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Theorem foimacnv 5658
Description: A reverse version of f1imacnv 5657. (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 5142 . 2  |-  ( ( F  |`  ( `' F " C ) )
" ( `' F " C ) )  =  ( F " ( `' F " C ) )
2 fofun 5621 . . . . . 6  |-  ( F : A -onto-> B  ->  Fun  F )
32adantr 465 . . . . 5  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  Fun  F )
4 funcnvres2 5489 . . . . 5  |-  ( Fun 
F  ->  `' ( `' F  |`  C )  =  ( F  |`  ( `' F " C ) ) )
53, 4syl 16 . . . 4  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  `' ( `' F  |`  C )  =  ( F  |`  ( `' F " C ) ) )
65imaeq1d 5168 . . 3  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  ( `' ( `' F  |`  C )
" ( `' F " C ) )  =  ( ( F  |`  ( `' F " C ) ) " ( `' F " C ) ) )
7 resss 5134 . . . . . . . . . . 11  |-  ( `' F  |`  C )  C_  `' F
8 cnvss 5012 . . . . . . . . . . 11  |-  ( ( `' F  |`  C ) 
C_  `' F  ->  `' ( `' F  |`  C )  C_  `' `' F )
97, 8ax-mp 5 . . . . . . . . . 10  |-  `' ( `' F  |`  C ) 
C_  `' `' F
10 cnvcnvss 5292 . . . . . . . . . 10  |-  `' `' F  C_  F
119, 10sstri 3365 . . . . . . . . 9  |-  `' ( `' F  |`  C ) 
C_  F
12 funss 5436 . . . . . . . . 9  |-  ( `' ( `' F  |`  C )  C_  F  ->  ( Fun  F  ->  Fun  `' ( `' F  |`  C ) ) )
1311, 2, 12mpsyl 63 . . . . . . . 8  |-  ( F : A -onto-> B  ->  Fun  `' ( `' F  |`  C ) )
1413adantr 465 . . . . . . 7  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  Fun  `' ( `' F  |`  C ) )
15 df-ima 4853 . . . . . . . 8  |-  ( `' F " C )  =  ran  ( `' F  |`  C )
16 df-rn 4851 . . . . . . . 8  |-  ran  ( `' F  |`  C )  =  dom  `' ( `' F  |`  C )
1715, 16eqtr2i 2464 . . . . . . 7  |-  dom  `' ( `' F  |`  C )  =  ( `' F " C )
1814, 17jctir 538 . . . . . 6  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  ( Fun  `' ( `' F  |`  C )  /\  dom  `' ( `' F  |`  C )  =  ( `' F " C ) ) )
19 df-fn 5421 . . . . . 6  |-  ( `' ( `' F  |`  C )  Fn  ( `' F " C )  <-> 
( Fun  `' ( `' F  |`  C )  /\  dom  `' ( `' F  |`  C )  =  ( `' F " C ) ) )
2018, 19sylibr 212 . . . . 5  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  `' ( `' F  |`  C )  Fn  ( `' F " C ) )
21 dfdm4 5032 . . . . . 6  |-  dom  ( `' F  |`  C )  =  ran  `' ( `' F  |`  C )
22 forn 5623 . . . . . . . . . 10  |-  ( F : A -onto-> B  ->  ran  F  =  B )
2322sseq2d 3384 . . . . . . . . 9  |-  ( F : A -onto-> B  -> 
( C  C_  ran  F  <-> 
C  C_  B )
)
2423biimpar 485 . . . . . . . 8  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  C  C_  ran  F )
25 df-rn 4851 . . . . . . . 8  |-  ran  F  =  dom  `' F
2624, 25syl6sseq 3402 . . . . . . 7  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  C  C_  dom  `' F )
27 ssdmres 5132 . . . . . . 7  |-  ( C 
C_  dom  `' F  <->  dom  ( `' F  |`  C )  =  C )
2826, 27sylib 196 . . . . . 6  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  dom  ( `' F  |`  C )  =  C )
2921, 28syl5eqr 2489 . . . . 5  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  ran  `' ( `' F  |`  C )  =  C )
30 df-fo 5424 . . . . 5  |-  ( `' ( `' F  |`  C ) : ( `' F " C )
-onto-> C  <->  ( `' ( `' F  |`  C )  Fn  ( `' F " C )  /\  ran  `' ( `' F  |`  C )  =  C ) )
3120, 29, 30sylanbrc 664 . . . 4  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  `' ( `' F  |`  C ) : ( `' F " C ) -onto-> C )
32 foima 5625 . . . 4  |-  ( `' ( `' F  |`  C ) : ( `' F " C )
-onto-> C  ->  ( `' ( `' F  |`  C )
" ( `' F " C ) )  =  C )
3331, 32syl 16 . . 3  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  ( `' ( `' F  |`  C )
" ( `' F " C ) )  =  C )
346, 33eqtr3d 2477 . 2  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  ( ( F  |`  ( `' F " C ) ) "
( `' F " C ) )  =  C )
351, 34syl5eqr 2489 1  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  ( F "
( `' F " C ) )  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    C_ wss 3328   `'ccnv 4839   dom cdm 4840   ran crn 4841    |` cres 4842   "cima 4843   Fun wfun 5412    Fn wfn 5413   -onto->wfo 5416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-br 4293  df-opab 4351  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-fun 5420  df-fn 5421  df-f 5422  df-fo 5424
This theorem is referenced by:  f1opw2  6313  imacosupp  6729  fopwdom  7419  f1opwfi  7615  enfin2i  8490  fin1a2lem7  8575  fsumss  13202  gicsubgen  15806  gsumval3OLD  16382  coe1mul2lem2  17722  cncmp  18995  cnconn  19026  qtoprest  19290  qtopomap  19291  qtopcmap  19292  hmeoimaf1o  19343  elfm3  19523  imasf1oxms  20064  mbfimaopnlem  21133  cvmsss2  27163  fprodss  27461  lnmepi  29438  pwfi2f1o  29451  diaintclN  34703  dibintclN  34812  dihintcl  34989
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