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Theorem fnssresd 6664
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 6663 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) Fn 𝐵)
41, 2, 3syl2anc 583 1 (𝜑 → (𝐹𝐵) Fn 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3940  cres 5668   Fn wfn 6528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-res 5678  df-fun 6535  df-fn 6536
This theorem is referenced by:  rescnvimafod  7065  fpwwe2lem7  10627  pfxccat1  14648  mdetrsca  22415  2ndresdju  32298  fdifsuppconst  32335  ply1gsumz  33101  dimkerim  33157  rmulccn  33363  subfacp1lem3  34628  satfn  34801  eqresfnbd  41513  tfsconcatrev  42553  ofoafg  42559  xlimconst2  45002  fcoreslem4  46227
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