Theorem fgss2 23903

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 23901	. . . . . 6 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹))
2	1	adantr 480	. . . . 5 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → 𝐹 ⊆ (𝑋filGen𝐹))
3	2	sseld 4007	. . . 4 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (𝑥 ∈ 𝐹 → 𝑥 ∈ (𝑋filGen𝐹)))
4		ssel2 4003	. . . . . 6 ⊢ (((𝑋filGen𝐹) ⊆ (𝑋filGen𝐺) ∧ 𝑥 ∈ (𝑋filGen𝐹)) → 𝑥 ∈ (𝑋filGen𝐺))
5		elfg 23900	. . . . . . . 8 ⊢ (𝐺 ∈ (fBas‘𝑋) → (𝑥 ∈ (𝑋filGen𝐺) ↔ (𝑥 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥)))
6		simpr 484	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥) → ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥)
7	5, 6	biimtrdi 253	. . . . . . 7 ⊢ (𝐺 ∈ (fBas‘𝑋) → (𝑥 ∈ (𝑋filGen𝐺) → ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥))
8	7	adantl 481	. . . . . 6 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (𝑥 ∈ (𝑋filGen𝐺) → ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥))
9	4, 8	syl5 34	. . . . 5 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (((𝑋filGen𝐹) ⊆ (𝑋filGen𝐺) ∧ 𝑥 ∈ (𝑋filGen𝐹)) → ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥))
10	9	expd 415	. . . 4 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ((𝑋filGen𝐹) ⊆ (𝑋filGen𝐺) → (𝑥 ∈ (𝑋filGen𝐹) → ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥)))
11	3, 10	syl5d 73	. . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ((𝑋filGen𝐹) ⊆ (𝑋filGen𝐺) → (𝑥 ∈ 𝐹 → ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥)))
12	11	ralrimdv 3158	. 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ((𝑋filGen𝐹) ⊆ (𝑋filGen𝐺) → ∀𝑥 ∈ 𝐹 ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥))
13		sseq2 4035	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑢 → (𝑦 ⊆ 𝑥 ↔ 𝑦 ⊆ 𝑢))
14	13	rexbidv 3185	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑢 → (∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥 ↔ ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑢))
15	14	rspcv 3631	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝐹 → (∀𝑥 ∈ 𝐹 ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥 → ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑢))
16	15	adantl 481	. . . . . . . . 9 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ 𝑢 ∈ 𝐹) → (∀𝑥 ∈ 𝐹 ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥 → ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑢))
17		sstr 4017	. . . . . . . . . . . . 13 ⊢ ((𝑦 ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑡) → 𝑦 ⊆ 𝑡)
18		sseq1 4034	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑦 → (𝑣 ⊆ 𝑡 ↔ 𝑦 ⊆ 𝑡))
19	18	rspcev 3635	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝐺 ∧ 𝑦 ⊆ 𝑡) → ∃𝑣 ∈ 𝐺 𝑣 ⊆ 𝑡)
20	19	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ 𝑢 ∈ 𝐹) ∧ (𝑦 ∈ 𝐺 ∧ 𝑦 ⊆ 𝑡)) → ∃𝑣 ∈ 𝐺 𝑣 ⊆ 𝑡)
21	20	a1d 25	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ 𝑢 ∈ 𝐹) ∧ (𝑦 ∈ 𝐺 ∧ 𝑦 ⊆ 𝑡)) → (𝑡 ⊆ 𝑋 → ∃𝑣 ∈ 𝐺 𝑣 ⊆ 𝑡))
22	17, 21	sylanr2 682	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ 𝑢 ∈ 𝐹) ∧ (𝑦 ∈ 𝐺 ∧ (𝑦 ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑡))) → (𝑡 ⊆ 𝑋 → ∃𝑣 ∈ 𝐺 𝑣 ⊆ 𝑡))
23	22	ancld 550	. . . . . . . . . . 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 438	. . . . . . . . . 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 3159	. . . . . . . . 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 47	. . . . . . . 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 451	. . . . . . 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 3159	. . . . . 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 411	. . . . 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 23900	. . . . . . 7 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑢 ∈ 𝐹 𝑢 ⊆ 𝑡)))
31	30	adantr 480	. . . . . 6 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑢 ∈ 𝐹 𝑢 ⊆ 𝑡)))
32	31	adantr 480	. . . . 5 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ ∀𝑥 ∈ 𝐹 ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑢 ∈ 𝐹 𝑢 ⊆ 𝑡)))
33		elfg 23900	. . . . . . 7 ⊢ (𝐺 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐺) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑣 ∈ 𝐺 𝑣 ⊆ 𝑡)))
34	33	adantl 481	. . . . . 6 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (𝑡 ∈ (𝑋filGen𝐺) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑣 ∈ 𝐺 𝑣 ⊆ 𝑡)))
35	34	adantr 480	. . . . 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 294	. . . 4 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ ∀𝑥 ∈ 𝐹 ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥) → (𝑡 ∈ (𝑋filGen𝐹) → 𝑡 ∈ (𝑋filGen𝐺)))
37	36	ssrdv 4014	. . 3 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ ∀𝑥 ∈ 𝐹 ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥) → (𝑋filGen𝐹) ⊆ (𝑋filGen𝐺))
38	37	ex 412	. 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥 ∈ 𝐹 ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥 → (𝑋filGen𝐹) ⊆ (𝑋filGen𝐺)))
39	12, 38	impbid 212	1 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ((𝑋filGen𝐹) ⊆ (𝑋filGen𝐺) ↔ ∀𝑥 ∈ 𝐹 ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076   ⊆ wss 3976  ‘cfv 6573  (class class class)co 7448  fBascfbas 21375  filGencfg 21376

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-fbas 21384  df-fg 21385

This theorem is referenced by: (None)