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Theorem funres 5484
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 5162 . 2  |-  ( F  |`  A )  C_  F
2 funss 5464 . 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 3312    |` cres 4872   Fun wfun 5440
This theorem is referenced by:  fnssresb  5549  fnresi  5554  fores  5654  respreima  5851  resfunexg  5949  funfvima  5965  funiunfv  5987  smores  6606  smores2  6608  frfnom  6684  sbthlem7  7215  ordtypelem4  7482  wdomima2g  7546  imadomg  8404  cnrest  17341  qtoptop2  17723  volf  19417  sspg  22219  ssps  22221  sspn  22227  hlimf  22732  wfrlem5  25534  frrlem5  25578  funcoressn  27958  afvelrn  27999  dmfcoafv  28006  afvco2  28007  aovmpt4g  28032  hashimarn  28141
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-in 3319  df-ss 3326  df-br 4205  df-opab 4259  df-rel 4877  df-cnv 4878  df-co 4879  df-res 4882  df-fun 5448
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