Theorem fnfco 6697

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 6494	. 2 ⊢ (𝐺:𝐵⟶𝐴 ↔ (𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴))
2		fnco 6608	. . 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 3890  ran crn 5623   ∘ ccom 5626   Fn wfn 6485  ⟶wf 6486

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 5231  ax-pr 5368

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 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-fun 6492  df-fn 6493  df-f 6494

This theorem is referenced by:  cocan1  7237  cocan2  7238  coof  7646  ofco  7647  1stcof  7963  2ndcof  7964  axcc3  10349  dmaf  17974  cdaf  17975  gsumzaddlem  19854  prdstopn  23571  xpstopnlem2  23754  prdstgpd  24068  prdsxmslem2  24472  uniiccdif  25523  uniiccvol  25525  uniioombllem2  25528  resinf1o  26485  jensen  26939  occllem  31363  nlelchi  32121  hmopidmchi  32211  ofrco  32672  constcof  32683  1arithidomlem2  33601  mplvrpmrhm  33696  esplyfval3  33721  iprodefisumlem  35928  brcoffn  44460  brcofffn  44461  stoweidlem27  46459  gricushgr  48351  fucoid  49781