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

Theorem fnresdm 5307
Description: A function does not change when restricted to its domain. (Contributed by NM, 5-Sep-2004.)
Assertion
Ref Expression
fnresdm  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )

Proof of Theorem fnresdm
StepHypRef Expression
1 fnrel 5296 . 2  |-  ( F  Fn  A  ->  Rel  F )
2 fndm 5297 . . 3  |-  ( F  Fn  A  ->  dom  F  =  A )
3 eqimss 3201 . . 3  |-  ( dom 
F  =  A  ->  dom  F  C_  A )
42, 3syl 14 . 2  |-  ( F  Fn  A  ->  dom  F 
C_  A )
5 relssres 4929 . 2  |-  ( ( Rel  F  /\  dom  F 
C_  A )  -> 
( F  |`  A )  =  F )
61, 4, 5syl2anc 409 1  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1348    C_ wss 3121   dom cdm 4611    |` cres 4613   Rel wrel 4616    Fn wfn 5193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-dm 4621  df-res 4623  df-fun 5200  df-fn 5201
This theorem is referenced by:  fnima  5316  fresin  5376  resasplitss  5377  fnsnsplitss  5695  fsnunfv  5697  fsnunres  5698  fnsnsplitdc  6484  fnfi  6914  fseq1p1m1  10050  facnn  10661  fac0  10662  dfrelog  13575
  Copyright terms: Public domain W3C validator