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Theorem fssresd 5406
Description: Restriction of a function with a subclass of its domain, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fssresd.1  |-  ( ph  ->  F : A --> B )
fssresd.2  |-  ( ph  ->  C  C_  A )
Assertion
Ref Expression
fssresd  |-  ( ph  ->  ( F  |`  C ) : C --> B )

Proof of Theorem fssresd
StepHypRef Expression
1 fssresd.1 . 2  |-  ( ph  ->  F : A --> B )
2 fssresd.2 . 2  |-  ( ph  ->  C  C_  A )
3 fssres 5405 . 2  |-  ( ( F : A --> B  /\  C  C_  A )  -> 
( F  |`  C ) : C --> B )
41, 2, 3syl2anc 411 1  |-  ( ph  ->  ( F  |`  C ) : C --> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    C_ wss 3143    |` cres 4642   -->wf 5226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-pow 4188  ax-pr 4223
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472  df-rex 2473  df-v 2753  df-un 3147  df-in 3149  df-ss 3156  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-br 4018  df-opab 4079  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-fun 5232  df-fn 5233  df-f 5234
This theorem is referenced by:  cnrest  14118  cnptopresti  14121  cnptoprest  14122  psmetres2  14216  xmetres2  14262  metres2  14264  xmetresbl  14323  rescncf  14451  trilpolemlt1  15173
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