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Theorem foimacnv 5284
Description: A reverse version of f1imacnv 5283. (Contributed by Jeff Hankins, 16-Jul-2009.)
Assertion
Ref Expression
foimacnv  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  ( F "
( `' F " C ) )  =  C )

Proof of Theorem foimacnv
StepHypRef Expression
1 resima 4758 . 2  |-  ( ( F  |`  ( `' F " C ) )
" ( `' F " C ) )  =  ( F " ( `' F " C ) )
2 fofun 5247 . . . . . 6  |-  ( F : A -onto-> B  ->  Fun  F )
32adantr 271 . . . . 5  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  Fun  F )
4 funcnvres2 5102 . . . . 5  |-  ( Fun 
F  ->  `' ( `' F  |`  C )  =  ( F  |`  ( `' F " C ) ) )
53, 4syl 14 . . . 4  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  `' ( `' F  |`  C )  =  ( F  |`  ( `' F " C ) ) )
65imaeq1d 4786 . . 3  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  ( `' ( `' F  |`  C )
" ( `' F " C ) )  =  ( ( F  |`  ( `' F " C ) ) " ( `' F " C ) ) )
7 resss 4750 . . . . . . . . . . 11  |-  ( `' F  |`  C )  C_  `' F
8 cnvss 4622 . . . . . . . . . . 11  |-  ( ( `' F  |`  C ) 
C_  `' F  ->  `' ( `' F  |`  C )  C_  `' `' F )
97, 8ax-mp 7 . . . . . . . . . 10  |-  `' ( `' F  |`  C ) 
C_  `' `' F
10 cnvcnvss 4898 . . . . . . . . . 10  |-  `' `' F  C_  F
119, 10sstri 3035 . . . . . . . . 9  |-  `' ( `' F  |`  C ) 
C_  F
12 funss 5047 . . . . . . . . 9  |-  ( `' ( `' F  |`  C )  C_  F  ->  ( Fun  F  ->  Fun  `' ( `' F  |`  C ) ) )
1311, 2, 12mpsyl 65 . . . . . . . 8  |-  ( F : A -onto-> B  ->  Fun  `' ( `' F  |`  C ) )
1413adantr 271 . . . . . . 7  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  Fun  `' ( `' F  |`  C ) )
15 df-ima 4465 . . . . . . . 8  |-  ( `' F " C )  =  ran  ( `' F  |`  C )
16 df-rn 4463 . . . . . . . 8  |-  ran  ( `' F  |`  C )  =  dom  `' ( `' F  |`  C )
1715, 16eqtr2i 2110 . . . . . . 7  |-  dom  `' ( `' F  |`  C )  =  ( `' F " C )
1814, 17jctir 307 . . . . . 6  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  ( Fun  `' ( `' F  |`  C )  /\  dom  `' ( `' F  |`  C )  =  ( `' F " C ) ) )
19 df-fn 5031 . . . . . 6  |-  ( `' ( `' F  |`  C )  Fn  ( `' F " C )  <-> 
( Fun  `' ( `' F  |`  C )  /\  dom  `' ( `' F  |`  C )  =  ( `' F " C ) ) )
2018, 19sylibr 133 . . . . 5  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  `' ( `' F  |`  C )  Fn  ( `' F " C ) )
21 dfdm4 4641 . . . . . 6  |-  dom  ( `' F  |`  C )  =  ran  `' ( `' F  |`  C )
22 forn 5249 . . . . . . . . . 10  |-  ( F : A -onto-> B  ->  ran  F  =  B )
2322sseq2d 3055 . . . . . . . . 9  |-  ( F : A -onto-> B  -> 
( C  C_  ran  F  <-> 
C  C_  B )
)
2423biimpar 292 . . . . . . . 8  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  C  C_  ran  F )
25 df-rn 4463 . . . . . . . 8  |-  ran  F  =  dom  `' F
2624, 25syl6sseq 3073 . . . . . . 7  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  C  C_  dom  `' F )
27 ssdmres 4748 . . . . . . 7  |-  ( C 
C_  dom  `' F  <->  dom  ( `' F  |`  C )  =  C )
2826, 27sylib 121 . . . . . 6  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  dom  ( `' F  |`  C )  =  C )
2921, 28syl5eqr 2135 . . . . 5  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  ran  `' ( `' F  |`  C )  =  C )
30 df-fo 5034 . . . . 5  |-  ( `' ( `' F  |`  C ) : ( `' F " C )
-onto-> C  <->  ( `' ( `' F  |`  C )  Fn  ( `' F " C )  /\  ran  `' ( `' F  |`  C )  =  C ) )
3120, 29, 30sylanbrc 409 . . . 4  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  `' ( `' F  |`  C ) : ( `' F " C ) -onto-> C )
32 foima 5251 . . . 4  |-  ( `' ( `' F  |`  C ) : ( `' F " C )
-onto-> C  ->  ( `' ( `' F  |`  C )
" ( `' F " C ) )  =  C )
3331, 32syl 14 . . 3  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  ( `' ( `' F  |`  C )
" ( `' F " C ) )  =  C )
346, 33eqtr3d 2123 . 2  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  ( ( F  |`  ( `' F " C ) ) "
( `' F " C ) )  =  C )
351, 34syl5eqr 2135 1  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  ( F "
( `' F " C ) )  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1290    C_ wss 3000   `'ccnv 4451   dom cdm 4452   ran crn 4453    |` cres 4454   "cima 4455   Fun wfun 5022    Fn wfn 5023   -onto->wfo 5026
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-fun 5030  df-fn 5031  df-f 5032  df-fo 5034
This theorem is referenced by:  f1opw2  5864  fopwdom  6606  fisumss  10845
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