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Theorem resima2 5141
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 4851 . 2  |-  ( ( A  |`  C ) " B )  =  ran  ( ( A  |`  C )  |`  B )
2 resres 5121 . . . 4  |-  ( ( A  |`  C )  |`  B )  =  ( A  |`  ( C  i^i  B ) )
32rneqi 5064 . . 3  |-  ran  (
( A  |`  C )  |`  B )  =  ran  ( A  |`  ( C  i^i  B ) )
4 df-ss 3340 . . . 4  |-  ( B 
C_  C  <->  ( B  i^i  C )  =  B )
5 incom 3541 . . . . . . . 8  |-  ( C  i^i  B )  =  ( B  i^i  C
)
65a1i 11 . . . . . . 7  |-  ( ( B  i^i  C )  =  B  ->  ( C  i^i  B )  =  ( B  i^i  C
) )
76reseq2d 5108 . . . . . 6  |-  ( ( B  i^i  C )  =  B  ->  ( A  |`  ( C  i^i  B ) )  =  ( A  |`  ( B  i^i  C ) ) )
87rneqd 5065 . . . . 5  |-  ( ( B  i^i  C )  =  B  ->  ran  ( A  |`  ( C  i^i  B ) )  =  ran  ( A  |`  ( B  i^i  C
) ) )
9 reseq2 5103 . . . . . . 7  |-  ( ( B  i^i  C )  =  B  ->  ( A  |`  ( B  i^i  C ) )  =  ( A  |`  B )
)
109rneqd 5065 . . . . . 6  |-  ( ( B  i^i  C )  =  B  ->  ran  ( A  |`  ( B  i^i  C ) )  =  ran  ( A  |`  B ) )
11 df-ima 4851 . . . . . 6  |-  ( A
" B )  =  ran  ( A  |`  B )
1210, 11syl6eqr 2491 . . . . 5  |-  ( ( B  i^i  C )  =  B  ->  ran  ( A  |`  ( B  i^i  C ) )  =  ( A " B ) )
138, 12eqtrd 2473 . . . 4  |-  ( ( B  i^i  C )  =  B  ->  ran  ( A  |`  ( C  i^i  B ) )  =  ( A " B ) )
144, 13sylbi 195 . . 3  |-  ( B 
C_  C  ->  ran  ( A  |`  ( C  i^i  B ) )  =  ( A " B ) )
153, 14syl5eq 2485 . 2  |-  ( B 
C_  C  ->  ran  ( ( A  |`  C )  |`  B )  =  ( A " B ) )
161, 15syl5eq 2485 1  |-  ( B 
C_  C  ->  (
( A  |`  C )
" B )  =  ( A " B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    i^i cin 3325    C_ wss 3326   ran crn 4839    |` cres 4840   "cima 4841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-br 4291  df-opab 4349  df-xp 4844  df-rel 4845  df-cnv 4846  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851
This theorem is referenced by:  ressuppss  6706  ressuppssdif  6708  marypha1lem  7681  ackbij2lem3  8408  dmdprdsplit2lem  16542  cnpresti  18890  cnprest  18891  limcflf  21354  limcresi  21358  limciun  21367  efopnlem2  22100  cvmopnlem  27165  cvmlift2lem9a  27190
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