Theorem fnfco 6069

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 5892	. 2 ⊢ (𝐺:𝐵⟶𝐴 ↔ (𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴))
2		fnco 5999	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵)
3	2	3expb 1266	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴)) → (𝐹 ∘ 𝐺) Fn 𝐵)
4	1, 3	sylan2b 492	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵)

Syntax hints:   → wi 4   ∧ wa 384   ⊆ wss 3574  ran crn 5115   ∘ ccom 5118   Fn wfn 5883  ⟶wf 5884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-fun 5890  df-fn 5891  df-f 5892

This theorem is referenced by:  cocan1  6546  cocan2  6547  ofco  6917  1stcof  7196  2ndcof  7197  axcc3  9260  dmaf  16699  cdaf  16700  gsumzaddlem  18321  prdstopn  21431  xpstopnlem2  21614  prdstgpd  21928  prdsxmslem2  22334  uniiccdif  23346  uniiccvol  23348  uniioombllem2  23351  resinf1o  24282  jensen  24715  occllem  28162  nlelchi  28920  hmopidmchi  29010  iprodefisumlem  31626  brcoffn  38328  brcofffn  38329  stoweidlem27  40244