MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funres11 Structured version   Visualization version   GIF version

Theorem funres11 6575
Description: The restriction of a one-to-one function is one-to-one. (Contributed by NM, 25-Mar-1998.)
Assertion
Ref Expression
funres11 (Fun 𝐹 → Fun (𝐹𝐴))

Proof of Theorem funres11
StepHypRef Expression
1 resss 5960 . 2 (𝐹𝐴) ⊆ 𝐹
2 cnvss 5826 . 2 ((𝐹𝐴) ⊆ 𝐹(𝐹𝐴) ⊆ 𝐹)
3 funss 6517 . 2 ((𝐹𝐴) ⊆ 𝐹 → (Fun 𝐹 → Fun (𝐹𝐴)))
41, 2, 3mp2b 10 1 (Fun 𝐹 → Fun (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3908  ccnv 5630  cres 5633  Fun wfun 6487
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3445  df-in 3915  df-ss 3925  df-br 5104  df-opab 5166  df-rel 5638  df-cnv 5639  df-co 5640  df-res 5643  df-fun 6495
This theorem is referenced by:  f1ssres  6743  resdif  6802  f1ssf1  6813  ssdomg  8936  sbthlem8  9030  spthispth  28560
  Copyright terms: Public domain W3C validator