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Theorem foimacnv 5490
Description: A reverse version of f1imacnv 5489. (Contributed by Jeffrey 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 4987 . 2  |-  ( ( F  |`  ( `' F " C ) )
" ( `' F " C ) )  =  ( F " ( `' F " C ) )
2 fofun 5452 . . . . . 6  |-  ( F : A -onto-> B  ->  Fun  F )
32adantr 451 . . . . 5  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  Fun  F )
4 funcnvres2 5323 . . . . 5  |-  ( Fun 
F  ->  `' ( `' F  |`  C )  =  ( F  |`  ( `' F " C ) ) )
53, 4syl 15 . . . 4  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  `' ( `' F  |`  C )  =  ( F  |`  ( `' F " C ) ) )
65imaeq1d 5011 . . 3  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  ( `' ( `' F  |`  C )
" ( `' F " C ) )  =  ( ( F  |`  ( `' F " C ) ) " ( `' F " C ) ) )
7 resss 4979 . . . . . . . . . . 11  |-  ( `' F  |`  C )  C_  `' F
8 cnvss 4854 . . . . . . . . . . 11  |-  ( ( `' F  |`  C ) 
C_  `' F  ->  `' ( `' F  |`  C )  C_  `' `' F )
97, 8ax-mp 8 . . . . . . . . . 10  |-  `' ( `' F  |`  C ) 
C_  `' `' F
10 cnvcnvss 5128 . . . . . . . . . 10  |-  `' `' F  C_  F
119, 10sstri 3188 . . . . . . . . 9  |-  `' ( `' F  |`  C ) 
C_  F
12 funss 5273 . . . . . . . . 9  |-  ( `' ( `' F  |`  C )  C_  F  ->  ( Fun  F  ->  Fun  `' ( `' F  |`  C ) ) )
1311, 2, 12mpsyl 59 . . . . . . . 8  |-  ( F : A -onto-> B  ->  Fun  `' ( `' F  |`  C ) )
1413adantr 451 . . . . . . 7  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  Fun  `' ( `' F  |`  C ) )
15 df-ima 4702 . . . . . . . 8  |-  ( `' F " C )  =  ran  ( `' F  |`  C )
16 df-rn 4700 . . . . . . . 8  |-  ran  ( `' F  |`  C )  =  dom  `' ( `' F  |`  C )
1715, 16eqtr2i 2304 . . . . . . 7  |-  dom  `' ( `' F  |`  C )  =  ( `' F " C )
1814, 17jctir 524 . . . . . 6  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  ( Fun  `' ( `' F  |`  C )  /\  dom  `' ( `' F  |`  C )  =  ( `' F " C ) ) )
19 df-fn 5258 . . . . . 6  |-  ( `' ( `' F  |`  C )  Fn  ( `' F " C )  <-> 
( Fun  `' ( `' F  |`  C )  /\  dom  `' ( `' F  |`  C )  =  ( `' F " C ) ) )
2018, 19sylibr 203 . . . . 5  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  `' ( `' F  |`  C )  Fn  ( `' F " C ) )
21 dfdm4 4872 . . . . . 6  |-  dom  ( `' F  |`  C )  =  ran  `' ( `' F  |`  C )
22 forn 5454 . . . . . . . . . 10  |-  ( F : A -onto-> B  ->  ran  F  =  B )
2322sseq2d 3206 . . . . . . . . 9  |-  ( F : A -onto-> B  -> 
( C  C_  ran  F  <-> 
C  C_  B )
)
2423biimpar 471 . . . . . . . 8  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  C  C_  ran  F )
25 df-rn 4700 . . . . . . . 8  |-  ran  F  =  dom  `' F
2624, 25syl6sseq 3224 . . . . . . 7  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  C  C_  dom  `' F )
27 ssdmres 4977 . . . . . . 7  |-  ( C 
C_  dom  `' F  <->  dom  ( `' F  |`  C )  =  C )
2826, 27sylib 188 . . . . . 6  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  dom  ( `' F  |`  C )  =  C )
2921, 28syl5eqr 2329 . . . . 5  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  ran  `' ( `' F  |`  C )  =  C )
30 df-fo 5261 . . . . 5  |-  ( `' ( `' F  |`  C ) : ( `' F " C )
-onto-> C  <->  ( `' ( `' F  |`  C )  Fn  ( `' F " C )  /\  ran  `' ( `' F  |`  C )  =  C ) )
3120, 29, 30sylanbrc 645 . . . 4  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  `' ( `' F  |`  C ) : ( `' F " C ) -onto-> C )
32 foima 5456 . . . 4  |-  ( `' ( `' F  |`  C ) : ( `' F " C )
-onto-> C  ->  ( `' ( `' F  |`  C )
" ( `' F " C ) )  =  C )
3331, 32syl 15 . . 3  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  ( `' ( `' F  |`  C )
" ( `' F " C ) )  =  C )
346, 33eqtr3d 2317 . 2  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  ( ( F  |`  ( `' F " C ) ) "
( `' F " C ) )  =  C )
351, 34syl5eqr 2329 1  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  ( F "
( `' F " C ) )  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    C_ wss 3152   `'ccnv 4688   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692   Fun wfun 5249    Fn wfn 5250   -onto->wfo 5253
This theorem is referenced by:  f1opw2  6071  fopwdom  6970  f1opwfi  7159  enfin2i  7947  fin1a2lem7  8032  fsumss  12198  gicsubgen  14742  gsumval3  15191  coe1mul2lem2  16345  cncmp  17119  cnconn  17148  qtoprest  17408  qtopomap  17409  qtopcmap  17410  hmeoimaf1o  17461  elfm3  17645  imasf1oxms  18035  mbfimaopnlem  19010  cvmsss2  23805  lnmepi  27183  pwfi2f1o  27260  diaintclN  31248  dibintclN  31357  dihintcl  31534
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261
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