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Theorem foimacnv 5831
Description: A reverse version of f1imacnv 5830. (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 5304 . 2  |-  ( ( F  |`  ( `' F " C ) )
" ( `' F " C ) )  =  ( F " ( `' F " C ) )
2 fofun 5794 . . . . . 6  |-  ( F : A -onto-> B  ->  Fun  F )
32adantr 465 . . . . 5  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  Fun  F )
4 funcnvres2 5657 . . . . 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 5334 . . 3  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  ( `' ( `' F  |`  C )
" ( `' F " C ) )  =  ( ( F  |`  ( `' F " C ) ) " ( `' F " C ) ) )
7 resss 5295 . . . . . . . . . . 11  |-  ( `' F  |`  C )  C_  `' F
8 cnvss 5173 . . . . . . . . . . 11  |-  ( ( `' F  |`  C ) 
C_  `' F  ->  `' ( `' F  |`  C )  C_  `' `' F )
97, 8ax-mp 5 . . . . . . . . . 10  |-  `' ( `' F  |`  C ) 
C_  `' `' F
10 cnvcnvss 5459 . . . . . . . . . 10  |-  `' `' F  C_  F
119, 10sstri 3513 . . . . . . . . 9  |-  `' ( `' F  |`  C ) 
C_  F
12 funss 5604 . . . . . . . . 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 5012 . . . . . . . 8  |-  ( `' F " C )  =  ran  ( `' F  |`  C )
16 df-rn 5010 . . . . . . . 8  |-  ran  ( `' F  |`  C )  =  dom  `' ( `' F  |`  C )
1715, 16eqtr2i 2497 . . . . . . 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 5589 . . . . . 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 5193 . . . . . 6  |-  dom  ( `' F  |`  C )  =  ran  `' ( `' F  |`  C )
22 forn 5796 . . . . . . . . . 10  |-  ( F : A -onto-> B  ->  ran  F  =  B )
2322sseq2d 3532 . . . . . . . . 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 5010 . . . . . . . 8  |-  ran  F  =  dom  `' F
2624, 25syl6sseq 3550 . . . . . . 7  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  C  C_  dom  `' F )
27 ssdmres 5293 . . . . . . 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 2522 . . . . 5  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  ran  `' ( `' F  |`  C )  =  C )
30 df-fo 5592 . . . . 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 5798 . . . 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 2510 . 2  |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  ( ( F  |`  ( `' F " C ) ) "
( `' F " C ) )  =  C )
351, 34syl5eqr 2522 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 1379    C_ wss 3476   `'ccnv 4998   dom cdm 4999   ran crn 5000    |` cres 5001   "cima 5002   Fun wfun 5580    Fn wfn 5581   -onto->wfo 5584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-fun 5588  df-fn 5589  df-f 5590  df-fo 5592
This theorem is referenced by:  f1opw2  6510  imacosupp  6937  fopwdom  7622  f1opwfi  7820  enfin2i  8697  fin1a2lem7  8782  fsumss  13503  gicsubgen  16118  gsumval3OLD  16696  coe1mul2lem2  18077  cncmp  19655  cnconn  19686  qtoprest  19950  qtopomap  19951  qtopcmap  19952  hmeoimaf1o  20003  elfm3  20183  imasf1oxms  20724  mbfimaopnlem  21794  cvmsss2  28356  fprodss  28654  lnmepi  30635  pwfi2f1o  30648  diaintclN  35855  dibintclN  35964  dihintcl  36141
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