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Theorem resima2 4988
Description: Image under a restricted class. (Contributed by FL, 31-Aug-2009.)
Assertion
Ref Expression
resima2  |-  ( B 
C_  C  ->  (
( A  |`  C )
" B )  =  ( A " B
) )

Proof of Theorem resima2
StepHypRef Expression
1 df-ima 4702 . 2  |-  ( ( A  |`  C ) " B )  =  ran  ( ( A  |`  C )  |`  B )
2 resres 4968 . . . 4  |-  ( ( A  |`  C )  |`  B )  =  ( A  |`  ( C  i^i  B ) )
32rneqi 4905 . . 3  |-  ran  (
( A  |`  C )  |`  B )  =  ran  ( A  |`  ( C  i^i  B ) )
4 df-ss 3166 . . . 4  |-  ( B 
C_  C  <->  ( B  i^i  C )  =  B )
5 incom 3361 . . . . . . . 8  |-  ( C  i^i  B )  =  ( B  i^i  C
)
65a1i 10 . . . . . . 7  |-  ( ( B  i^i  C )  =  B  ->  ( C  i^i  B )  =  ( B  i^i  C
) )
76reseq2d 4955 . . . . . 6  |-  ( ( B  i^i  C )  =  B  ->  ( A  |`  ( C  i^i  B ) )  =  ( A  |`  ( B  i^i  C ) ) )
87rneqd 4906 . . . . 5  |-  ( ( B  i^i  C )  =  B  ->  ran  ( A  |`  ( C  i^i  B ) )  =  ran  ( A  |`  ( B  i^i  C
) ) )
9 reseq2 4950 . . . . . . 7  |-  ( ( B  i^i  C )  =  B  ->  ( A  |`  ( B  i^i  C ) )  =  ( A  |`  B )
)
109rneqd 4906 . . . . . 6  |-  ( ( B  i^i  C )  =  B  ->  ran  ( A  |`  ( B  i^i  C ) )  =  ran  ( A  |`  B ) )
11 df-ima 4702 . . . . . 6  |-  ( A
" B )  =  ran  ( A  |`  B )
1210, 11syl6eqr 2333 . . . . 5  |-  ( ( B  i^i  C )  =  B  ->  ran  ( A  |`  ( B  i^i  C ) )  =  ( A " B ) )
138, 12eqtrd 2315 . . . 4  |-  ( ( B  i^i  C )  =  B  ->  ran  ( A  |`  ( C  i^i  B ) )  =  ( A " B ) )
144, 13sylbi 187 . . 3  |-  ( B 
C_  C  ->  ran  ( A  |`  ( C  i^i  B ) )  =  ( A " B ) )
153, 14syl5eq 2327 . 2  |-  ( B 
C_  C  ->  ran  ( ( A  |`  C )  |`  B )  =  ( A " B ) )
161, 15syl5eq 2327 1  |-  ( B 
C_  C  ->  (
( A  |`  C )
" B )  =  ( A " B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    i^i cin 3151    C_ wss 3152   ran crn 4690    |` cres 4691   "cima 4692
This theorem is referenced by:  marypha1lem  7186  ackbij2lem3  7867  dmdprdsplit2lem  15280  cnpresti  17016  cnprest  17017  limcflf  19231  limcresi  19235  limciun  19244  efopnlem2  20004  cvmopnlem  23809  cvmlift2lem9a  23834  imrestr  25098
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702
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