Theorem fco 5488

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 5321	. . 3 ⊢ (𝐹:𝐵⟶𝐶 ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶))
2		df-f 5321	. . 3 ⊢ (𝐺:𝐴⟶𝐵 ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐵))
3		fnco 5430	. . . . . . 7 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐺 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐵) → (𝐹 ∘ 𝐺) Fn 𝐴)
4	3	3expib 1230	. . . . . 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 4994	. . . . . . 7 ⊢ ran (𝐹 ∘ 𝐺) ⊆ ran 𝐹
7		sstr 3232	. . . . . . 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 5321	. 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 3197  ran crn 4719   ∘ ccom 4722   Fn wfn 5312  ⟶wf 5313

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-fun 5319  df-fn 5320  df-f 5321

This theorem is referenced by:  fco2  5489  f1co  5542  foco  5558  mapen  7003  ctm  7272  enomnilem  7301  enmkvlem  7324  enwomnilem  7332  fnn0nninf  10655  seqf1oglem2  10737  fsumcl2lem  11904  fsumadd  11912  fprodmul  12097  algcvg  12565  mhmco  13518  gsumwmhm  13526  gsumfzreidx  13869  gsumfzmhm  13875  psrnegcl  14641  cnco  14889  cnptopco  14890  lmtopcnp  14918  cnmpt11  14951  cnmpt21  14959  comet  15167  cnmet  15198  cnfldms  15204  cncfco  15259  limccnpcntop  15343  dvcoapbr  15375  dvcjbr  15376  dvcj  15377