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Theorem funres 5150
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 4886 . 2  |-  ( F  |`  A )  C_  F
2 funss 5131 . 2  |-  ( ( F  |`  A )  C_  F  ->  ( Fun  F  ->  Fun  ( F  |`  A ) ) )
31, 2ax-mp 10 1  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    C_ wss 3078    |` cres 4582   Fun wfun 4586
This theorem is referenced by:  fnssresb  5213  fnresi  5218  fores  5317  respreima  5506  resfunexg  5589  funfvima  5605  funiunfv  5626  smores  6255  smores2  6257  frfnom  6333  sbthlem7  6862  ordtypelem4  7120  wdomima2g  7184  imadomg  8043  cnrest  16845  qtoptop2  17222  volf  18720  sspg  21134  ssps  21136  sspn  21142  hlimf  21647  wfrlem5  23428  frrlem5  23453  domrancur1clem  24367  domrancur1c  24368  tartarmap  25054
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-v 2729  df-in 3085  df-ss 3089  df-br 3921  df-opab 3975  df-rel 4595  df-cnv 4596  df-co 4597  df-res 4600  df-fun 4602
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