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Theorem fssresd 5411
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 5410 . 2 ((𝐹:𝐴𝐵𝐶𝐴) → (𝐹𝐶):𝐶𝐵)
41, 2, 3syl2anc 411 1 (𝜑 → (𝐹𝐶):𝐶𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wss 3144  cres 4646  wf 5231
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-fun 5237  df-fn 5238  df-f 5239
This theorem is referenced by:  cnrest  14192  cnptopresti  14195  cnptoprest  14196  psmetres2  14290  xmetres2  14336  metres2  14338  xmetresbl  14397  rescncf  14525  trilpolemlt1  15248
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