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Theorem resima2 5307
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 5012 . 2  |-  ( ( A  |`  C ) " B )  =  ran  ( ( A  |`  C )  |`  B )
2 resres 5286 . . . 4  |-  ( ( A  |`  C )  |`  B )  =  ( A  |`  ( C  i^i  B ) )
32rneqi 5229 . . 3  |-  ran  (
( A  |`  C )  |`  B )  =  ran  ( A  |`  ( C  i^i  B ) )
4 df-ss 3490 . . . 4  |-  ( B 
C_  C  <->  ( B  i^i  C )  =  B )
5 incom 3691 . . . . . . . 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 5273 . . . . . 6  |-  ( ( B  i^i  C )  =  B  ->  ( A  |`  ( C  i^i  B ) )  =  ( A  |`  ( B  i^i  C ) ) )
87rneqd 5230 . . . . 5  |-  ( ( B  i^i  C )  =  B  ->  ran  ( A  |`  ( C  i^i  B ) )  =  ran  ( A  |`  ( B  i^i  C
) ) )
9 reseq2 5268 . . . . . . 7  |-  ( ( B  i^i  C )  =  B  ->  ( A  |`  ( B  i^i  C ) )  =  ( A  |`  B )
)
109rneqd 5230 . . . . . 6  |-  ( ( B  i^i  C )  =  B  ->  ran  ( A  |`  ( B  i^i  C ) )  =  ran  ( A  |`  B ) )
11 df-ima 5012 . . . . . 6  |-  ( A
" B )  =  ran  ( A  |`  B )
1210, 11syl6eqr 2526 . . . . 5  |-  ( ( B  i^i  C )  =  B  ->  ran  ( A  |`  ( B  i^i  C ) )  =  ( A " B ) )
138, 12eqtrd 2508 . . . 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 2520 . 2  |-  ( B 
C_  C  ->  ran  ( ( A  |`  C )  |`  B )  =  ( A " B ) )
161, 15syl5eq 2520 1  |-  ( B 
C_  C  ->  (
( A  |`  C )
" B )  =  ( A " B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    i^i cin 3475    C_ wss 3476   ran crn 5000    |` cres 5001   "cima 5002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012
This theorem is referenced by:  ressuppss  6919  ressuppssdif  6921  marypha1lem  7893  ackbij2lem3  8621  dmdprdsplit2lem  16896  cnpresti  19583  cnprest  19584  limcflf  22048  limcresi  22052  limciun  22061  efopnlem2  22794  cvmopnlem  28391  cvmlift2lem9a  28416
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