Theorem fnfco 6707

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 6504	. 2 ⊢ (𝐺:𝐵⟶𝐴 ↔ (𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴))
2		fnco 6618	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵)
3	2	3expb 1121	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴)) → (𝐹 ∘ 𝐺) Fn 𝐵)
4	1, 3	sylan2b 595	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ⊆ wss 3903  ran crn 5633   ∘ ccom 5636   Fn wfn 6495  ⟶wf 6496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-fun 6502  df-fn 6503  df-f 6504

This theorem is referenced by:  cocan1  7247  cocan2  7248  coof  7656  ofco  7657  1stcof  7973  2ndcof  7974  axcc3  10360  dmaf  17985  cdaf  17986  gsumzaddlem  19862  prdstopn  23584  xpstopnlem2  23767  prdstgpd  24081  prdsxmslem2  24485  uniiccdif  25547  uniiccvol  25549  uniioombllem2  25552  resinf1o  26513  jensen  26967  occllem  31390  nlelchi  32148  hmopidmchi  32238  ofrco  32699  constcof  32710  1arithidomlem2  33628  mplvrpmrhm  33723  esplyfval3  33748  iprodefisumlem  35953  brcoffn  44383  brcofffn  44384  stoweidlem27  46382  gricushgr  48274  fucoid  49704