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Theorem fgss2 22410
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 22408 . . . . . 6 (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹))
21adantr 481 . . . . 5 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → 𝐹 ⊆ (𝑋filGen𝐹))
32sseld 3963 . . . 4 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (𝑥𝐹𝑥 ∈ (𝑋filGen𝐹)))
4 ssel2 3959 . . . . . 6 (((𝑋filGen𝐹) ⊆ (𝑋filGen𝐺) ∧ 𝑥 ∈ (𝑋filGen𝐹)) → 𝑥 ∈ (𝑋filGen𝐺))
5 elfg 22407 . . . . . . . 8 (𝐺 ∈ (fBas‘𝑋) → (𝑥 ∈ (𝑋filGen𝐺) ↔ (𝑥𝑋 ∧ ∃𝑦𝐺 𝑦𝑥)))
6 simpr 485 . . . . . . . 8 ((𝑥𝑋 ∧ ∃𝑦𝐺 𝑦𝑥) → ∃𝑦𝐺 𝑦𝑥)
75, 6syl6bi 254 . . . . . . 7 (𝐺 ∈ (fBas‘𝑋) → (𝑥 ∈ (𝑋filGen𝐺) → ∃𝑦𝐺 𝑦𝑥))
87adantl 482 . . . . . 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 416 . . . 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 3185 . 2 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ((𝑋filGen𝐹) ⊆ (𝑋filGen𝐺) → ∀𝑥𝐹𝑦𝐺 𝑦𝑥))
13 sseq2 3990 . . . . . . . . . . . 12 (𝑥 = 𝑢 → (𝑦𝑥𝑦𝑢))
1413rexbidv 3294 . . . . . . . . . . 11 (𝑥 = 𝑢 → (∃𝑦𝐺 𝑦𝑥 ↔ ∃𝑦𝐺 𝑦𝑢))
1514rspcv 3615 . . . . . . . . . 10 (𝑢𝐹 → (∀𝑥𝐹𝑦𝐺 𝑦𝑥 → ∃𝑦𝐺 𝑦𝑢))
1615adantl 482 . . . . . . . . 9 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ 𝑢𝐹) → (∀𝑥𝐹𝑦𝐺 𝑦𝑥 → ∃𝑦𝐺 𝑦𝑢))
17 sstr 3972 . . . . . . . . . . . . 13 ((𝑦𝑢𝑢𝑡) → 𝑦𝑡)
18 sseq1 3989 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑦 → (𝑣𝑡𝑦𝑡))
1918rspcev 3620 . . . . . . . . . . . . . . 15 ((𝑦𝐺𝑦𝑡) → ∃𝑣𝐺 𝑣𝑡)
2019adantl 482 . . . . . . . . . . . . . 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 679 . . . . . . . . . . . 12 ((((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ 𝑢𝐹) ∧ (𝑦𝐺 ∧ (𝑦𝑢𝑢𝑡))) → (𝑡𝑋 → ∃𝑣𝐺 𝑣𝑡))
2322ancld 551 . . . . . . . . . . 11 ((((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ 𝑢𝐹) ∧ (𝑦𝐺 ∧ (𝑦𝑢𝑢𝑡))) → (𝑡𝑋 → (𝑡𝑋 ∧ ∃𝑣𝐺 𝑣𝑡)))
2423exp45 439 . . . . . . . . . 10 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ 𝑢𝐹) → (𝑦𝐺 → (𝑦𝑢 → (𝑢𝑡 → (𝑡𝑋 → (𝑡𝑋 ∧ ∃𝑣𝐺 𝑣𝑡))))))
2524rexlimdv 3280 . . . . . . . . 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 452 . . . . . . 7 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ ∀𝑥𝐹𝑦𝐺 𝑦𝑥) → (𝑢𝐹 → (𝑢𝑡 → (𝑡𝑋 → (𝑡𝑋 ∧ ∃𝑣𝐺 𝑣𝑡)))))
2827rexlimdv 3280 . . . . . 6 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ ∀𝑥𝐹𝑦𝐺 𝑦𝑥) → (∃𝑢𝐹 𝑢𝑡 → (𝑡𝑋 → (𝑡𝑋 ∧ ∃𝑣𝐺 𝑣𝑡))))
2928impcomd 412 . . . . 5 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ ∀𝑥𝐹𝑦𝐺 𝑦𝑥) → ((𝑡𝑋 ∧ ∃𝑢𝐹 𝑢𝑡) → (𝑡𝑋 ∧ ∃𝑣𝐺 𝑣𝑡)))
30 elfg 22407 . . . . . . 7 (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡𝑋 ∧ ∃𝑢𝐹 𝑢𝑡)))
3130adantr 481 . . . . . 6 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡𝑋 ∧ ∃𝑢𝐹 𝑢𝑡)))
3231adantr 481 . . . . 5 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ ∀𝑥𝐹𝑦𝐺 𝑦𝑥) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡𝑋 ∧ ∃𝑢𝐹 𝑢𝑡)))
33 elfg 22407 . . . . . . 7 (𝐺 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐺) ↔ (𝑡𝑋 ∧ ∃𝑣𝐺 𝑣𝑡)))
3433adantl 482 . . . . . 6 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (𝑡 ∈ (𝑋filGen𝐺) ↔ (𝑡𝑋 ∧ ∃𝑣𝐺 𝑣𝑡)))
3534adantr 481 . . . . 5 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ ∀𝑥𝐹𝑦𝐺 𝑦𝑥) → (𝑡 ∈ (𝑋filGen𝐺) ↔ (𝑡𝑋 ∧ ∃𝑣𝐺 𝑣𝑡)))
3629, 32, 353imtr4d 295 . . . 4 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ ∀𝑥𝐹𝑦𝐺 𝑦𝑥) → (𝑡 ∈ (𝑋filGen𝐹) → 𝑡 ∈ (𝑋filGen𝐺)))
3736ssrdv 3970 . . 3 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ ∀𝑥𝐹𝑦𝐺 𝑦𝑥) → (𝑋filGen𝐹) ⊆ (𝑋filGen𝐺))
3837ex 413 . 2 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥𝐹𝑦𝐺 𝑦𝑥 → (𝑋filGen𝐹) ⊆ (𝑋filGen𝐺)))
3912, 38impbid 213 1 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ((𝑋filGen𝐹) ⊆ (𝑋filGen𝐺) ↔ ∀𝑥𝐹𝑦𝐺 𝑦𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wcel 2105  wral 3135  wrex 3136  wss 3933  cfv 6348  (class class class)co 7145  fBascfbas 20461  filGencfg 20462
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-fbas 20470  df-fg 20471
This theorem is referenced by: (None)
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