Theorem fnco 4950

Description: Composition of two functions. (Contributed by NM, 22-May-2006.)

Assertion

Ref	Expression
fnco	⊢ ((𝐹 Fn A ∧ 𝐺 Fn B ∧ ran 𝐺 ⊆ A) → (𝐹 ∘ 𝐺) Fn B)

Proof of Theorem fnco

Step	Hyp	Ref	Expression
1		fnfun 4939	. . . 4 ⊢ (𝐹 Fn A → Fun 𝐹)
2		fnfun 4939	. . . 4 ⊢ (𝐺 Fn B → Fun 𝐺)
3		funco 4883	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺))
4	1, 2, 3	syl2an 273	. . 3 ⊢ ((𝐹 Fn A ∧ 𝐺 Fn B) → Fun (𝐹 ∘ 𝐺))
5	4	3adant3 923	. 2 ⊢ ((𝐹 Fn A ∧ 𝐺 Fn B ∧ ran 𝐺 ⊆ A) → Fun (𝐹 ∘ 𝐺))
6		fndm 4941	. . . . . . 7 ⊢ (𝐹 Fn A → dom 𝐹 = A)
7	6	sseq2d 2967	. . . . . 6 ⊢ (𝐹 Fn A → (ran 𝐺 ⊆ dom 𝐹 ↔ ran 𝐺 ⊆ A))
8	7	biimpar 281	. . . . 5 ⊢ ((𝐹 Fn A ∧ ran 𝐺 ⊆ A) → ran 𝐺 ⊆ dom 𝐹)
9		dmcosseq 4546	. . . . 5 ⊢ (ran 𝐺 ⊆ dom 𝐹 → dom (𝐹 ∘ 𝐺) = dom 𝐺)
10	8, 9	syl 14	. . . 4 ⊢ ((𝐹 Fn A ∧ ran 𝐺 ⊆ A) → dom (𝐹 ∘ 𝐺) = dom 𝐺)
11	10	3adant2 922	. . 3 ⊢ ((𝐹 Fn A ∧ 𝐺 Fn B ∧ ran 𝐺 ⊆ A) → dom (𝐹 ∘ 𝐺) = dom 𝐺)
12		fndm 4941	. . . 4 ⊢ (𝐺 Fn B → dom 𝐺 = B)
13	12	3ad2ant2 925	. . 3 ⊢ ((𝐹 Fn A ∧ 𝐺 Fn B ∧ ran 𝐺 ⊆ A) → dom 𝐺 = B)
14	11, 13	eqtrd 2069	. 2 ⊢ ((𝐹 Fn A ∧ 𝐺 Fn B ∧ ran 𝐺 ⊆ A) → dom (𝐹 ∘ 𝐺) = B)
15		df-fn 4848	. 2 ⊢ ((𝐹 ∘ 𝐺) Fn B ↔ (Fun (𝐹 ∘ 𝐺) ∧ dom (𝐹 ∘ 𝐺) = B))
16	5, 14, 15	sylanbrc 394	1 ⊢ ((𝐹 Fn A ∧ 𝐺 Fn B ∧ ran 𝐺 ⊆ A) → (𝐹 ∘ 𝐺) Fn B)

Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 884   = wceq 1242   ⊆ wss 2911  dom cdm 4288  ran crn 4289   ∘ ccom 4292  Fun wfun 4839   Fn wfn 4840

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848

This theorem is referenced by:  fco  4999  fnfco  5008