq12i 3245 . 2  |-  ( ( A " C )  i^i  ( B " C ) )  =  ( ran  ( A  |`  C )  i^i  ran  ( B  |`  C ) )
91, 5, 83sstr4i 3108 1  |-  ( ( A  i^i  B )
" C )  C_  ( ( A " C )  i^i  ( B " C ) )
Colors of variables: wff set class
Syntax hints:    i^i cin 3040    C_ wss 3041   ran crn 4510    |` cres 4511   "cima 4512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-rel 4516  df-cnv 4517  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522
This theorem is referenced by: (None)
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