MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnssresd Structured version   Visualization version   GIF version

Theorem fnssresd 6668
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 6667 . 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 3943  cres 5671   Fn wfn 6532
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 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
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 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-res 5681  df-fun 6539  df-fn 6540
This theorem is referenced by:  rescnvimafod  7069  fpwwe2lem7  10634  pfxccat1  14658  mdetrsca  22460  2ndresdju  32383  fdifsuppconst  32418  ply1gsumz  33174  dimkerim  33230  rmulccn  33438  subfacp1lem3  34701  satfn  34874  eqresfnbd  41612  tfsconcatrev  42671  ofoafg  42677  xlimconst2  45120  fcoreslem4  46345
  Copyright terms: Public domain W3C validator