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

Theorem fnfco 5065
Description: Composition of two functions. (Contributed by NM, 22-May-2006.)
Assertion
Ref Expression
fnfco ((𝐹 Fn 𝐴𝐺:𝐵𝐴) → (𝐹𝐺) Fn 𝐵)

Proof of Theorem fnfco
StepHypRef Expression
1 df-f 4906 . 2 (𝐺:𝐵𝐴 ↔ (𝐺 Fn 𝐵 ∧ ran 𝐺𝐴))
2 fnco 5007 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ran 𝐺𝐴) → (𝐹𝐺) Fn 𝐵)
323expb 1105 . 2 ((𝐹 Fn 𝐴 ∧ (𝐺 Fn 𝐵 ∧ ran 𝐺𝐴)) → (𝐹𝐺) Fn 𝐵)
41, 3sylan2b 271 1 ((𝐹 Fn 𝐴𝐺:𝐵𝐴) → (𝐹𝐺) Fn 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wss 2917  ran crn 4346  ccom 4349   Fn wfn 4897  wf 4898
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-fun 4904  df-fn 4905  df-f 4906
This theorem is referenced by:  cocan1  5427  cocan2  5428  ofco  5729  1stcof  5790  2ndcof  5791
  Copyright terms: Public domain W3C validator