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Theorem funres 5134
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 4813 . 2  |-  ( F  |`  A )  C_  F
2 funss 5112 . 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 3041    |` cres 4511   Fun wfun 5087
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047  df-ss 3054  df-br 3900  df-opab 3960  df-rel 4516  df-cnv 4517  df-co 4518  df-res 4521  df-fun 5095
This theorem is referenced by:  fnssresb  5205  fnresi  5210  fores  5324  respreima  5516  resfunexg  5609  funfvima  5617  smores  6157  smores2  6159  frecfun  6260  sbthlem7  6819  setsfun  11921  setsfun0  11922
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