Theorem fgss2 23378

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 23376	. . . . . 6 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹))
2	1	adantr 482	. . . . 5 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → 𝐹 ⊆ (𝑋filGen𝐹))
3	2	sseld 3982	. . . 4 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (𝑥 ∈ 𝐹 → 𝑥 ∈ (𝑋filGen𝐹)))
4		ssel2 3978	. . . . . 6 ⊢ (((𝑋filGen𝐹) ⊆ (𝑋filGen𝐺) ∧ 𝑥 ∈ (𝑋filGen𝐹)) → 𝑥 ∈ (𝑋filGen𝐺))
5		elfg 23375	. . . . . . . 8 ⊢ (𝐺 ∈ (fBas‘𝑋) → (𝑥 ∈ (𝑋filGen𝐺) ↔ (𝑥 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥)))
6		simpr 486	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥) → ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥)
7	5, 6	syl6bi 253	. . . . . . 7 ⊢ (𝐺 ∈ (fBas‘𝑋) → (𝑥 ∈ (𝑋filGen𝐺) → ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥))
8	7	adantl 483	. . . . . 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 417	. . . 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 3153	. 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ((𝑋filGen𝐹) ⊆ (𝑋filGen𝐺) → ∀𝑥 ∈ 𝐹 ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥))
13		sseq2 4009	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑢 → (𝑦 ⊆ 𝑥 ↔ 𝑦 ⊆ 𝑢))
14	13	rexbidv 3179	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑢 → (∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥 ↔ ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑢))
15	14	rspcv 3609	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝐹 → (∀𝑥 ∈ 𝐹 ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥 → ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑢))
16	15	adantl 483	. . . . . . . . 9 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ 𝑢 ∈ 𝐹) → (∀𝑥 ∈ 𝐹 ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥 → ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑢))
17		sstr 3991	. . . . . . . . . . . . 13 ⊢ ((𝑦 ⊆ 𝑢 ∧ 𝑢 ⊆ 𝑡) → 𝑦 ⊆ 𝑡)
18		sseq1 4008	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑦 → (𝑣 ⊆ 𝑡 ↔ 𝑦 ⊆ 𝑡))
19	18	rspcev 3613	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝐺 ∧ 𝑦 ⊆ 𝑡) → ∃𝑣 ∈ 𝐺 𝑣 ⊆ 𝑡)
20	19	adantl 483	. . . . . . . . . . . . . 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 552	. . . . . . . . . . 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 440	. . . . . . . . . 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 3154	. . . . . . . . 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 453	. . . . . . 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 3154	. . . . . 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 413	. . . . 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 23375	. . . . . . 7 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑢 ∈ 𝐹 𝑢 ⊆ 𝑡)))
31	30	adantr 482	. . . . . 6 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑢 ∈ 𝐹 𝑢 ⊆ 𝑡)))
32	31	adantr 482	. . . . 5 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ ∀𝑥 ∈ 𝐹 ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑢 ∈ 𝐹 𝑢 ⊆ 𝑡)))
33		elfg 23375	. . . . . . 7 ⊢ (𝐺 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐺) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑣 ∈ 𝐺 𝑣 ⊆ 𝑡)))
34	33	adantl 483	. . . . . 6 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (𝑡 ∈ (𝑋filGen𝐺) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑣 ∈ 𝐺 𝑣 ⊆ 𝑡)))
35	34	adantr 482	. . . . 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 3989	. . 3 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ ∀𝑥 ∈ 𝐹 ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥) → (𝑋filGen𝐹) ⊆ (𝑋filGen𝐺))
38	37	ex 414	. 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥 ∈ 𝐹 ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥 → (𝑋filGen𝐹) ⊆ (𝑋filGen𝐺)))
39	12, 38	impbid 211	1 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ((𝑋filGen𝐹) ⊆ (𝑋filGen𝐺) ↔ ∀𝑥 ∈ 𝐹 ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071   ⊆ wss 3949  ‘cfv 6544  (class class class)co 7409  fBascfbas 20932  filGencfg 20933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-fbas 20941  df-fg 20942

This theorem is referenced by: (None)