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Theorem fssresd 5346
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 5345 . 2  |-  ( ( F : A --> B  /\  C  C_  A )  -> 
( F  |`  C ) : C --> B )
41, 2, 3syl2anc 409 1  |-  ( ph  ->  ( F  |`  C ) : C --> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    C_ wss 3102    |` cres 4588   -->wf 5166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-fun 5172  df-fn 5173  df-f 5174
This theorem is referenced by:  cnrest  12635  cnptopresti  12638  cnptoprest  12639  psmetres2  12733  xmetres2  12779  metres2  12781  xmetresbl  12840  rescncf  12968  trilpolemlt1  13612
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