Theorem fnresfnco 48111

Description: Composition of two functions, similar to fnco 6657. (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 6639	. . 3 ⊢ ((𝐹 ↾ ran 𝐺) Fn ran 𝐺 → Fun (𝐹 ↾ ran 𝐺))
2		fnfun 6639	. . 3 ⊢ (𝐺 Fn 𝐵 → Fun 𝐺)
3		funresfunco 6581	. . 3 ⊢ ((Fun (𝐹 ↾ ran 𝐺) ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺))
4	1, 2, 3	syl2an 608	. 2 ⊢ (((𝐹 ↾ ran 𝐺) Fn ran 𝐺 ∧ 𝐺 Fn 𝐵) → Fun (𝐹 ∘ 𝐺))
5		fndm 6642	. . . . . 6 ⊢ ((𝐹 ↾ ran 𝐺) Fn ran 𝐺 → dom (𝐹 ↾ ran 𝐺) = ran 𝐺)
6		dmres 6002	. . . . . . . 8 ⊢ dom (𝐹 ↾ ran 𝐺) = (ran 𝐺 ∩ dom 𝐹)
7	6	eqeq1i 2765	. . . . . . 7 ⊢ (dom (𝐹 ↾ ran 𝐺) = ran 𝐺 ↔ (ran 𝐺 ∩ dom 𝐹) = ran 𝐺)
8		dfss2 4094	. . . . . . 7 ⊢ (ran 𝐺 ⊆ dom 𝐹 ↔ (ran 𝐺 ∩ dom 𝐹) = ran 𝐺)
9	7, 8	sylbb2 241	. . . . . 6 ⊢ (dom (𝐹 ↾ ran 𝐺) = ran 𝐺 → ran 𝐺 ⊆ dom 𝐹)
10	5, 9	syl 18	. . . . 5 ⊢ ((𝐹 ↾ ran 𝐺) Fn ran 𝐺 → ran 𝐺 ⊆ dom 𝐹)
11	10	adantr 486	. . . 4 ⊢ (((𝐹 ↾ ran 𝐺) Fn ran 𝐺 ∧ 𝐺 Fn 𝐵) → ran 𝐺 ⊆ dom 𝐹)
12		dmcosseq 5959	. . . 4 ⊢ (ran 𝐺 ⊆ dom 𝐹 → dom (𝐹 ∘ 𝐺) = dom 𝐺)
13	11, 12	syl 18	. . 3 ⊢ (((𝐹 ↾ ran 𝐺) Fn ran 𝐺 ∧ 𝐺 Fn 𝐵) → dom (𝐹 ∘ 𝐺) = dom 𝐺)
14		fndm 6642	. . . 4 ⊢ (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
15	14	adantl 487	. . 3 ⊢ (((𝐹 ↾ ran 𝐺) Fn ran 𝐺 ∧ 𝐺 Fn 𝐵) → dom 𝐺 = 𝐵)
16	13, 15	eqtrd 2795	. 2 ⊢ (((𝐹 ↾ ran 𝐺) Fn ran 𝐺 ∧ 𝐺 Fn 𝐵) → dom (𝐹 ∘ 𝐺) = 𝐵)
17		df-fn 6541	. 2 ⊢ ((𝐹 ∘ 𝐺) Fn 𝐵 ↔ (Fun (𝐹 ∘ 𝐺) ∧ dom (𝐹 ∘ 𝐺) = 𝐵))
18	4, 16, 17	sylanbrc 595	1 ⊢ (((𝐹 ↾ ran 𝐺) Fn ran 𝐺 ∧ 𝐺 Fn 𝐵) → (𝐹 ∘ 𝐺) Fn 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ⊆ wss 3895   ∩ cin 4089  dom cdm 5650  ran crn 5651   ↾ cres 5652   ∘ ccom 5654  Fun wfun 6532   Fn wfn 6533

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-fun 6540  df-fn 6541

This theorem is used by:  funcoressn  48112