Theorem fmco 23456

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 1193	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → 𝐵 ∈ (fBas‘𝑍))
2		ssfg 23367	. . . . . . . . . . 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 3980	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → (𝑢 ∈ 𝐵 → 𝑢 ∈ (𝑍filGen𝐵)))
5		simpl2 1192	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → 𝑌 ∈ 𝑊)
6		simprr 771	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → 𝐺:𝑍⟶𝑌)
7		eqid 2732	. . . . . . . . . . . 12 ⊢ (𝑍filGen𝐵) = (𝑍filGen𝐵)
8	7	imaelfm 23446	. . . . . . . . . . 11 ⊢ (((𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍) ∧ 𝐺:𝑍⟶𝑌) ∧ 𝑢 ∈ (𝑍filGen𝐵)) → (𝐺 “ 𝑢) ∈ ((𝑌 FilMap 𝐺)‘𝐵))
9	8	ex 413	. . . . . . . . . 10 ⊢ ((𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍) ∧ 𝐺:𝑍⟶𝑌) → (𝑢 ∈ (𝑍filGen𝐵) → (𝐺 “ 𝑢) ∈ ((𝑌 FilMap 𝐺)‘𝐵)))
10	5, 1, 6, 9	syl3anc 1371	. . . . . . . . 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 47	. . . . . . . 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 407	. . . . . . 7 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) ∧ 𝑢 ∈ 𝐵) → (𝐺 “ 𝑢) ∈ ((𝑌 FilMap 𝐺)‘𝐵))
13		imaeq2 6053	. . . . . . . . . . 11 ⊢ (𝑡 = (𝐺 “ 𝑢) → (𝐹 “ 𝑡) = (𝐹 “ (𝐺 “ 𝑢)))
14		imaco 6247	. . . . . . . . . . 11 ⊢ ((𝐹 ∘ 𝐺) “ 𝑢) = (𝐹 “ (𝐺 “ 𝑢))
15	13, 14	eqtr4di 2790	. . . . . . . . . 10 ⊢ (𝑡 = (𝐺 “ 𝑢) → (𝐹 “ 𝑡) = ((𝐹 ∘ 𝐺) “ 𝑢))
16	15	sseq1d 4012	. . . . . . . . 9 ⊢ (𝑡 = (𝐺 “ 𝑢) → ((𝐹 “ 𝑡) ⊆ 𝑠 ↔ ((𝐹 ∘ 𝐺) “ 𝑢) ⊆ 𝑠))
17	16	rspcev 3612	. . . . . . . 8 ⊢ (((𝐺 “ 𝑢) ∈ ((𝑌 FilMap 𝐺)‘𝐵) ∧ ((𝐹 ∘ 𝐺) “ 𝑢) ⊆ 𝑠) → ∃𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵)(𝐹 “ 𝑡) ⊆ 𝑠)
18	17	ex 413	. . . . . . 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 3155	. . . . 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 23442	. . . . . . . 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 1371	. . . . . . 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		sstr2 3988	. . . . . . . . . . 11 ⊢ (((𝐹 ∘ 𝐺) “ 𝑢) ⊆ (𝐹 “ 𝑡) → ((𝐹 “ 𝑡) ⊆ 𝑠 → ((𝐹 ∘ 𝐺) “ 𝑢) ⊆ 𝑠))
24		imass2 6098	. . . . . . . . . . . 12 ⊢ ((𝐺 “ 𝑢) ⊆ 𝑡 → (𝐹 “ (𝐺 “ 𝑢)) ⊆ (𝐹 “ 𝑡))
25	14, 24	eqsstrid 4029	. . . . . . . . . . 11 ⊢ ((𝐺 “ 𝑢) ⊆ 𝑡 → ((𝐹 ∘ 𝐺) “ 𝑢) ⊆ (𝐹 “ 𝑡))
26	23, 25	syl11 33	. . . . . . . . . 10 ⊢ ((𝐹 “ 𝑡) ⊆ 𝑠 → ((𝐺 “ 𝑢) ⊆ 𝑡 → ((𝐹 ∘ 𝐺) “ 𝑢) ⊆ 𝑠))
27	26	reximdv 3170	. . . . . . . . 9 ⊢ ((𝐹 “ 𝑡) ⊆ 𝑠 → (∃𝑢 ∈ 𝐵 (𝐺 “ 𝑢) ⊆ 𝑡 → ∃𝑢 ∈ 𝐵 ((𝐹 ∘ 𝐺) “ 𝑢) ⊆ 𝑠))
28	27	com12 32	. . . . . . . 8 ⊢ (∃𝑢 ∈ 𝐵 (𝐺 “ 𝑢) ⊆ 𝑡 → ((𝐹 “ 𝑡) ⊆ 𝑠 → ∃𝑢 ∈ 𝐵 ((𝐹 ∘ 𝐺) “ 𝑢) ⊆ 𝑠))
29	28	adantl 482	. . . . . . 7 ⊢ ((𝑡 ⊆ 𝑌 ∧ ∃𝑢 ∈ 𝐵 (𝐺 “ 𝑢) ⊆ 𝑡) → ((𝐹 “ 𝑡) ⊆ 𝑠 → ∃𝑢 ∈ 𝐵 ((𝐹 ∘ 𝐺) “ 𝑢) ⊆ 𝑠))
30	22, 29	syl6bi 252	. . . . . 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)))
31	30	rexlimdv 3153	. . . . 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))
32	20, 31	impbid 211	. . . 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))
33	32	anbi2d 629	. . 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)))
34		simpl1 1191	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → 𝑋 ∈ 𝑉)
35		fco 6738	. . . . 5 ⊢ ((𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌) → (𝐹 ∘ 𝐺):𝑍⟶𝑋)
36	35	adantl 482	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → (𝐹 ∘ 𝐺):𝑍⟶𝑋)
37		elfm 23442	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐵 ∈ (fBas‘𝑍) ∧ (𝐹 ∘ 𝐺):𝑍⟶𝑋) → (𝑠 ∈ ((𝑋 FilMap (𝐹 ∘ 𝐺))‘𝐵) ↔ (𝑠 ⊆ 𝑋 ∧ ∃𝑢 ∈ 𝐵 ((𝐹 ∘ 𝐺) “ 𝑢) ⊆ 𝑠)))
38	34, 1, 36, 37	syl3anc 1371	. . 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)))
39		fmfil 23439	. . . . . 6 ⊢ ((𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍) ∧ 𝐺:𝑍⟶𝑌) → ((𝑌 FilMap 𝐺)‘𝐵) ∈ (Fil‘𝑌))
40	5, 1, 6, 39	syl3anc 1371	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → ((𝑌 FilMap 𝐺)‘𝐵) ∈ (Fil‘𝑌))
41		filfbas 23343	. . . . 5 ⊢ (((𝑌 FilMap 𝐺)‘𝐵) ∈ (Fil‘𝑌) → ((𝑌 FilMap 𝐺)‘𝐵) ∈ (fBas‘𝑌))
42	40, 41	syl 17	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → ((𝑌 FilMap 𝐺)‘𝐵) ∈ (fBas‘𝑌))
43		simprl 769	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → 𝐹:𝑌⟶𝑋)
44		elfm 23442	. . . 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)))
45	34, 42, 43, 44	syl3anc 1371	. . 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)))
46	33, 38, 45	3bitr4d 310	. 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))))
47	46	eqrdv 2730	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 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∃wrex 3070   ⊆ wss 3947   “ cima 5678   ∘ ccom 5679  ⟶wf 6536  ‘cfv 6540  (class class class)co 7405  fBascfbas 20924  filGencfg 20925  Filcfil 23340   FilMap cfm 23428

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-fbas 20933  df-fg 20934  df-fil 23341  df-fm 23433

This theorem is referenced by:  ufldom  23457  flfcnp  23499