Theorem fgss2 24000

Description: A condition for a filter to be finer than another involving their filter bases. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Assertion

Ref	Expression
fgss2	⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ((𝑋filGen𝐹) ⊆ (𝑋filGen𝐺) ↔ ∀𝑥 ∈ 𝐹 ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥))

Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝐺,𝑦   𝑥,𝑋,𝑦

Proof of Theorem fgss2

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

Step	Hyp	Ref	Expression
1		ssfg 23998	. . . . . 6 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹))
2	1	adantr 485	. . . . 5 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → 𝐹 ⊆ (𝑋filGen𝐹))
3	2	sseld 3944	. . . 4 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (𝑥 ∈ 𝐹 → 𝑥 ∈ (𝑋filGen𝐹)))
4		ssel2 3940	. . . . . 6 ⊢ (((𝑋filGen𝐹) ⊆ (𝑋filGen𝐺) ∧ 𝑥 ∈ (𝑋filGen𝐹)) → 𝑥 ∈ (𝑋filGen𝐺))
5		elfg 23997	. . . . . . . 8 ⊢ (𝐺 ∈ (fBas‘𝑋) → (𝑥 ∈ (𝑋filGen𝐺) ↔ (𝑥 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥)))
6		simpr 489	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥) → ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥)
7	5, 6	biimtrdi 256	. . . . . . 7 ⊢ (𝐺 ∈ (fBas‘𝑋) → (𝑥 ∈ (𝑋filGen𝐺) → ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥))
8	7	adantl 486	. . . . . 6 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (𝑥 ∈ (𝑋filGen𝐺) → ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥))
9	4, 8	syl5 35	. . . . 5 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (((𝑋filGen𝐹) ⊆ (𝑋filGen𝐺) ∧ 𝑥 ∈ (𝑋filGen𝐹)) → ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥))
10	9	expd 420	. . . 4 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ((𝑋filGen𝐹) ⊆ (𝑋filGen𝐺) → (𝑥 ∈ (𝑋filGen𝐹) → ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥)))
11	3, 10	syl5d 74	. . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ((𝑋filGen𝐹) ⊆ (𝑋filGen𝐺) → (𝑥 ∈ 𝐹 → ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥)))
12	11	ralrimdv 3169	. 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ((𝑋filGen𝐹) ⊆ (𝑋filGen𝐺) → ∀𝑥 ∈ 𝐹 ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥))
13		sseq2 3971	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑢 → (𝑦 ⊆ 𝑥 ↔ 𝑦 ⊆ 𝑢))
14	13	rexbidv 3195	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑢 → (∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥 ↔ ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑢))
15	14	rspcv 3586	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝐹 → (∀𝑥 ∈ 𝐹 ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥 → ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑢))
16	15	adantl 486	. . . . . . . . 9 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ 𝑢 ∈ 𝐹) → (∀𝑥 ∈ 𝐹 ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥 → ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑢))
17		sstr 3953	. . . . . . . . . . . . 13 ⊢ ((𝑦 ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑡) → 𝑦 ⊆ 𝑡)
18		sseq1 3970	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑦 → (𝑣 ⊆ 𝑡 ↔ 𝑦 ⊆ 𝑡))
19	18	rspcev 3590	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝐺 ∧ 𝑦 ⊆ 𝑡) → ∃𝑣 ∈ 𝐺 𝑣 ⊆ 𝑡)
20	19	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ 𝑢 ∈ 𝐹) ∧ (𝑦 ∈ 𝐺 ∧ 𝑦 ⊆ 𝑡)) → ∃𝑣 ∈ 𝐺 𝑣 ⊆ 𝑡)
21	20	a1d 26	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ 𝑢 ∈ 𝐹) ∧ (𝑦 ∈ 𝐺 ∧ 𝑦 ⊆ 𝑡)) → (𝑡 ⊆ 𝑋 → ∃𝑣 ∈ 𝐺 𝑣 ⊆ 𝑡))
22	17, 21	sylanr2 695	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ 𝑢 ∈ 𝐹) ∧ (𝑦 ∈ 𝐺 ∧ (𝑦 ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑡))) → (𝑡 ⊆ 𝑋 → ∃𝑣 ∈ 𝐺 𝑣 ⊆ 𝑡))
23	22	ancld 559	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ 𝑢 ∈ 𝐹) ∧ (𝑦 ∈ 𝐺 ∧ (𝑦 ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑡))) → (𝑡 ⊆ 𝑋 → (𝑡 ⊆ 𝑋 ∧ ∃𝑣 ∈ 𝐺 𝑣 ⊆ 𝑡)))
24	23	exp45 443	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ 𝑢 ∈ 𝐹) → (𝑦 ∈ 𝐺 → (𝑦 ⊆ 𝑢 → (𝑢 ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → (𝑡 ⊆ 𝑋 ∧ ∃𝑣 ∈ 𝐺 𝑣 ⊆ 𝑡))))))
25	24	rexlimdv 3170	. . . . . . . . 9 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ 𝑢 ∈ 𝐹) → (∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑢 → (𝑢 ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → (𝑡 ⊆ 𝑋 ∧ ∃𝑣 ∈ 𝐺 𝑣 ⊆ 𝑡)))))
26	16, 25	syld 48	. . . . . . . 8 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ 𝑢 ∈ 𝐹) → (∀𝑥 ∈ 𝐹 ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥 → (𝑢 ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → (𝑡 ⊆ 𝑋 ∧ ∃𝑣 ∈ 𝐺 𝑣 ⊆ 𝑡)))))
27	26	impancom 456	. . . . . . 7 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ ∀𝑥 ∈ 𝐹 ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥) → (𝑢 ∈ 𝐹 → (𝑢 ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → (𝑡 ⊆ 𝑋 ∧ ∃𝑣 ∈ 𝐺 𝑣 ⊆ 𝑡)))))
28	27	rexlimdv 3170	. . . . . 6 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ ∀𝑥 ∈ 𝐹 ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥) → (∃𝑢 ∈ 𝐹 𝑢 ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → (𝑡 ⊆ 𝑋 ∧ ∃𝑣 ∈ 𝐺 𝑣 ⊆ 𝑡))))
29	28	impcomd 416	. . . . 5 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ ∀𝑥 ∈ 𝐹 ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥) → ((𝑡 ⊆ 𝑋 ∧ ∃𝑢 ∈ 𝐹 𝑢 ⊆ 𝑡) → (𝑡 ⊆ 𝑋 ∧ ∃𝑣 ∈ 𝐺 𝑣 ⊆ 𝑡)))
30		elfg 23997	. . . . . . 7 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑢 ∈ 𝐹 𝑢 ⊆ 𝑡)))
31	30	adantr 485	. . . . . 6 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑢 ∈ 𝐹 𝑢 ⊆ 𝑡)))
32	31	adantr 485	. . . . 5 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ ∀𝑥 ∈ 𝐹 ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑢 ∈ 𝐹 𝑢 ⊆ 𝑡)))
33		elfg 23997	. . . . . . 7 ⊢ (𝐺 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐺) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑣 ∈ 𝐺 𝑣 ⊆ 𝑡)))
34	33	adantl 486	. . . . . 6 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (𝑡 ∈ (𝑋filGen𝐺) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑣 ∈ 𝐺 𝑣 ⊆ 𝑡)))
35	34	adantr 485	. . . . 5 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ ∀𝑥 ∈ 𝐹 ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥) → (𝑡 ∈ (𝑋filGen𝐺) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑣 ∈ 𝐺 𝑣 ⊆ 𝑡)))
36	29, 32, 35	3imtr4d 297	. . . 4 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ ∀𝑥 ∈ 𝐹 ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥) → (𝑡 ∈ (𝑋filGen𝐹) → 𝑡 ∈ (𝑋filGen𝐺)))
37	36	ssrdv 3951	. . 3 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ ∀𝑥 ∈ 𝐹 ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥) → (𝑋filGen𝐹) ⊆ (𝑋filGen𝐺))
38	37	ex 417	. 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥 ∈ 𝐹 ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥 → (𝑋filGen𝐹) ⊆ (𝑋filGen𝐺)))
39	12, 38	impbid 215	1 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ((𝑋filGen𝐹) ⊆ (𝑋filGen𝐺) ↔ ∀𝑥 ∈ 𝐹 ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095   ⊆ wss 3913  ‘cfv 6537  (class class class)co 7411  fBascfbas 21479  filGencfg 21480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-fbas 21488  df-fg 21489

This theorem is referenced by: (None)