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Theorem funres 5609
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 5285 . 2  |-  ( F  |`  A )  C_  F
2 funss 5588 . 2  |-  ( ( F  |`  A )  C_  F  ->  ( Fun  F  ->  Fun  ( F  |`  A ) ) )
31, 2ax-mp 5 1  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    C_ wss 3461    |` cres 4990   Fun wfun 5564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108  df-in 3468  df-ss 3475  df-br 4440  df-opab 4498  df-rel 4995  df-cnv 4996  df-co 4997  df-res 5000  df-fun 5572
This theorem is referenced by:  fnssresb  5675  fnresi  5680  fores  5786  respreima  5992  resfunexg  6112  funfvima  6122  funiunfv  6135  smores  7015  smores2  7017  frfnom  7092  sbthlem7  7626  fsuppres  7846  ordtypelem4  7938  wdomima2g  8004  imadomg  8903  hashimarn  12480  lubfun  15809  glbfun  15822  gsumzadd  17134  gsum2dlem2  17194  qtoptop2  20366  volf  22106  sspg  25839  ssps  25841  sspn  25847  hlimf  26353  fresf1o  27692  eulerpartlemmf  28578  eulerpartlemgvv  28579  wfrlem5  29587  frrlem5  29631  funcoressn  32451  afvelrn  32492  dmfcoafv  32499  afvco2  32500  aovmpt4g  32525
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