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Theorem fgabs 24015

Description: Absorption law for filter generation. (Contributed by Mario Carneiro, 15-Oct-2015.)

**Assertion**
Ref	Expression
fgabs	⊢ ((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) → (𝑋filGen(𝑌filGen𝐹)) = (𝑋filGen𝐹))

Proof of Theorem fgabs Dummy variables 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 778	. . . . . . . . 9 ⊢ (((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) → 𝐹 ∈ (fBas‘𝑌))
2		fgcl 24014	. . . . . . . . 9 ⊢ (𝐹 ∈ (fBas‘𝑌) → (𝑌filGen𝐹) ∈ (Fil‘𝑌))
3		filfbas 23984	. . . . . . . . 9 ⊢ ((𝑌filGen𝐹) ∈ (Fil‘𝑌) → (𝑌filGen𝐹) ∈ (fBas‘𝑌))
4	1, 2, 3	3syl 19	. . . . . . . 8 ⊢ (((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) → (𝑌filGen𝐹) ∈ (fBas‘𝑌))
5		fbsspw 23968	. . . . . . . . . 10 ⊢ ((𝑌filGen𝐹) ∈ (fBas‘𝑌) → (𝑌filGen𝐹) ⊆ 𝒫 𝑌)
6	4, 5	syl 18	. . . . . . . . 9 ⊢ (((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) → (𝑌filGen𝐹) ⊆ 𝒫 𝑌)
7		simplr 780	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) → 𝑌 ⊆ 𝑋)
8	7	sspwd 4574	. . . . . . . . 9 ⊢ (((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) → 𝒫 𝑌 ⊆ 𝒫 𝑋)
9	6, 8	sstrd 3946	. . . . . . . 8 ⊢ (((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) → (𝑌filGen𝐹) ⊆ 𝒫 𝑋)
10		simpr 489	. . . . . . . 8 ⊢ (((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) → 𝑋 ∈ V)
11		fbasweak 24001	. . . . . . . 8 ⊢ (((𝑌filGen𝐹) ∈ (fBas‘𝑌) ∧ (𝑌filGen𝐹) ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ V) → (𝑌filGen𝐹) ∈ (fBas‘𝑋))
12	4, 9, 10, 11	syl3anc 1396	. . . . . . 7 ⊢ (((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) → (𝑌filGen𝐹) ∈ (fBas‘𝑋))
13		elfg 24007	. . . . . . 7 ⊢ ((𝑌filGen𝐹) ∈ (fBas‘𝑋) → (𝑥 ∈ (𝑋filGen(𝑌filGen𝐹)) ↔ (𝑥 ⊆ 𝑋 ∧ ∃𝑦 ∈ (𝑌filGen𝐹)𝑦 ⊆ 𝑥)))
14	12, 13	syl 18	. . . . . 6 ⊢ (((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) → (𝑥 ∈ (𝑋filGen(𝑌filGen𝐹)) ↔ (𝑥 ⊆ 𝑋 ∧ ∃𝑦 ∈ (𝑌filGen𝐹)𝑦 ⊆ 𝑥)))
15	1	adantr 485	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) ∧ 𝑥 ⊆ 𝑋) → 𝐹 ∈ (fBas‘𝑌))
16		elfg 24007	. . . . . . . . . 10 ⊢ (𝐹 ∈ (fBas‘𝑌) → (𝑦 ∈ (𝑌filGen𝐹) ↔ (𝑦 ⊆ 𝑌 ∧ ∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑦)))
17	15, 16	syl 18	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) ∧ 𝑥 ⊆ 𝑋) → (𝑦 ∈ (𝑌filGen𝐹) ↔ (𝑦 ⊆ 𝑌 ∧ ∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑦)))
18		fbsspw 23968	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 ∈ (fBas‘𝑌) → 𝐹 ⊆ 𝒫 𝑌)
19	1, 18	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) → 𝐹 ⊆ 𝒫 𝑌)
20	19, 8	sstrd 3946	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) → 𝐹 ⊆ 𝒫 𝑋)
21		fbasweak 24001	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 ∈ (fBas‘𝑌) ∧ 𝐹 ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ V) → 𝐹 ∈ (fBas‘𝑋))
22	1, 20, 10, 21	syl3anc 1396	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) → 𝐹 ∈ (fBas‘𝑋))
23		fgcl 24014	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑋filGen𝐹) ∈ (Fil‘𝑋))
24	22, 23	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) → (𝑋filGen𝐹) ∈ (Fil‘𝑋))
25	24	ad2antrr 738	. . . . . . . . . . . . . 14 ⊢ (((((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) ∧ (𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑌)) ∧ ((𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ 𝑦) ∧ 𝑦 ⊆ 𝑥)) → (𝑋filGen𝐹) ∈ (Fil‘𝑋))
26		ssfg 24008	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹))
27	22, 26	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) → 𝐹 ⊆ (𝑋filGen𝐹))
28	27	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) ∧ (𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑌)) → 𝐹 ⊆ (𝑋filGen𝐹))
29	28	sselda 3936	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) ∧ (𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑌)) ∧ 𝑧 ∈ 𝐹) → 𝑧 ∈ (𝑋filGen𝐹))
30	29	adantrr 729	. . . . . . . . . . . . . . 15 ⊢ (((((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) ∧ (𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑌)) ∧ (𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ 𝑦)) → 𝑧 ∈ (𝑋filGen𝐹))
31	30	adantrr 729	. . . . . . . . . . . . . 14 ⊢ (((((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) ∧ (𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑌)) ∧ ((𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ 𝑦) ∧ 𝑦 ⊆ 𝑥)) → 𝑧 ∈ (𝑋filGen𝐹))
32		simplrl 788	. . . . . . . . . . . . . 14 ⊢ (((((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) ∧ (𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑌)) ∧ ((𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ 𝑦) ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ⊆ 𝑋)
33		simprlr 791	. . . . . . . . . . . . . . 15 ⊢ (((((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) ∧ (𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑌)) ∧ ((𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ 𝑦) ∧ 𝑦 ⊆ 𝑥)) → 𝑧 ⊆ 𝑦)
34		simprr 784	. . . . . . . . . . . . . . 15 ⊢ (((((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) ∧ (𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑌)) ∧ ((𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ 𝑦) ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ⊆ 𝑥)
35	33, 34	sstrd 3946	. . . . . . . . . . . . . 14 ⊢ (((((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) ∧ (𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑌)) ∧ ((𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ 𝑦) ∧ 𝑦 ⊆ 𝑥)) → 𝑧 ⊆ 𝑥)
36		filss 23989	. . . . . . . . . . . . . 14 ⊢ (((𝑋filGen𝐹) ∈ (Fil‘𝑋) ∧ (𝑧 ∈ (𝑋filGen𝐹) ∧ 𝑥 ⊆ 𝑋 ∧ 𝑧 ⊆ 𝑥)) → 𝑥 ∈ (𝑋filGen𝐹))
37	25, 31, 32, 35, 36	syl13anc 1397	. . . . . . . . . . . . 13 ⊢ (((((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) ∧ (𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑌)) ∧ ((𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ 𝑦) ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ∈ (𝑋filGen𝐹))
38	37	expr 461	. . . . . . . . . . . 12 ⊢ (((((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) ∧ (𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑌)) ∧ (𝑧 ∈ 𝐹 ∧ 𝑧 ⊆ 𝑦)) → (𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝑋filGen𝐹)))
39	38	rexlimdvaa 3165	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) ∧ (𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑌)) → (∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑦 → (𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝑋filGen𝐹))))
40	39	anassrs 472	. . . . . . . . . 10 ⊢ (((((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) ∧ 𝑥 ⊆ 𝑋) ∧ 𝑦 ⊆ 𝑌) → (∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑦 → (𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝑋filGen𝐹))))
41	40	expimpd 458	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) ∧ 𝑥 ⊆ 𝑋) → ((𝑦 ⊆ 𝑌 ∧ ∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑦) → (𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝑋filGen𝐹))))
42	17, 41	sylbid 243	. . . . . . . 8 ⊢ ((((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) ∧ 𝑥 ⊆ 𝑋) → (𝑦 ∈ (𝑌filGen𝐹) → (𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝑋filGen𝐹))))
43	42	rexlimdv 3162	. . . . . . 7 ⊢ ((((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) ∧ 𝑥 ⊆ 𝑋) → (∃𝑦 ∈ (𝑌filGen𝐹)𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝑋filGen𝐹)))
44	43	expimpd 458	. . . . . 6 ⊢ (((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) → ((𝑥 ⊆ 𝑋 ∧ ∃𝑦 ∈ (𝑌filGen𝐹)𝑦 ⊆ 𝑥) → 𝑥 ∈ (𝑋filGen𝐹)))
45	14, 44	sylbid 243	. . . . 5 ⊢ (((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) → (𝑥 ∈ (𝑋filGen(𝑌filGen𝐹)) → 𝑥 ∈ (𝑋filGen𝐹)))
46	45	ssrdv 3942	. . . 4 ⊢ (((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) → (𝑋filGen(𝑌filGen𝐹)) ⊆ (𝑋filGen𝐹))
47		ssfg 24008	. . . . . 6 ⊢ (𝐹 ∈ (fBas‘𝑌) → 𝐹 ⊆ (𝑌filGen𝐹))
48	47	ad2antrr 738	. . . . 5 ⊢ (((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) → 𝐹 ⊆ (𝑌filGen𝐹))
49		fgss 24009	. . . . 5 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ (𝑌filGen𝐹) ∈ (fBas‘𝑋) ∧ 𝐹 ⊆ (𝑌filGen𝐹)) → (𝑋filGen𝐹) ⊆ (𝑋filGen(𝑌filGen𝐹)))
50	22, 12, 48, 49	syl3anc 1396	. . . 4 ⊢ (((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) → (𝑋filGen𝐹) ⊆ (𝑋filGen(𝑌filGen𝐹)))
51	46, 50	eqssd 3953	. . 3 ⊢ (((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑋 ∈ V) → (𝑋filGen(𝑌filGen𝐹)) = (𝑋filGen𝐹))
52	51	ex 417	. 2 ⊢ ((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) → (𝑋 ∈ V → (𝑋filGen(𝑌filGen𝐹)) = (𝑋filGen𝐹)))
53		df-fg 21499	. . . . 5 ⊢ filGen = (𝑤 ∈ V, 𝑥 ∈ (fBas‘𝑤) ↦ {𝑦 ∈ 𝒫 𝑤 ∣ (𝑥 ∩ 𝒫 𝑦) ≠ ∅})
54	53	reldmmpo 7544	. . . 4 ⊢ Rel dom filGen
55	54	ovprc1 7449	. . 3 ⊢ (¬ 𝑋 ∈ V → (𝑋filGen(𝑌filGen𝐹)) = ∅)
56	54	ovprc1 7449	. . 3 ⊢ (¬ 𝑋 ∈ V → (𝑋filGen𝐹) = ∅)
57	55, 56	eqtr4d 2799	. 2 ⊢ (¬ 𝑋 ∈ V → (𝑋filGen(𝑌filGen𝐹)) = (𝑋filGen𝐹))
58	52, 57	pm2.61d1 182	1 ⊢ ((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) → (𝑋filGen(𝑌filGen𝐹)) = (𝑋filGen𝐹))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1568 ∈ wcel 2141 ≠ wne 2956 ∃wrex 3087 {crab 3414 Vcvv 3453 ∩ cin 3903 ⊆ wss 3904 ∅c0 4285 𝒫 cpw 4561 ‘cfv 6536 (class class class)co 7410 fBascfbas 21489 filGencfg 21490 Filcfil 23981

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5256 ax-nul 5268 ax-pow 5336 ax-pr 5404

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rab 3415 df-v 3455 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-br 5109 df-opab 5173 df-mpt 5192 df-id 5556 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-iota 6492 df-fun 6538 df-fv 6544 df-ov 7413 df-oprab 7414 df-mpo 7415 df-fbas 21498 df-fg 21499 df-fil 23982

This theorem is referenced by: minveclem4a 25568

W3C validator