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

Theorem fnssres 5127
Description: Restriction of a function with a subclass of its domain. (Contributed by NM, 2-Aug-1994.)
Assertion
Ref Expression
fnssres ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) Fn 𝐵)

Proof of Theorem fnssres
StepHypRef Expression
1 fnssresb 5126 . 2 (𝐹 Fn 𝐴 → ((𝐹𝐵) Fn 𝐵𝐵𝐴))
21biimpar 291 1 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) Fn 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wss 2999  cres 4440   Fn wfn 5010
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-res 4450  df-fun 5017  df-fn 5018
This theorem is referenced by:  fnresin1  5128  fnresin2  5129  fssres  5186  fvreseq  5403  fnreseql  5409  ffvresb  5461  fnressn  5483  ofres  5869  tfrlem1  6073  frecrdg  6173  resfnfinfinss  6649  iseqfeq2  9891  seq3feq2  9893  reeff1  10991  eucialg  11319
  Copyright terms: Public domain W3C validator