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Theorem funres 5451
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 5129 . 2  |-  ( F  |`  A )  C_  F
2 funss 5431 . 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 3280    |` cres 4839   Fun wfun 5407
This theorem is referenced by:  fnssresb  5516  fnresi  5521  fores  5621  respreima  5818  resfunexg  5916  funfvima  5932  funiunfv  5954  smores  6573  smores2  6575  frfnom  6651  sbthlem7  7182  ordtypelem4  7446  wdomima2g  7510  imadomg  8368  cnrest  17303  qtoptop2  17684  volf  19378  sspg  22180  ssps  22182  sspn  22188  hlimf  22693  wfrlem5  25474  frrlem5  25499  funcoressn  27858  afvelrn  27899  dmfcoafv  27906  afvco2  27907  aovmpt4g  27932  hashimarn  27994
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-in 3287  df-ss 3294  df-br 4173  df-opab 4227  df-rel 4844  df-cnv 4845  df-co 4846  df-res 4849  df-fun 5415
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