Theorem fmco 24169

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 1212	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → 𝐵 ∈ (fBas‘𝑍))
2		ssfg 24080	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (fBas‘𝑍) → 𝐵 ⊆ (𝑍filGen𝐵))
3	1, 2	syl 18	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → 𝐵 ⊆ (𝑍filGen𝐵))
4	3	sseld 3937	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → (𝑢 ∈ 𝐵 → 𝑢 ∈ (𝑍filGen𝐵)))
5		simpl2 1211	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → 𝑌 ∈ 𝑊)
6		simprr 785	. . . . . . . . . 10 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → 𝐺:𝑍⟶𝑌)
7		eqid 2765	. . . . . . . . . . . 12 ⊢ (𝑍filGen𝐵) = (𝑍filGen𝐵)
8	7	imaelfm 24159	. . . . . . . . . . 11 ⊢ (((𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍) ∧ 𝐺:𝑍⟶𝑌) ∧ 𝑢 ∈ (𝑍filGen𝐵)) → (𝐺 “ 𝑢) ∈ ((𝑌 FilMap 𝐺)‘𝐵))
9	8	ex 418	. . . . . . . . . 10 ⊢ ((𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍) ∧ 𝐺:𝑍⟶𝑌) → (𝑢 ∈ (𝑍filGen𝐵) → (𝐺 “ 𝑢) ∈ ((𝑌 FilMap 𝐺)‘𝐵)))
10	5, 1, 6, 9	syl3anc 1398	. . . . . . . . 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 48	. . . . . . . 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 412	. . . . . . 7 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) ∧ 𝑢 ∈ 𝐵) → (𝐺 “ 𝑢) ∈ ((𝑌 FilMap 𝐺)‘𝐵))
13		imaeq2 6060	. . . . . . . . . . 11 ⊢ (𝑡 = (𝐺 “ 𝑢) → (𝐹 “ 𝑡) = (𝐹 “ (𝐺 “ 𝑢)))
14		imaco 6254	. . . . . . . . . . 11 ⊢ ((𝐹 ∘ 𝐺) “ 𝑢) = (𝐹 “ (𝐺 “ 𝑢))
15	13, 14	eqtr4di 2818	. . . . . . . . . 10 ⊢ (𝑡 = (𝐺 “ 𝑢) → (𝐹 “ 𝑡) = ((𝐹 ∘ 𝐺) “ 𝑢))
16	15	sseq1d 3969	. . . . . . . . 9 ⊢ (𝑡 = (𝐺 “ 𝑢) → ((𝐹 “ 𝑡) ⊆ 𝑠 ↔ ((𝐹 ∘ 𝐺) “ 𝑢) ⊆ 𝑠))
17	16	rspcev 3583	. . . . . . . 8 ⊢ (((𝐺 “ 𝑢) ∈ ((𝑌 FilMap 𝐺)‘𝐵) ∧ ((𝐹 ∘ 𝐺) “ 𝑢) ⊆ 𝑠) → ∃𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵)(𝐹 “ 𝑡) ⊆ 𝑠)
18	17	ex 418	. . . . . . 7 ⊢ ((𝐺 “ 𝑢) ∈ ((𝑌 FilMap 𝐺)‘𝐵) → (((𝐹 ∘ 𝐺) “ 𝑢) ⊆ 𝑠 → ∃𝑡 ∈ ((𝑌 FilMap 𝐺)‘𝐵)(𝐹 “ 𝑡) ⊆ 𝑠))
19	12, 18	syl 18	. . . . . 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 3168	. . . . 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 24155	. . . . . . . 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 1398	. . . . . . 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 3945	. . . . . . . . . . 11 ⊢ (((𝐹 ∘ 𝐺) “ 𝑢) ⊆ (𝐹 “ 𝑡) → ((𝐹 “ 𝑡) ⊆ 𝑠 → ((𝐹 ∘ 𝐺) “ 𝑢) ⊆ 𝑠))
24		imass2 6106	. . . . . . . . . . . 12 ⊢ ((𝐺 “ 𝑢) ⊆ 𝑡 → (𝐹 “ (𝐺 “ 𝑢)) ⊆ (𝐹 “ 𝑡))
25	14, 24	eqsstrid 3976	. . . . . . . . . . 11 ⊢ ((𝐺 “ 𝑢) ⊆ 𝑡 → ((𝐹 ∘ 𝐺) “ 𝑢) ⊆ (𝐹 “ 𝑡))
26	23, 25	syl11 34	. . . . . . . . . 10 ⊢ ((𝐹 “ 𝑡) ⊆ 𝑠 → ((𝐺 “ 𝑢) ⊆ 𝑡 → ((𝐹 ∘ 𝐺) “ 𝑢) ⊆ 𝑠))
27	26	reximdv 3182	. . . . . . . . 9 ⊢ ((𝐹 “ 𝑡) ⊆ 𝑠 → (∃𝑢 ∈ 𝐵 (𝐺 “ 𝑢) ⊆ 𝑡 → ∃𝑢 ∈ 𝐵 ((𝐹 ∘ 𝐺) “ 𝑢) ⊆ 𝑠))
28	27	com12 33	. . . . . . . 8 ⊢ (∃𝑢 ∈ 𝐵 (𝐺 “ 𝑢) ⊆ 𝑡 → ((𝐹 “ 𝑡) ⊆ 𝑠 → ∃𝑢 ∈ 𝐵 ((𝐹 ∘ 𝐺) “ 𝑢) ⊆ 𝑠))
29	28	adantl 487	. . . . . . 7 ⊢ ((𝑡 ⊆ 𝑌 ∧ ∃𝑢 ∈ 𝐵 (𝐺 “ 𝑢) ⊆ 𝑡) → ((𝐹 “ 𝑡) ⊆ 𝑠 → ∃𝑢 ∈ 𝐵 ((𝐹 ∘ 𝐺) “ 𝑢) ⊆ 𝑠))
30	22, 29	biimtrdi 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 3166	. . . . 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 642	. . 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 1210	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → 𝑋 ∈ 𝑉)
35		fco 6734	. . . . 5 ⊢ ((𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌) → (𝐹 ∘ 𝐺):𝑍⟶𝑋)
36	35	adantl 487	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → (𝐹 ∘ 𝐺):𝑍⟶𝑋)
37		elfm 24155	. . . 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 1398	. . 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 24152	. . . . . 6 ⊢ ((𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍) ∧ 𝐺:𝑍⟶𝑌) → ((𝑌 FilMap 𝐺)‘𝐵) ∈ (Fil‘𝑌))
40	5, 1, 6, 39	syl3anc 1398	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → ((𝑌 FilMap 𝐺)‘𝐵) ∈ (Fil‘𝑌))
41		filfbas 24056	. . . . 5 ⊢ (((𝑌 FilMap 𝐺)‘𝐵) ∈ (Fil‘𝑌) → ((𝑌 FilMap 𝐺)‘𝐵) ∈ (fBas‘𝑌))
42	40, 41	syl 18	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → ((𝑌 FilMap 𝐺)‘𝐵) ∈ (fBas‘𝑌))
43		simprl 783	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → 𝐹:𝑌⟶𝑋)
44		elfm 24155	. . . 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 1398	. . 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 2763	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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∃wrex 3091   ⊆ wss 3906   “ cima 5666   ∘ ccom 5667  ⟶wf 6536  ‘cfv 6540  (class class class)co 7419  fBascfbas 21560  filGencfg 21561  Filcfil 24053   FilMap cfm 24141

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-fbas 21569  df-fg 21570  df-fil 24054  df-fm 24146

This theorem is used by:  ufldom  24170  flfcnp  24212