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Theorem fgss2 23897
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) ↔ ∀𝑥𝐹𝑦𝐺 𝑦𝑥))
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.
StepHypRef Expression
1 ssfg 23895 . . . . . 6 (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹))
21adantr 480 . . . . 5 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → 𝐹 ⊆ (𝑋filGen𝐹))
32sseld 3993 . . . 4 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (𝑥𝐹𝑥 ∈ (𝑋filGen𝐹)))
4 ssel2 3989 . . . . . 6 (((𝑋filGen𝐹) ⊆ (𝑋filGen𝐺) ∧ 𝑥 ∈ (𝑋filGen𝐹)) → 𝑥 ∈ (𝑋filGen𝐺))
5 elfg 23894 . . . . . . . 8 (𝐺 ∈ (fBas‘𝑋) → (𝑥 ∈ (𝑋filGen𝐺) ↔ (𝑥𝑋 ∧ ∃𝑦𝐺 𝑦𝑥)))
6 simpr 484 . . . . . . . 8 ((𝑥𝑋 ∧ ∃𝑦𝐺 𝑦𝑥) → ∃𝑦𝐺 𝑦𝑥)
75, 6biimtrdi 253 . . . . . . 7 (𝐺 ∈ (fBas‘𝑋) → (𝑥 ∈ (𝑋filGen𝐺) → ∃𝑦𝐺 𝑦𝑥))
87adantl 481 . . . . . 6 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (𝑥 ∈ (𝑋filGen𝐺) → ∃𝑦𝐺 𝑦𝑥))
94, 8syl5 34 . . . . 5 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (((𝑋filGen𝐹) ⊆ (𝑋filGen𝐺) ∧ 𝑥 ∈ (𝑋filGen𝐹)) → ∃𝑦𝐺 𝑦𝑥))
109expd 415 . . . 4 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ((𝑋filGen𝐹) ⊆ (𝑋filGen𝐺) → (𝑥 ∈ (𝑋filGen𝐹) → ∃𝑦𝐺 𝑦𝑥)))
113, 10syl5d 73 . . 3 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ((𝑋filGen𝐹) ⊆ (𝑋filGen𝐺) → (𝑥𝐹 → ∃𝑦𝐺 𝑦𝑥)))
1211ralrimdv 3149 . 2 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ((𝑋filGen𝐹) ⊆ (𝑋filGen𝐺) → ∀𝑥𝐹𝑦𝐺 𝑦𝑥))
13 sseq2 4021 . . . . . . . . . . . 12 (𝑥 = 𝑢 → (𝑦𝑥𝑦𝑢))
1413rexbidv 3176 . . . . . . . . . . 11 (𝑥 = 𝑢 → (∃𝑦𝐺 𝑦𝑥 ↔ ∃𝑦𝐺 𝑦𝑢))
1514rspcv 3617 . . . . . . . . . 10 (𝑢𝐹 → (∀𝑥𝐹𝑦𝐺 𝑦𝑥 → ∃𝑦𝐺 𝑦𝑢))
1615adantl 481 . . . . . . . . 9 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ 𝑢𝐹) → (∀𝑥𝐹𝑦𝐺 𝑦𝑥 → ∃𝑦𝐺 𝑦𝑢))
17 sstr 4003 . . . . . . . . . . . . 13 ((𝑦𝑢𝑢𝑡) → 𝑦𝑡)
18 sseq1 4020 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑦 → (𝑣𝑡𝑦𝑡))
1918rspcev 3621 . . . . . . . . . . . . . . 15 ((𝑦𝐺𝑦𝑡) → ∃𝑣𝐺 𝑣𝑡)
2019adantl 481 . . . . . . . . . . . . . 14 ((((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ 𝑢𝐹) ∧ (𝑦𝐺𝑦𝑡)) → ∃𝑣𝐺 𝑣𝑡)
2120a1d 25 . . . . . . . . . . . . 13 ((((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ 𝑢𝐹) ∧ (𝑦𝐺𝑦𝑡)) → (𝑡𝑋 → ∃𝑣𝐺 𝑣𝑡))
2217, 21sylanr2 683 . . . . . . . . . . . 12 ((((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ 𝑢𝐹) ∧ (𝑦𝐺 ∧ (𝑦𝑢𝑢𝑡))) → (𝑡𝑋 → ∃𝑣𝐺 𝑣𝑡))
2322ancld 550 . . . . . . . . . . 11 ((((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ 𝑢𝐹) ∧ (𝑦𝐺 ∧ (𝑦𝑢𝑢𝑡))) → (𝑡𝑋 → (𝑡𝑋 ∧ ∃𝑣𝐺 𝑣𝑡)))
2423exp45 438 . . . . . . . . . 10 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ 𝑢𝐹) → (𝑦𝐺 → (𝑦𝑢 → (𝑢𝑡 → (𝑡𝑋 → (𝑡𝑋 ∧ ∃𝑣𝐺 𝑣𝑡))))))
2524rexlimdv 3150 . . . . . . . . 9 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ 𝑢𝐹) → (∃𝑦𝐺 𝑦𝑢 → (𝑢𝑡 → (𝑡𝑋 → (𝑡𝑋 ∧ ∃𝑣𝐺 𝑣𝑡)))))
2616, 25syld 47 . . . . . . . 8 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ 𝑢𝐹) → (∀𝑥𝐹𝑦𝐺 𝑦𝑥 → (𝑢𝑡 → (𝑡𝑋 → (𝑡𝑋 ∧ ∃𝑣𝐺 𝑣𝑡)))))
2726impancom 451 . . . . . . 7 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ ∀𝑥𝐹𝑦𝐺 𝑦𝑥) → (𝑢𝐹 → (𝑢𝑡 → (𝑡𝑋 → (𝑡𝑋 ∧ ∃𝑣𝐺 𝑣𝑡)))))
2827rexlimdv 3150 . . . . . 6 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ ∀𝑥𝐹𝑦𝐺 𝑦𝑥) → (∃𝑢𝐹 𝑢𝑡 → (𝑡𝑋 → (𝑡𝑋 ∧ ∃𝑣𝐺 𝑣𝑡))))
2928impcomd 411 . . . . 5 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ ∀𝑥𝐹𝑦𝐺 𝑦𝑥) → ((𝑡𝑋 ∧ ∃𝑢𝐹 𝑢𝑡) → (𝑡𝑋 ∧ ∃𝑣𝐺 𝑣𝑡)))
30 elfg 23894 . . . . . . 7 (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡𝑋 ∧ ∃𝑢𝐹 𝑢𝑡)))
3130adantr 480 . . . . . 6 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡𝑋 ∧ ∃𝑢𝐹 𝑢𝑡)))
3231adantr 480 . . . . 5 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ ∀𝑥𝐹𝑦𝐺 𝑦𝑥) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡𝑋 ∧ ∃𝑢𝐹 𝑢𝑡)))
33 elfg 23894 . . . . . . 7 (𝐺 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐺) ↔ (𝑡𝑋 ∧ ∃𝑣𝐺 𝑣𝑡)))
3433adantl 481 . . . . . 6 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (𝑡 ∈ (𝑋filGen𝐺) ↔ (𝑡𝑋 ∧ ∃𝑣𝐺 𝑣𝑡)))
3534adantr 480 . . . . 5 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ ∀𝑥𝐹𝑦𝐺 𝑦𝑥) → (𝑡 ∈ (𝑋filGen𝐺) ↔ (𝑡𝑋 ∧ ∃𝑣𝐺 𝑣𝑡)))
3629, 32, 353imtr4d 294 . . . 4 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ ∀𝑥𝐹𝑦𝐺 𝑦𝑥) → (𝑡 ∈ (𝑋filGen𝐹) → 𝑡 ∈ (𝑋filGen𝐺)))
3736ssrdv 4000 . . 3 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ ∀𝑥𝐹𝑦𝐺 𝑦𝑥) → (𝑋filGen𝐹) ⊆ (𝑋filGen𝐺))
3837ex 412 . 2 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥𝐹𝑦𝐺 𝑦𝑥 → (𝑋filGen𝐹) ⊆ (𝑋filGen𝐺)))
3912, 38impbid 212 1 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ((𝑋filGen𝐹) ⊆ (𝑋filGen𝐺) ↔ ∀𝑥𝐹𝑦𝐺 𝑦𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2105  wral 3058  wrex 3067  wss 3962  cfv 6562  (class class class)co 7430  fBascfbas 21369  filGencfg 21370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-fbas 21378  df-fg 21379
This theorem is referenced by: (None)
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