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Theorem fimacnv 5646
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 4982 . . 3  |-  ( `' F " B ) 
C_  ran  `' F
2 dfdm4 4820 . . . 4  |-  dom  F  =  ran  `' F
3 fdm 5372 . . . . 5  |-  ( F : A --> B  ->  dom  F  =  A )
4 ssid 3176 . . . . 5  |-  A  C_  A
53, 4eqsstrdi 3208 . . . 4  |-  ( F : A --> B  ->  dom  F  C_  A )
62, 5eqsstrrid 3203 . . 3  |-  ( F : A --> B  ->  ran  `' F  C_  A )
71, 6sstrid 3167 . 2  |-  ( F : A --> B  -> 
( `' F " B )  C_  A
)
8 imassrn 4982 . . . 4  |-  ( F
" A )  C_  ran  F
9 frn 5375 . . . 4  |-  ( F : A --> B  ->  ran  F  C_  B )
108, 9sstrid 3167 . . 3  |-  ( F : A --> B  -> 
( F " A
)  C_  B )
11 ffun 5369 . . . 4  |-  ( F : A --> B  ->  Fun  F )
124, 3sseqtrrid 3207 . . . 4  |-  ( F : A --> B  ->  A  C_  dom  F )
13 funimass3 5633 . . . 4  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( ( F " A )  C_  B  <->  A 
C_  ( `' F " B ) ) )
1411, 12, 13syl2anc 411 . . 3  |-  ( F : A --> B  -> 
( ( F " A )  C_  B  <->  A 
C_  ( `' F " B ) ) )
1510, 14mpbid 147 . 2  |-  ( F : A --> B  ->  A  C_  ( `' F " B ) )
167, 15eqssd 3173 1  |-  ( F : A --> B  -> 
( `' F " B )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1353    C_ wss 3130   `'ccnv 4626   dom cdm 4627   ran crn 4628   "cima 4630   Fun wfun 5211   -->wf 5213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fv 5225
This theorem is referenced by:  fmpt  5667  nn0supp  9228  cnclima  13726
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