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Theorem resima2 5095
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 4804 . 2  |-  ( ( A  |`  C ) " B )  =  ran  ( ( A  |`  C )  |`  B )
2 resres 5074 . . . 4  |-  ( ( A  |`  C )  |`  B )  =  ( A  |`  ( C  i^i  B ) )
32rneqi 5018 . . 3  |-  ran  (
( A  |`  C )  |`  B )  =  ran  ( A  |`  ( C  i^i  B ) )
4 df-ss 3388 . . . 4  |-  ( B 
C_  C  <->  ( B  i^i  C )  =  B )
5 incom 3593 . . . . . . . 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 5062 . . . . . 6  |-  ( ( B  i^i  C )  =  B  ->  ( A  |`  ( C  i^i  B ) )  =  ( A  |`  ( B  i^i  C ) ) )
87rneqd 5019 . . . . 5  |-  ( ( B  i^i  C )  =  B  ->  ran  ( A  |`  ( C  i^i  B ) )  =  ran  ( A  |`  ( B  i^i  C
) ) )
9 reseq2 5057 . . . . . . 7  |-  ( ( B  i^i  C )  =  B  ->  ( A  |`  ( B  i^i  C ) )  =  ( A  |`  B )
)
109rneqd 5019 . . . . . 6  |-  ( ( B  i^i  C )  =  B  ->  ran  ( A  |`  ( B  i^i  C ) )  =  ran  ( A  |`  B ) )
11 df-ima 4804 . . . . . 6  |-  ( A
" B )  =  ran  ( A  |`  B )
1210, 11syl6eqr 2475 . . . . 5  |-  ( ( B  i^i  C )  =  B  ->  ran  ( A  |`  ( B  i^i  C ) )  =  ( A " B ) )
138, 12eqtrd 2457 . . . 4  |-  ( ( B  i^i  C )  =  B  ->  ran  ( A  |`  ( C  i^i  B ) )  =  ( A " B ) )
144, 13sylbi 198 . . 3  |-  ( B 
C_  C  ->  ran  ( A  |`  ( C  i^i  B ) )  =  ( A " B ) )
153, 14syl5eq 2469 . 2  |-  ( B 
C_  C  ->  ran  ( ( A  |`  C )  |`  B )  =  ( A " B ) )
161, 15syl5eq 2469 1  |-  ( B 
C_  C  ->  (
( A  |`  C )
" B )  =  ( A " B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    i^i cin 3373    C_ wss 3374   ran crn 4792    |` cres 4793   "cima 4794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-sep 4484  ax-nul 4493  ax-pr 4598
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-ral 2714  df-rex 2715  df-rab 2718  df-v 3019  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-nul 3700  df-if 3850  df-sn 3937  df-pr 3939  df-op 3943  df-br 4362  df-opab 4421  df-xp 4797  df-rel 4798  df-cnv 4799  df-dm 4801  df-rn 4802  df-res 4803  df-ima 4804
This theorem is referenced by:  ressuppss  6884  ressuppssdif  6886  marypha1lem  7895  ackbij2lem3  8617  dmdprdsplit2lem  17616  cnpresti  20241  cnprest  20242  limcflf  22773  limcresi  22777  limciun  22786  efopnlem2  23539  cvmopnlem  29948  cvmlift2lem9a  29973  poimirlem4  31851
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