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Theorem funres 5164
Description: A restriction of a function is a function. Compare Exercise 18 of [TakeutiZaring] p. 25. (Contributed by NM, 16-Aug-1994.)
Assertion
Ref Expression
funres  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )

Proof of Theorem funres
StepHypRef Expression
1 resss 4843 . 2  |-  ( F  |`  A )  C_  F
2 funss 5142 . 2  |-  ( ( F  |`  A )  C_  F  ->  ( Fun  F  ->  Fun  ( F  |`  A ) ) )
31, 2ax-mp 5 1  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    C_ wss 3071    |` cres 4541   Fun wfun 5117
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077  df-ss 3084  df-br 3930  df-opab 3990  df-rel 4546  df-cnv 4547  df-co 4548  df-res 4551  df-fun 5125
This theorem is referenced by:  fnssresb  5235  fnresi  5240  fores  5354  respreima  5548  resfunexg  5641  funfvima  5649  smores  6189  smores2  6191  frecfun  6292  sbthlem7  6851  setsfun  12004  setsfun0  12005
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