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Theorem fnssresd 6459
 Description: Restriction of a function to a subclass of its domain. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
Hypotheses
Ref Expression
fnssresd.1 (𝜑𝐹 Fn 𝐴)
fnssresd.2 (𝜑𝐵𝐴)
Assertion
Ref Expression
fnssresd (𝜑 → (𝐹𝐵) Fn 𝐵)

Proof of Theorem fnssresd
StepHypRef Expression
1 fnssresd.1 . 2 (𝜑𝐹 Fn 𝐴)
2 fnssresd.2 . 2 (𝜑𝐵𝐴)
3 fnssres 6458 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) Fn 𝐵)
41, 2, 3syl2anc 587 1 (𝜑 → (𝐹𝐵) Fn 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ⊆ wss 3860   ↾ cres 5530   Fn wfn 6335 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-br 5037  df-opab 5099  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-res 5540  df-fun 6342  df-fn 6343 This theorem is referenced by:  fpwwe2lem7  10110  pfxccat1  14124  2ndresdju  30522  fdifsuppconst  30560  subfacp1lem3  32673  satfn  32846  xlimconst2  42888
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