ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fssres2 GIF version

Theorem fssres2 5010
Description: Restriction of a restricted function with a subclass of its domain. (Contributed by NM, 21-Jul-2005.)
Assertion
Ref Expression
fssres2 (((𝐹A):AB 𝐶A) → (𝐹𝐶):𝐶B)

Proof of Theorem fssres2
StepHypRef Expression
1 fssres 5009 . 2 (((𝐹A):AB 𝐶A) → ((𝐹A) ↾ 𝐶):𝐶B)
2 resabs1 4583 . . . 4 (𝐶A → ((𝐹A) ↾ 𝐶) = (𝐹𝐶))
32feq1d 4977 . . 3 (𝐶A → (((𝐹A) ↾ 𝐶):𝐶B ↔ (𝐹𝐶):𝐶B))
43adantl 262 . 2 (((𝐹A):AB 𝐶A) → (((𝐹A) ↾ 𝐶):𝐶B ↔ (𝐹𝐶):𝐶B))
51, 4mpbid 135 1 (((𝐹A):AB 𝐶A) → (𝐹𝐶):𝐶B)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wss 2911  cres 4290  wf 4841
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-fun 4847  df-fn 4848  df-f 4849
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator