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Theorem fnssresd 39977
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 6161 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) Fn 𝐵)
41, 2, 3syl2anc 696 1 (𝜑 → (𝐹𝐵) Fn 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3711  cres 5264   Fn wfn 6040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929  ax-nul 4937  ax-pr 5051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ral 3051  df-rex 3052  df-rab 3055  df-v 3338  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4801  df-opab 4861  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-res 5274  df-fun 6047  df-fn 6048
This theorem is referenced by:  xlimconst2  40560
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