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Theorem resima2 4672
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 4386 . 2  |-  ( ( A  |`  C ) " B )  =  ran  ( ( A  |`  C )  |`  B )
2 resres 4652 . . . 4  |-  ( ( A  |`  C )  |`  B )  =  ( A  |`  ( C  i^i  B ) )
32rneqi 4590 . . 3  |-  ran  (
( A  |`  C )  |`  B )  =  ran  ( A  |`  ( C  i^i  B ) )
4 df-ss 2959 . . . 4  |-  ( B 
C_  C  <->  ( B  i^i  C )  =  B )
5 incom 3157 . . . . . . . 8  |-  ( C  i^i  B )  =  ( B  i^i  C
)
65a1i 9 . . . . . . 7  |-  ( ( B  i^i  C )  =  B  ->  ( C  i^i  B )  =  ( B  i^i  C
) )
76reseq2d 4640 . . . . . 6  |-  ( ( B  i^i  C )  =  B  ->  ( A  |`  ( C  i^i  B ) )  =  ( A  |`  ( B  i^i  C ) ) )
87rneqd 4591 . . . . 5  |-  ( ( B  i^i  C )  =  B  ->  ran  ( A  |`  ( C  i^i  B ) )  =  ran  ( A  |`  ( B  i^i  C
) ) )
9 reseq2 4635 . . . . . . 7  |-  ( ( B  i^i  C )  =  B  ->  ( A  |`  ( B  i^i  C ) )  =  ( A  |`  B )
)
109rneqd 4591 . . . . . 6  |-  ( ( B  i^i  C )  =  B  ->  ran  ( A  |`  ( B  i^i  C ) )  =  ran  ( A  |`  B ) )
11 df-ima 4386 . . . . . 6  |-  ( A
" B )  =  ran  ( A  |`  B )
1210, 11syl6eqr 2106 . . . . 5  |-  ( ( B  i^i  C )  =  B  ->  ran  ( A  |`  ( B  i^i  C ) )  =  ( A " B ) )
138, 12eqtrd 2088 . . . 4  |-  ( ( B  i^i  C )  =  B  ->  ran  ( A  |`  ( C  i^i  B ) )  =  ( A " B ) )
144, 13sylbi 118 . . 3  |-  ( B 
C_  C  ->  ran  ( A  |`  ( C  i^i  B ) )  =  ( A " B ) )
153, 14syl5eq 2100 . 2  |-  ( B 
C_  C  ->  ran  ( ( A  |`  C )  |`  B )  =  ( A " B ) )
161, 15syl5eq 2100 1  |-  ( B 
C_  C  ->  (
( A  |`  C )
" B )  =  ( A " B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1259    i^i cin 2944    C_ wss 2945   ran crn 4374    |` cres 4375   "cima 4376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-xp 4379  df-rel 4380  df-cnv 4381  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386
This theorem is referenced by: (None)
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