Theorem fnresfnco 46956

Description: Composition of two functions, similar to fnco 6697. (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 6679	. . 3 ⊢ ((𝐹 ↾ ran 𝐺) Fn ran 𝐺 → Fun (𝐹 ↾ ran 𝐺))
2		fnfun 6679	. . 3 ⊢ (𝐺 Fn 𝐵 → Fun 𝐺)
3		funresfunco 6619	. . 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 6682	. . . . . 6 ⊢ ((𝐹 ↾ ran 𝐺) Fn ran 𝐺 → dom (𝐹 ↾ ran 𝐺) = ran 𝐺)
6		dmres 6041	. . . . . . . 8 ⊢ dom (𝐹 ↾ ran 𝐺) = (ran 𝐺 ∩ dom 𝐹)
7	6	eqeq1i 2745	. . . . . . 7 ⊢ (dom (𝐹 ↾ ran 𝐺) = ran 𝐺 ↔ (ran 𝐺 ∩ dom 𝐹) = ran 𝐺)
8		dfss2 3994	. . . . . . 7 ⊢ (ran 𝐺 ⊆ dom 𝐹 ↔ (ran 𝐺 ∩ dom 𝐹) = ran 𝐺)
9	7, 8	sylbb2 238	. . . . . 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 5999	. . . 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 6682	. . . 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 2780	. 2 ⊢ (((𝐹 ↾ ran 𝐺) Fn ran 𝐺 ∧ 𝐺 Fn 𝐵) → dom (𝐹 ∘ 𝐺) = 𝐵)
17		df-fn 6576	. 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 1537   ∩ cin 3975   ⊆ wss 3976  dom cdm 5700  ran crn 5701   ↾ cres 5702   ∘ ccom 5704  Fun wfun 6567   Fn wfn 6568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-fun 6575  df-fn 6576

This theorem is referenced by:  funcoressn  46957