Theorem fnfco 5085

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 4926	. 2 ⊢ (𝐺:𝐵⟶𝐴 ↔ (𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴))
2		fnco 5027	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵)
3	2	3expb 1139	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴)) → (𝐹 ∘ 𝐺) Fn 𝐵)
4	1, 3	sylan2b 281	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵)

Syntax hints:   → wi 4   ∧ wa 102   ⊆ wss 2973  ran crn 4364   ∘ ccom 4367   Fn wfn 4917  ⟶wf 4918

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-fun 4924  df-fn 4925  df-f 4926

This theorem is referenced by:  cocan1  5447  cocan2  5448  ofco  5749  1stcof  5810  2ndcof  5811