Theorem fmco 22566

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 1190	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → 𝐵 ∈ (fBas‘𝑍))
2		ssfg 22477	. . . . . . . . . . 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 3914	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → (𝑢 ∈ 𝐵 → 𝑢 ∈ (𝑍filGen𝐵)))
5		simpl2 1189	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → 𝑌 ∈ 𝑊)
6		simprr 772	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → 𝐺:𝑍⟶𝑌)
7		eqid 2798	. . . . . . . . . . . 12 ⊢ (𝑍filGen𝐵) = (𝑍filGen𝐵)
8	7	imaelfm 22556	. . . . . . . . . . 11 ⊢ (((𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍) ∧ 𝐺:𝑍⟶𝑌) ∧ 𝑢 ∈ (𝑍filGen𝐵)) → (𝐺 “ 𝑢) ∈ ((𝑌 FilMap 𝐺)‘𝐵))
9	8	ex 416	. . . . . . . . . 10 ⊢ ((𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍) ∧ 𝐺:𝑍⟶𝑌) → (𝑢 ∈ (𝑍filGen𝐵) → (𝐺 “ 𝑢) ∈ ((𝑌 FilMap 𝐺)‘𝐵)))
10	5, 1, 6, 9	syl3anc 1368	. . . . . . . . 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 410	. . . . . . 7 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) ∧ 𝑢 ∈ 𝐵) → (𝐺 “ 𝑢) ∈ ((𝑌 FilMap 𝐺)‘𝐵))
13		imaeq2 5892	. . . . . . . . . . 11 ⊢ (𝑡 = (𝐺 “ 𝑢) → (𝐹 “ 𝑡) = (𝐹 “ (𝐺 “ 𝑢)))
14		imaco 6071	. . . . . . . . . . 11 ⊢ ((𝐹 ∘ 𝐺) “ 𝑢) = (𝐹 “ (𝐺 “ 𝑢))
15	13, 14	eqtr4di 2851	. . . . . . . . . 10 ⊢ (𝑡 = (𝐺 “ 𝑢) → (𝐹 “ 𝑡) = ((𝐹 ∘ 𝐺) “ 𝑢))
16	15	sseq1d 3946	. . . . . . . . 9 ⊢ (𝑡 = (𝐺 “ 𝑢) → ((𝐹 “ 𝑡) ⊆ 𝑠 ↔ ((𝐹 ∘ 𝐺) “ 𝑢) ⊆ 𝑠))
17	16	rspcev 3571	. . . . . . . 8 ⊢ (((𝐺 “ 𝑢) ∈ ((𝑌 FilMap 𝐺)‘𝐵) ∧ ((𝐹 ∘ 𝐺) “ 𝑢) ⊆ 𝑠) → ∃𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵)(𝐹 “ 𝑡) ⊆ 𝑠)
18	17	ex 416	. . . . . . 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 3243	. . . . 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 22552	. . . . . . . 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 1368	. . . . . . 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 3922	. . . . . . . . . . 11 ⊢ (((𝐹 ∘ 𝐺) “ 𝑢) ⊆ (𝐹 “ 𝑡) → ((𝐹 “ 𝑡) ⊆ 𝑠 → ((𝐹 ∘ 𝐺) “ 𝑢) ⊆ 𝑠))
24		imass2 5932	. . . . . . . . . . . 12 ⊢ ((𝐺 “ 𝑢) ⊆ 𝑡 → (𝐹 “ (𝐺 “ 𝑢)) ⊆ (𝐹 “ 𝑡))
25	14, 24	eqsstrid 3963	. . . . . . . . . . 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 3232	. . . . . . . . 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 485	. . . . . . 7 ⊢ ((𝑡 ⊆ 𝑌 ∧ ∃𝑢 ∈ 𝐵 (𝐺 “ 𝑢) ⊆ 𝑡) → ((𝐹 “ 𝑡) ⊆ 𝑠 → ∃𝑢 ∈ 𝐵 ((𝐹 ∘ 𝐺) “ 𝑢) ⊆ 𝑠))
30	22, 29	syl6bi 256	. . . . . 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 3242	. . . . 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 215	. . . 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 631	. . 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 1188	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → 𝑋 ∈ 𝑉)
35		fco 6505	. . . . 5 ⊢ ((𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌) → (𝐹 ∘ 𝐺):𝑍⟶𝑋)
36	35	adantl 485	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → (𝐹 ∘ 𝐺):𝑍⟶𝑋)
37		elfm 22552	. . . 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 1368	. . 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 22549	. . . . . 6 ⊢ ((𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍) ∧ 𝐺:𝑍⟶𝑌) → ((𝑌 FilMap 𝐺)‘𝐵) ∈ (Fil‘𝑌))
40	5, 1, 6, 39	syl3anc 1368	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → ((𝑌 FilMap 𝐺)‘𝐵) ∈ (Fil‘𝑌))
41		filfbas 22453	. . . . 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 770	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → 𝐹:𝑌⟶𝑋)
44		elfm 22552	. . . 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 1368	. . 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 314	. 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 2796	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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∃wrex 3107   ⊆ wss 3881   “ cima 5522   ∘ ccom 5523  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  fBascfbas 20079  filGencfg 20080  Filcfil 22450   FilMap cfm 22538

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-fbas 20088  df-fg 20089  df-fil 22451  df-fm 22543

This theorem is referenced by:  ufldom  22567  flfcnp  22609