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Theorem foimacnv 5172
Description: A reverse version of f1imacnv 5171. (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 4671 . 2  |-  ( ( F  |`  ( `' F " C ) )
" ( `' F " C ) )  =  ( F " ( `' F " C ) )
2 fofun 5135 . . . . . 6  |-  ( F : A -onto-> B  ->  Fun  F )
32adantr 265 . . . . 5  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  Fun  F )
4 funcnvres2 5002 . . . . 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 4695 . . 3  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  ( `' ( `' F  |`  C )
" ( `' F " C ) )  =  ( ( F  |`  ( `' F " C ) ) " ( `' F " C ) ) )
7 resss 4663 . . . . . . . . . . 11  |-  ( `' F  |`  C )  C_  `' F
8 cnvss 4536 . . . . . . . . . . 11  |-  ( ( `' F  |`  C ) 
C_  `' F  ->  `' ( `' F  |`  C )  C_  `' `' F )
97, 8ax-mp 7 . . . . . . . . . 10  |-  `' ( `' F  |`  C ) 
C_  `' `' F
10 cnvcnvss 4803 . . . . . . . . . 10  |-  `' `' F  C_  F
119, 10sstri 2982 . . . . . . . . 9  |-  `' ( `' F  |`  C ) 
C_  F
12 funss 4948 . . . . . . . . 9  |-  ( `' ( `' F  |`  C )  C_  F  ->  ( Fun  F  ->  Fun  `' ( `' F  |`  C ) ) )
1311, 2, 12mpsyl 63 . . . . . . . 8  |-  ( F : A -onto-> B  ->  Fun  `' ( `' F  |`  C ) )
1413adantr 265 . . . . . . 7  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  Fun  `' ( `' F  |`  C ) )
15 df-ima 4386 . . . . . . . 8  |-  ( `' F " C )  =  ran  ( `' F  |`  C )
16 df-rn 4384 . . . . . . . 8  |-  ran  ( `' F  |`  C )  =  dom  `' ( `' F  |`  C )
1715, 16eqtr2i 2077 . . . . . . 7  |-  dom  `' ( `' F  |`  C )  =  ( `' F " C )
1814, 17jctir 300 . . . . . 6  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  ( Fun  `' ( `' F  |`  C )  /\  dom  `' ( `' F  |`  C )  =  ( `' F " C ) ) )
19 df-fn 4933 . . . . . 6  |-  ( `' ( `' F  |`  C )  Fn  ( `' F " C )  <-> 
( Fun  `' ( `' F  |`  C )  /\  dom  `' ( `' F  |`  C )  =  ( `' F " C ) ) )
2018, 19sylibr 141 . . . . 5  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  `' ( `' F  |`  C )  Fn  ( `' F " C ) )
21 dfdm4 4555 . . . . . 6  |-  dom  ( `' F  |`  C )  =  ran  `' ( `' F  |`  C )
22 forn 5137 . . . . . . . . . 10  |-  ( F : A -onto-> B  ->  ran  F  =  B )
2322sseq2d 3001 . . . . . . . . 9  |-  ( F : A -onto-> B  -> 
( C  C_  ran  F  <-> 
C  C_  B )
)
2423biimpar 285 . . . . . . . 8  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  C  C_  ran  F )
25 df-rn 4384 . . . . . . . 8  |-  ran  F  =  dom  `' F
2624, 25syl6sseq 3019 . . . . . . 7  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  C  C_  dom  `' F )
27 ssdmres 4661 . . . . . . 7  |-  ( C 
C_  dom  `' F  <->  dom  ( `' F  |`  C )  =  C )
2826, 27sylib 131 . . . . . 6  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  dom  ( `' F  |`  C )  =  C )
2921, 28syl5eqr 2102 . . . . 5  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  ran  `' ( `' F  |`  C )  =  C )
30 df-fo 4936 . . . . 5  |-  ( `' ( `' F  |`  C ) : ( `' F " C )
-onto-> C  <->  ( `' ( `' F  |`  C )  Fn  ( `' F " C )  /\  ran  `' ( `' F  |`  C )  =  C ) )
3120, 29, 30sylanbrc 402 . . . 4  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  `' ( `' F  |`  C ) : ( `' F " C ) -onto-> C )
32 foima 5139 . . . 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 2090 . 2  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  ( ( F  |`  ( `' F " C ) ) "
( `' F " C ) )  =  C )
351, 34syl5eqr 2102 1  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  ( F "
( `' F " C ) )  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    = wceq 1259    C_ wss 2945   `'ccnv 4372   dom cdm 4373   ran crn 4374    |` cres 4375   "cima 4376   Fun wfun 4924    Fn wfn 4925   -onto->wfo 4928
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-eu 1919  df-mo 1920  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-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-fun 4932  df-fn 4933  df-f 4934  df-fo 4936
This theorem is referenced by:  f1opw2  5734  fopwdom  6341
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