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Theorem funres 5293
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 4979 . 2  |-  ( F  |`  A )  C_  F
2 funss 5273 . 2  |-  ( ( F  |`  A )  C_  F  ->  ( Fun  F  ->  Fun  ( F  |`  A ) ) )
31, 2ax-mp 8 1  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    C_ wss 3152    |` cres 4691   Fun wfun 5249
This theorem is referenced by:  fnssresb  5356  fnresi  5361  fores  5460  respreima  5654  resfunexg  5737  funfvima  5753  funiunfv  5774  smores  6369  smores2  6371  frfnom  6447  sbthlem7  6977  ordtypelem4  7236  wdomima2g  7300  imadomg  8159  cnrest  17013  qtoptop2  17390  volf  18888  sspg  21304  ssps  21306  sspn  21312  hlimf  21817  wfrlem5  24260  frrlem5  24285  domrancur1clem  25201  domrancur1c  25202  tartarmap  25888  funcoressn  27990  afvelrn  28030  dmfcoafv  28036  afvco2  28037  aovmpt4g  28061
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159  df-ss 3166  df-br 4024  df-opab 4078  df-rel 4696  df-cnv 4697  df-co 4698  df-res 4701  df-fun 5257
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