ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fnresin1 GIF version

Theorem fnresin1 5038
Description: Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)
Assertion
Ref Expression
fnresin1 (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴𝐵)) Fn (𝐴𝐵))

Proof of Theorem fnresin1
StepHypRef Expression
1 inss1 3187 . 2 (𝐴𝐵) ⊆ 𝐴
2 fnssres 5037 . 2 ((𝐹 Fn 𝐴 ∧ (𝐴𝐵) ⊆ 𝐴) → (𝐹 ↾ (𝐴𝐵)) Fn (𝐴𝐵))
31, 2mpan2 416 1 (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴𝐵)) Fn (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  cin 2973  wss 2974  cres 4367   Fn wfn 4921
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788  df-opab 3842  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-res 4377  df-fun 4928  df-fn 4929
This theorem is referenced by:  frecfnom  6044
  Copyright terms: Public domain W3C validator