Theorem fco 2760

Description: Composition of two mappings.

Assertion

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

Proof of Theorem fco

Step	Hyp	Ref	Expression
1		funco 2696	. . . . . 6 ⊢ ((Fun F ∧ Fun G) → Fun (F ∘ G))
2		ffun 2754	. . . . . 6 ⊢ (F:B–→C → Fun F)
3		ffun 2754	. . . . . 6 ⊢ (G:A–→B → Fun G)
4	1, 2, 3	syl2an 349	. . . . 5 ⊢ ((F:B–→C ∧ G:A–→B) → Fun (F ∘ G))
5		fdm 2756	. . . . . . . . . 10 ⊢ (F:B–→C → dom F = B)
6	5	sseq2d 1528	. . . . . . . . 9 ⊢ (F:B–→C → (ran G ⊆ dom F ↔ ran G ⊆ B))
7		frn 2757	. . . . . . . . 9 ⊢ (G:A–→B → ran G ⊆ B)
8	6, 7	syl5bir 184	. . . . . . . 8 ⊢ (F:B–→C → (G:A–→B → ran G ⊆ dom F))
9	8	imp 277	. . . . . . 7 ⊢ ((F:B–→C ∧ G:A–→B) → ran G ⊆ dom F)
10		dmcosseq 2572	. . . . . . 7 ⊢ (ran G ⊆ dom F → dom (F ∘ G) = dom G)
11	9, 10	syl 12	. . . . . 6 ⊢ ((F:B–→C ∧ G:A–→B) → dom (F ∘ G) = dom G)
12		fdm 2756	. . . . . . 7 ⊢ (G:A–→B → dom G = A)
13	12	adantl 305	. . . . . 6 ⊢ ((F:B–→C ∧ G:A–→B) → dom G = A)
14	11, 13	eqtrd 1128	. . . . 5 ⊢ ((F:B–→C ∧ G:A–→B) → dom (F ∘ G) = A)
15	4, 14	jca 236	. . . 4 ⊢ ((F:B–→C ∧ G:A–→B) → (Fun (F ∘ G) ∧ dom (F ∘ G) = A))
16		df-fn 2433	. . . 4 ⊢ ((F ∘ G) Fn A ↔ (Fun (F ∘ G) ∧ dom (F ∘ G) = A))
17	15, 16	sylibr 175	. . 3 &38866; ((F:B–→C ∧ G:A–→B) → (F ∘ G) Fn A)
18		rnco 2571	. . . . 5 ⊢ ran (F ∘ G) ⊆ ran F
19		sstr2 1510	. . . . . 6 ⊢ (ran (F ∘ G) ⊆ ran F → (ran F ⊆ C → ran (F ∘ G) ⊆ C))
20		frn 2757	. . . . . 6 ⊢ (F:B–→C → ran F ⊆ C)
21	19, 20	syl5 22	. . . . 5 ⊢ (ran (F ∘ G) ⊆ ran F → (F:B–→C → ran (F ∘ G) ⊆ C))
22	18, 21	ax-mp 6	. . . 4 ⊢ (F:B–→C → ran (F ∘ G) ⊆ C)
23	22	adantr 306	. . 3 ⊢ ((F:B–→C ∧ G:A–→B) → ran (F ∘ G) ⊆ C)
24	17, 23	jca 236	. 2 ⊢ ((F:B–→C ∧ G:A–→B) → ((F ∘ G) Fn A ∧ ran (F ∘ G) ⊆ C))
25		df-f 2434	. 2 ⊢ ((F ∘ G):A–→C ↔ ((F ∘ G) Fn A ∧ ran (F ∘ G) ⊆ C))
26	24, 25	sylibr 175	1 ⊢ ((F:B–→C ∧ G:A–→B) → (F ∘ G):A–→C)

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ⊆ wss 1487  dom cdm 2410  ran crn 2411   ∘ ccom 2414  Fun wfun 2416   Fn wfn 2417  –→wf 2418

This theorem is referenced by:  f1co 2783  mapenlem1 3384  mapenlem2 3385  ac6lem 3575  ruclem17 4901  hocof 5600

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-fun 2432  df-fn 2433  df-f 2434

metamath.org