Theorem fnfco 6688

Description: Composition of two functions. (Contributed by NM, 22-May-2006.)

Assertion

Ref	Expression
fnfco	⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵)

Proof of Theorem fnfco

Step	Hyp	Ref	Expression
1		df-f 6485	. 2 ⊢ (𝐺:𝐵⟶𝐴 ↔ (𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴))
2		fnco 6599	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵)
3	2	3expb 1120	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴)) → (𝐹 ∘ 𝐺) Fn 𝐵)
4	1, 3	sylan2b 594	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ⊆ wss 3902  ran crn 5617   ∘ ccom 5620   Fn wfn 6476  ⟶wf 6477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-fun 6483  df-fn 6484  df-f 6485

This theorem is referenced by:  cocan1  7225  cocan2  7226  coof  7634  ofco  7635  1stcof  7951  2ndcof  7952  axcc3  10329  dmaf  17956  cdaf  17957  gsumzaddlem  19834  prdstopn  23544  xpstopnlem2  23727  prdstgpd  24041  prdsxmslem2  24445  uniiccdif  25507  uniiccvol  25509  uniioombllem2  25512  resinf1o  26473  jensen  26927  occllem  31281  nlelchi  32039  hmopidmchi  32129  ofrco  32591  constcof  32602  1arithidomlem2  33499  mplvrpmrhm  33575  iprodefisumlem  35782  brcoffn  44069  brcofffn  44070  stoweidlem27  46071  gricushgr  47954  fucoid  49386