Theorem fmco 21575

Description: Composition of image filters. (Contributed by Mario Carneiro, 27-Aug-2015.)

Assertion

Ref	Expression
fmco	⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → ((𝑋 FilMap (𝐹 ∘ 𝐺))‘𝐵) = ((𝑋 FilMap 𝐹)‘((𝑌 FilMap 𝐺)‘𝐵)))

Proof of Theorem fmco

Dummy variables 𝑡 𝑠 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl3 1059	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → 𝐵 ∈ (fBas‘𝑍))
2		ssfg 21486	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (fBas‘𝑍) → 𝐵 ⊆ (𝑍filGen𝐵))
3	1, 2	syl 17	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → 𝐵 ⊆ (𝑍filGen𝐵))
4	3	sseld 3567	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → (𝑢 ∈ 𝐵 → 𝑢 ∈ (𝑍filGen𝐵)))
5		simpl2 1058	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → 𝑌 ∈ 𝑊)
6		simprr 792	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → 𝐺:𝑍⟶𝑌)
7		eqid 2610	. . . . . . . . . . . 12 ⊢ (𝑍filGen𝐵) = (𝑍filGen𝐵)
8	7	imaelfm 21565	. . . . . . . . . . 11 ⊢ (((𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍) ∧ 𝐺:𝑍⟶𝑌) ∧ 𝑢 ∈ (𝑍filGen𝐵)) → (𝐺 “ 𝑢) ∈ ((𝑌 FilMap 𝐺)‘𝐵))
9	8	ex 449	. . . . . . . . . 10 ⊢ ((𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍) ∧ 𝐺:𝑍⟶𝑌) → (𝑢 ∈ (𝑍filGen𝐵) → (𝐺 “ 𝑢) ∈ ((𝑌 FilMap 𝐺)‘𝐵)))
10	5, 1, 6, 9	syl3anc 1318	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → (𝑢 ∈ (𝑍filGen𝐵) → (𝐺 “ 𝑢) ∈ ((𝑌 FilMap 𝐺)‘𝐵)))
11	4, 10	syld 46	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → (𝑢 ∈ 𝐵 → (𝐺 “ 𝑢) ∈ ((𝑌 FilMap 𝐺)‘𝐵)))
12	11	imp 444	. . . . . . 7 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) ∧ 𝑢 ∈ 𝐵) → (𝐺 “ 𝑢) ∈ ((𝑌 FilMap 𝐺)‘𝐵))
13		imaeq2 5381	. . . . . . . . . . 11 ⊢ (𝑡 = (𝐺 “ 𝑢) → (𝐹 “ 𝑡) = (𝐹 “ (𝐺 “ 𝑢)))
14		imaco 5557	. . . . . . . . . . 11 ⊢ ((𝐹 ∘ 𝐺) “ 𝑢) = (𝐹 “ (𝐺 “ 𝑢))
15	13, 14	syl6eqr 2662	. . . . . . . . . 10 ⊢ (𝑡 = (𝐺 “ 𝑢) → (𝐹 “ 𝑡) = ((𝐹 ∘ 𝐺) “ 𝑢))
16	15	sseq1d 3595	. . . . . . . . 9 ⊢ (𝑡 = (𝐺 “ 𝑢) → ((𝐹 “ 𝑡) ⊆ 𝑠 ↔ ((𝐹 ∘ 𝐺) “ 𝑢) ⊆ 𝑠))
17	16	rspcev 3282	. . . . . . . 8 ⊢ (((𝐺 “ 𝑢) ∈ ((𝑌 FilMap 𝐺)‘𝐵) ∧ ((𝐹 ∘ 𝐺) “ 𝑢) ⊆ 𝑠) → ∃𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵)(𝐹 “ 𝑡) ⊆ 𝑠)
18	17	ex 449	. . . . . . 7 ⊢ ((𝐺 “ 𝑢) ∈ ((𝑌 FilMap 𝐺)‘𝐵) → (((𝐹 ∘ 𝐺) “ 𝑢) ⊆ 𝑠 → ∃𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵)(𝐹 “ 𝑡) ⊆ 𝑠))
19	12, 18	syl 17	. . . . . 6 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) ∧ 𝑢 ∈ 𝐵) → (((𝐹 ∘ 𝐺) “ 𝑢) ⊆ 𝑠 → ∃𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵)(𝐹 “ 𝑡) ⊆ 𝑠))
20	19	rexlimdva 3013	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → (∃𝑢 ∈ 𝐵 ((𝐹 ∘ 𝐺) “ 𝑢) ⊆ 𝑠 → ∃𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵)(𝐹 “ 𝑡) ⊆ 𝑠))
21		elfm 21561	. . . . . . . 8 ⊢ ((𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍) ∧ 𝐺:𝑍⟶𝑌) → (𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵) ↔ (𝑡 ⊆ 𝑌 ∧ ∃𝑢 ∈ 𝐵 (𝐺 “ 𝑢) ⊆ 𝑡)))
22	5, 1, 6, 21	syl3anc 1318	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → (𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵) ↔ (𝑡 ⊆ 𝑌 ∧ ∃𝑢 ∈ 𝐵 (𝐺 “ 𝑢) ⊆ 𝑡)))
23		imass2 5420	. . . . . . . . . . . . 13 ⊢ ((𝐺 “ 𝑢) ⊆ 𝑡 → (𝐹 “ (𝐺 “ 𝑢)) ⊆ (𝐹 “ 𝑡))
24	14, 23	syl5eqss 3612	. . . . . . . . . . . 12 ⊢ ((𝐺 “ 𝑢) ⊆ 𝑡 → ((𝐹 ∘ 𝐺) “ 𝑢) ⊆ (𝐹 “ 𝑡))
25		sstr2 3575	. . . . . . . . . . . 12 ⊢ (((𝐹 ∘ 𝐺) “ 𝑢) ⊆ (𝐹 “ 𝑡) → ((𝐹 “ 𝑡) ⊆ 𝑠 → ((𝐹 ∘ 𝐺) “ 𝑢) ⊆ 𝑠))
26	24, 25	syl 17	. . . . . . . . . . 11 ⊢ ((𝐺 “ 𝑢) ⊆ 𝑡 → ((𝐹 “ 𝑡) ⊆ 𝑠 → ((𝐹 ∘ 𝐺) “ 𝑢) ⊆ 𝑠))
27	26	com12 32	. . . . . . . . . 10 ⊢ ((𝐹 “ 𝑡) ⊆ 𝑠 → ((𝐺 “ 𝑢) ⊆ 𝑡 → ((𝐹 ∘ 𝐺) “ 𝑢) ⊆ 𝑠))
28	27	reximdv 2999	. . . . . . . . 9 ⊢ ((𝐹 “ 𝑡) ⊆ 𝑠 → (∃𝑢 ∈ 𝐵 (𝐺 “ 𝑢) ⊆ 𝑡 → ∃𝑢 ∈ 𝐵 ((𝐹 ∘ 𝐺) “ 𝑢) ⊆ 𝑠))
29	28	com12 32	. . . . . . . 8 ⊢ (∃𝑢 ∈ 𝐵 (𝐺 “ 𝑢) ⊆ 𝑡 → ((𝐹 “ 𝑡) ⊆ 𝑠 → ∃𝑢 ∈ 𝐵 ((𝐹 ∘ 𝐺) “ 𝑢) ⊆ 𝑠))
30	29	adantl 481	. . . . . . 7 ⊢ ((𝑡 ⊆ 𝑌 ∧ ∃𝑢 ∈ 𝐵 (𝐺 “ 𝑢) ⊆ 𝑡) → ((𝐹 “ 𝑡) ⊆ 𝑠 → ∃𝑢 ∈ 𝐵 ((𝐹 ∘ 𝐺) “ 𝑢) ⊆ 𝑠))
31	22, 30	syl6bi 242	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → (𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵) → ((𝐹 “ 𝑡) ⊆ 𝑠 → ∃𝑢 ∈ 𝐵 ((𝐹 ∘ 𝐺) “ 𝑢) ⊆ 𝑠)))
32	31	rexlimdv 3012	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → (∃𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵)(𝐹 “ 𝑡) ⊆ 𝑠 → ∃𝑢 ∈ 𝐵 ((𝐹 ∘ 𝐺) “ 𝑢) ⊆ 𝑠))
33	20, 32	impbid 201	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → (∃𝑢 ∈ 𝐵 ((𝐹 ∘ 𝐺) “ 𝑢) ⊆ 𝑠 ↔ ∃𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵)(𝐹 “ 𝑡) ⊆ 𝑠))
34	33	anbi2d 736	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → ((𝑠 ⊆ 𝑋 ∧ ∃𝑢 ∈ 𝐵 ((𝐹 ∘ 𝐺) “ 𝑢) ⊆ 𝑠) ↔ (𝑠 ⊆ 𝑋 ∧ ∃𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵)(𝐹 “ 𝑡) ⊆ 𝑠)))
35		simpl1 1057	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → 𝑋 ∈ 𝑉)
36		fco 5971	. . . . 5 ⊢ ((𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌) → (𝐹 ∘ 𝐺):𝑍⟶𝑋)
37	36	adantl 481	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → (𝐹 ∘ 𝐺):𝑍⟶𝑋)
38		elfm 21561	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐵 ∈ (fBas‘𝑍) ∧ (𝐹 ∘ 𝐺):𝑍⟶𝑋) → (𝑠 ∈ ((𝑋 FilMap (𝐹 ∘ 𝐺))‘𝐵) ↔ (𝑠 ⊆ 𝑋 ∧ ∃𝑢 ∈ 𝐵 ((𝐹 ∘ 𝐺) “ 𝑢) ⊆ 𝑠)))
39	35, 1, 37, 38	syl3anc 1318	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → (𝑠 ∈ ((𝑋 FilMap (𝐹 ∘ 𝐺))‘𝐵) ↔ (𝑠 ⊆ 𝑋 ∧ ∃𝑢 ∈ 𝐵 ((𝐹 ∘ 𝐺) “ 𝑢) ⊆ 𝑠)))
40		fmfil 21558	. . . . . 6 ⊢ ((𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍) ∧ 𝐺:𝑍⟶𝑌) → ((𝑌 FilMap 𝐺)‘𝐵) ∈ (Fil‘𝑌))
41	5, 1, 6, 40	syl3anc 1318	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → ((𝑌 FilMap 𝐺)‘𝐵) ∈ (Fil‘𝑌))
42		filfbas 21462	. . . . 5 ⊢ (((𝑌 FilMap 𝐺)‘𝐵) ∈ (Fil‘𝑌) → ((𝑌 FilMap 𝐺)‘𝐵) ∈ (fBas‘𝑌))
43	41, 42	syl 17	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → ((𝑌 FilMap 𝐺)‘𝐵) ∈ (fBas‘𝑌))
44		simprl 790	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → 𝐹:𝑌⟶𝑋)
45		elfm 21561	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ ((𝑌 FilMap 𝐺)‘𝐵) ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑠 ∈ ((𝑋 FilMap 𝐹)‘((𝑌 FilMap 𝐺)‘𝐵)) ↔ (𝑠 ⊆ 𝑋 ∧ ∃𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵)(𝐹 “ 𝑡) ⊆ 𝑠)))
46	35, 43, 44, 45	syl3anc 1318	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → (𝑠 ∈ ((𝑋 FilMap 𝐹)‘((𝑌 FilMap 𝐺)‘𝐵)) ↔ (𝑠 ⊆ 𝑋 ∧ ∃𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵)(𝐹 “ 𝑡) ⊆ 𝑠)))
47	34, 39, 46	3bitr4d 299	. 2 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → (𝑠 ∈ ((𝑋 FilMap (𝐹 ∘ 𝐺))‘𝐵) ↔ 𝑠 ∈ ((𝑋 FilMap 𝐹)‘((𝑌 FilMap 𝐺)‘𝐵))))
48	47	eqrdv 2608	1 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → ((𝑋 FilMap (𝐹 ∘ 𝐺))‘𝐵) = ((𝑋 FilMap 𝐹)‘((𝑌 FilMap 𝐺)‘𝐵)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   ⊆ wss 3540   “ cima 5041   ∘ ccom 5042  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  fBascfbas 19555  filGencfg 19556  Filcfil 21459   FilMap cfm 21547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-fbas 19564  df-fg 19565  df-fil 21460  df-fm 21552

This theorem is referenced by:  ufldom  21576  flfcnp  21618