Theorem fnresfnco 44422

Description: Composition of two functions, similar to fnco 6533. (Contributed by Alexander van der Vekens, 25-Jul-2017.)

Assertion

Ref	Expression
fnresfnco	⊢ (((𝐹 ↾ ran 𝐺) Fn ran 𝐺 ∧ 𝐺 Fn 𝐵) → (𝐹 ∘ 𝐺) Fn 𝐵)

Proof of Theorem fnresfnco

Step	Hyp	Ref	Expression
1		fnfun 6517	. . 3 ⊢ ((𝐹 ↾ ran 𝐺) Fn ran 𝐺 → Fun (𝐹 ↾ ran 𝐺))
2		fnfun 6517	. . 3 ⊢ (𝐺 Fn 𝐵 → Fun 𝐺)
3		funresfunco 6459	. . 3 ⊢ ((Fun (𝐹 ↾ ran 𝐺) ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺))
4	1, 2, 3	syl2an 595	. 2 ⊢ (((𝐹 ↾ ran 𝐺) Fn ran 𝐺 ∧ 𝐺 Fn 𝐵) → Fun (𝐹 ∘ 𝐺))
5		fndm 6520	. . . . . 6 ⊢ ((𝐹 ↾ ran 𝐺) Fn ran 𝐺 → dom (𝐹 ↾ ran 𝐺) = ran 𝐺)
6		dmres 5902	. . . . . . . 8 ⊢ dom (𝐹 ↾ ran 𝐺) = (ran 𝐺 ∩ dom 𝐹)
7	6	eqeq1i 2743	. . . . . . 7 ⊢ (dom (𝐹 ↾ ran 𝐺) = ran 𝐺 ↔ (ran 𝐺 ∩ dom 𝐹) = ran 𝐺)
8		df-ss 3900	. . . . . . 7 ⊢ (ran 𝐺 ⊆ dom 𝐹 ↔ (ran 𝐺 ∩ dom 𝐹) = ran 𝐺)
9	7, 8	sylbb2 237	. . . . . 6 ⊢ (dom (𝐹 ↾ ran 𝐺) = ran 𝐺 → ran 𝐺 ⊆ dom 𝐹)
10	5, 9	syl 17	. . . . 5 ⊢ ((𝐹 ↾ ran 𝐺) Fn ran 𝐺 → ran 𝐺 ⊆ dom 𝐹)
11	10	adantr 480	. . . 4 ⊢ (((𝐹 ↾ ran 𝐺) Fn ran 𝐺 ∧ 𝐺 Fn 𝐵) → ran 𝐺 ⊆ dom 𝐹)
12		dmcosseq 5871	. . . 4 ⊢ (ran 𝐺 ⊆ dom 𝐹 → dom (𝐹 ∘ 𝐺) = dom 𝐺)
13	11, 12	syl 17	. . 3 ⊢ (((𝐹 ↾ ran 𝐺) Fn ran 𝐺 ∧ 𝐺 Fn 𝐵) → dom (𝐹 ∘ 𝐺) = dom 𝐺)
14		fndm 6520	. . . 4 ⊢ (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
15	14	adantl 481	. . 3 ⊢ (((𝐹 ↾ ran 𝐺) Fn ran 𝐺 ∧ 𝐺 Fn 𝐵) → dom 𝐺 = 𝐵)
16	13, 15	eqtrd 2778	. 2 ⊢ (((𝐹 ↾ ran 𝐺) Fn ran 𝐺 ∧ 𝐺 Fn 𝐵) → dom (𝐹 ∘ 𝐺) = 𝐵)
17		df-fn 6421	. 2 ⊢ ((𝐹 ∘ 𝐺) Fn 𝐵 ↔ (Fun (𝐹 ∘ 𝐺) ∧ dom (𝐹 ∘ 𝐺) = 𝐵))
18	4, 16, 17	sylanbrc 582	1 ⊢ (((𝐹 ↾ ran 𝐺) Fn ran 𝐺 ∧ 𝐺 Fn 𝐵) → (𝐹 ∘ 𝐺) Fn 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∩ cin 3882   ⊆ wss 3883  dom cdm 5580  ran crn 5581   ↾ cres 5582   ∘ ccom 5584  Fun wfun 6412   Fn wfn 6413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-fun 6420  df-fn 6421

This theorem is referenced by:  funcoressn  44423