ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fimacnv Unicode version

Theorem fimacnv 5428
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 4785 . . 3  |-  ( `' F " B ) 
C_  ran  `' F
2 dfdm4 4628 . . . 4  |-  dom  F  =  ran  `' F
3 fdm 5166 . . . . 5  |-  ( F : A --> B  ->  dom  F  =  A )
4 ssid 3044 . . . . 5  |-  A  C_  A
53, 4syl6eqss 3076 . . . 4  |-  ( F : A --> B  ->  dom  F  C_  A )
62, 5syl5eqssr 3071 . . 3  |-  ( F : A --> B  ->  ran  `' F  C_  A )
71, 6syl5ss 3036 . 2  |-  ( F : A --> B  -> 
( `' F " B )  C_  A
)
8 imassrn 4785 . . . 4  |-  ( F
" A )  C_  ran  F
9 frn 5169 . . . 4  |-  ( F : A --> B  ->  ran  F  C_  B )
108, 9syl5ss 3036 . . 3  |-  ( F : A --> B  -> 
( F " A
)  C_  B )
11 ffun 5164 . . . 4  |-  ( F : A --> B  ->  Fun  F )
124, 3syl5sseqr 3075 . . . 4  |-  ( F : A --> B  ->  A  C_  dom  F )
13 funimass3 5415 . . . 4  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( ( F " A )  C_  B  <->  A 
C_  ( `' F " B ) ) )
1411, 12, 13syl2anc 403 . . 3  |-  ( F : A --> B  -> 
( ( F " A )  C_  B  <->  A 
C_  ( `' F " B ) ) )
1510, 14mpbid 145 . 2  |-  ( F : A --> B  ->  A  C_  ( `' F " B ) )
167, 15eqssd 3042 1  |-  ( F : A --> B  -> 
( `' F " B )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1289    C_ wss 2999   `'ccnv 4437   dom cdm 4438   ran crn 4439   "cima 4441   Fun wfun 5009   -->wf 5011
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fv 5023
This theorem is referenced by:  fmpt  5449  nn0supp  8723
  Copyright terms: Public domain W3C validator