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Theorem fnresdm 5039
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 5028 . 2  |-  ( F  Fn  A  ->  Rel  F )
2 fndm 5029 . . 3  |-  ( F  Fn  A  ->  dom  F  =  A )
3 eqimss 3052 . . 3  |-  ( dom 
F  =  A  ->  dom  F  C_  A )
42, 3syl 14 . 2  |-  ( F  Fn  A  ->  dom  F 
C_  A )
5 relssres 4676 . 2  |-  ( ( Rel  F  /\  dom  F 
C_  A )  -> 
( F  |`  A )  =  F )
61, 4, 5syl2anc 403 1  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    C_ wss 2974   dom cdm 4371    |` cres 4373   Rel wrel 4376    Fn wfn 4927
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-xp 4377  df-rel 4378  df-dm 4381  df-res 4383  df-fun 4934  df-fn 4935
This theorem is referenced by:  fnima  5048  fresin  5099  resasplitss  5100  fsnunfv  5395  fsnunres  5396  fnfi  6446  fseq1p1m1  9187  facnn  9751  fac0  9752
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