ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  funres11 GIF version

Theorem funres11 5165
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 4813 . 2 (𝐹𝐴) ⊆ 𝐹
2 cnvss 4682 . 2 ((𝐹𝐴) ⊆ 𝐹(𝐹𝐴) ⊆ 𝐹)
3 funss 5112 . 2 ((𝐹𝐴) ⊆ 𝐹 → (Fun 𝐹 → Fun (𝐹𝐴)))
41, 2, 3mp2b 8 1 (Fun 𝐹 → Fun (𝐹𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wss 3041  ccnv 4508  cres 4511  Fun wfun 5087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047  df-ss 3054  df-br 3900  df-opab 3960  df-rel 4516  df-cnv 4517  df-co 4518  df-res 4521  df-fun 5095
This theorem is referenced by:  f1ssres  5307  resdif  5357  ssdomg  6640  sbthlemi8  6820
  Copyright terms: Public domain W3C validator