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Theorem fimacnv 5995
Description: The preimage of the codomain of a mapping is the mapping's domain. (Contributed by FL, 25-Jan-2007.)
Assertion
Ref Expression
fimacnv  |-  ( F : A --> B  -> 
( `' F " B )  =  A )

Proof of Theorem fimacnv
StepHypRef Expression
1 imassrn 5336 . . 3  |-  ( `' F " B ) 
C_  ran  `' F
2 dfdm4 5184 . . . 4  |-  dom  F  =  ran  `' F
3 fdm 5717 . . . . 5  |-  ( F : A --> B  ->  dom  F  =  A )
4 ssid 3508 . . . . 5  |-  A  C_  A
53, 4syl6eqss 3539 . . . 4  |-  ( F : A --> B  ->  dom  F  C_  A )
62, 5syl5eqssr 3534 . . 3  |-  ( F : A --> B  ->  ran  `' F  C_  A )
71, 6syl5ss 3500 . 2  |-  ( F : A --> B  -> 
( `' F " B )  C_  A
)
8 imassrn 5336 . . . 4  |-  ( F
" A )  C_  ran  F
9 frn 5719 . . . 4  |-  ( F : A --> B  ->  ran  F  C_  B )
108, 9syl5ss 3500 . . 3  |-  ( F : A --> B  -> 
( F " A
)  C_  B )
11 ffun 5715 . . . 4  |-  ( F : A --> B  ->  Fun  F )
124, 3syl5sseqr 3538 . . . 4  |-  ( F : A --> B  ->  A  C_  dom  F )
13 funimass3 5979 . . . 4  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( ( F " A )  C_  B  <->  A 
C_  ( `' F " B ) ) )
1411, 12, 13syl2anc 659 . . 3  |-  ( F : A --> B  -> 
( ( F " A )  C_  B  <->  A 
C_  ( `' F " B ) ) )
1510, 14mpbid 210 . 2  |-  ( F : A --> B  ->  A  C_  ( `' F " B ) )
167, 15eqssd 3506 1  |-  ( F : A --> B  -> 
( `' F " B )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1398    C_ wss 3461   `'ccnv 4987   dom cdm 4988   ran crn 4989   "cima 4991   Fun wfun 5564   -->wf 5566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578
This theorem is referenced by:  fmpt  6028  frnsuppeq  6903  fin1a2lem7  8777  nn0suppOLD  10846  cnclima  19936  iscncl  19937  cnindis  19960  cncmp  20059  ptrescn  20306  qtopuni  20369  qtopcld  20380  qtopcmap  20386  ordthmeolem  20468  rnelfmlem  20619  mbfdm  22201  ismbf  22203  mbfimaicc  22206  ismbf2d  22214  ismbf3d  22227  mbfimaopn2  22230  i1fd  22254  plyeq0  22774  fimacnvinrn  27696  fsumcvg4  28167  zrhunitpreima  28193  imambfm  28470  carsggect  28526  dstrvprob  28674  dvtanlem  30304  dvtan  30305  fsuppeq  31282
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