Theorem fco 5527

Description: Composition of two mappings. (Contributed by NM, 29-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Assertion

Ref	Expression
fco	⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶𝐶)

Proof of Theorem fco

Step	Hyp	Ref	Expression
1		df-f 5356	. . 3 ⊢ (𝐹:𝐵⟶𝐶 ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶))
2		df-f 5356	. . 3 ⊢ (𝐺:𝐴⟶𝐵 ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐵))
3		fnco 5466	. . . . . . 7 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐺 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐵) → (𝐹 ∘ 𝐺) Fn 𝐴)
4	3	3expib 1233	. . . . . 6 ⊢ (𝐹 Fn 𝐵 → ((𝐺 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐵) → (𝐹 ∘ 𝐺) Fn 𝐴))
5	4	adantr 276	. . . . 5 ⊢ ((𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶) → ((𝐺 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐵) → (𝐹 ∘ 𝐺) Fn 𝐴))
6		rncoss 5028	. . . . . . 7 ⊢ ran (𝐹 ∘ 𝐺) ⊆ ran 𝐹
7		sstr 3246	. . . . . . 7 ⊢ ((ran (𝐹 ∘ 𝐺) ⊆ ran 𝐹 ∧ ran 𝐹 ⊆ 𝐶) → ran (𝐹 ∘ 𝐺) ⊆ 𝐶)
8	6, 7	mpan 424	. . . . . 6 ⊢ (ran 𝐹 ⊆ 𝐶 → ran (𝐹 ∘ 𝐺) ⊆ 𝐶)
9	8	adantl 277	. . . . 5 ⊢ ((𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶) → ran (𝐹 ∘ 𝐺) ⊆ 𝐶)
10	5, 9	jctird 317	. . . 4 ⊢ ((𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶) → ((𝐺 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐵) → ((𝐹 ∘ 𝐺) Fn 𝐴 ∧ ran (𝐹 ∘ 𝐺) ⊆ 𝐶)))
11	10	imp 124	. . 3 ⊢ (((𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶) ∧ (𝐺 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐵)) → ((𝐹 ∘ 𝐺) Fn 𝐴 ∧ ran (𝐹 ∘ 𝐺) ⊆ 𝐶))
12	1, 2, 11	syl2anb 291	. 2 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → ((𝐹 ∘ 𝐺) Fn 𝐴 ∧ ran (𝐹 ∘ 𝐺) ⊆ 𝐶))
13		df-f 5356	. 2 ⊢ ((𝐹 ∘ 𝐺):𝐴⟶𝐶 ↔ ((𝐹 ∘ 𝐺) Fn 𝐴 ∧ ran (𝐹 ∘ 𝐺) ⊆ 𝐶))
14	12, 13	sylibr 134	1 ⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶𝐶)

Syntax hints:   → wi 4   ∧ wa 104   ⊆ wss 3211  ran crn 4750   ∘ ccom 4753   Fn wfn 5347  ⟶wf 5348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-fun 5354  df-fn 5355  df-f 5356

This theorem is referenced by:  fcod  5528  fco2  5529  f1co  5585  foco  5601  mapen  7099  ctm  7400  enomnilem  7429  enmkvlem  7452  enwomnilem  7460  fnn0nninf  10800  seqf1oglem2  10882  fsumcl2lem  12084  fsumadd  12092  fprodmul  12277  algcvg  12745  mhmco  13703  gsumwmhm  13711  gsumfzreidx  14054  gsumfzmhm  14060  psrnegcl  14838  cnco  15086  cnptopco  15087  lmtopcnp  15115  cnmpt11  15148  cnmpt21  15156  comet  15364  cnmet  15395  cnfldms  15401  cncfco  15456  limccnpcntop  15540  dvcoapbr  15572  dvcjbr  15573  dvcj  15574  gfsumval  16862  gsumgfsumlem  16865  gfsump1  16868