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Theorem fssresd 5299
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 (𝜑𝐹:𝐴𝐵)
fssresd.2 (𝜑𝐶𝐴)
Assertion
Ref Expression
fssresd (𝜑 → (𝐹𝐶):𝐶𝐵)

Proof of Theorem fssresd
StepHypRef Expression
1 fssresd.1 . 2 (𝜑𝐹:𝐴𝐵)
2 fssresd.2 . 2 (𝜑𝐶𝐴)
3 fssres 5298 . 2 ((𝐹:𝐴𝐵𝐶𝐴) → (𝐹𝐶):𝐶𝐵)
41, 2, 3syl2anc 408 1 (𝜑 → (𝐹𝐶):𝐶𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wss 3071  cres 4541  wf 5119
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-fun 5125  df-fn 5126  df-f 5127
This theorem is referenced by:  cnrest  12418  cnptopresti  12421  cnptoprest  12422  psmetres2  12516  xmetres2  12562  metres2  12564  xmetresbl  12623  rescncf  12751  trilpolemlt1  13341
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