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Theorem fgss2 21879
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 21877 . . . . . 6 (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹))
21adantr 472 . . . . 5 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → 𝐹 ⊆ (𝑋filGen𝐹))
32sseld 3743 . . . 4 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (𝑥𝐹𝑥 ∈ (𝑋filGen𝐹)))
4 ssel2 3739 . . . . . 6 (((𝑋filGen𝐹) ⊆ (𝑋filGen𝐺) ∧ 𝑥 ∈ (𝑋filGen𝐹)) → 𝑥 ∈ (𝑋filGen𝐺))
5 elfg 21876 . . . . . . . 8 (𝐺 ∈ (fBas‘𝑋) → (𝑥 ∈ (𝑋filGen𝐺) ↔ (𝑥𝑋 ∧ ∃𝑦𝐺 𝑦𝑥)))
6 simpr 479 . . . . . . . 8 ((𝑥𝑋 ∧ ∃𝑦𝐺 𝑦𝑥) → ∃𝑦𝐺 𝑦𝑥)
75, 6syl6bi 243 . . . . . . 7 (𝐺 ∈ (fBas‘𝑋) → (𝑥 ∈ (𝑋filGen𝐺) → ∃𝑦𝐺 𝑦𝑥))
87adantl 473 . . . . . 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 451 . . . 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 3106 . 2 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ((𝑋filGen𝐹) ⊆ (𝑋filGen𝐺) → ∀𝑥𝐹𝑦𝐺 𝑦𝑥))
13 sseq2 3768 . . . . . . . . . . . . 13 (𝑥 = 𝑢 → (𝑦𝑥𝑦𝑢))
1413rexbidv 3190 . . . . . . . . . . . 12 (𝑥 = 𝑢 → (∃𝑦𝐺 𝑦𝑥 ↔ ∃𝑦𝐺 𝑦𝑢))
1514rspcv 3445 . . . . . . . . . . 11 (𝑢𝐹 → (∀𝑥𝐹𝑦𝐺 𝑦𝑥 → ∃𝑦𝐺 𝑦𝑢))
1615adantl 473 . . . . . . . . . 10 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ 𝑢𝐹) → (∀𝑥𝐹𝑦𝐺 𝑦𝑥 → ∃𝑦𝐺 𝑦𝑢))
17 sstr 3752 . . . . . . . . . . . . . 14 ((𝑦𝑢𝑢𝑡) → 𝑦𝑡)
18 sseq1 3767 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝑦 → (𝑣𝑡𝑦𝑡))
1918rspcev 3449 . . . . . . . . . . . . . . . 16 ((𝑦𝐺𝑦𝑡) → ∃𝑣𝐺 𝑣𝑡)
2019adantl 473 . . . . . . . . . . . . . . 15 ((((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ 𝑢𝐹) ∧ (𝑦𝐺𝑦𝑡)) → ∃𝑣𝐺 𝑣𝑡)
2120a1d 25 . . . . . . . . . . . . . 14 ((((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ 𝑢𝐹) ∧ (𝑦𝐺𝑦𝑡)) → (𝑡𝑋 → ∃𝑣𝐺 𝑣𝑡))
2217, 21sylanr2 688 . . . . . . . . . . . . 13 ((((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ 𝑢𝐹) ∧ (𝑦𝐺 ∧ (𝑦𝑢𝑢𝑡))) → (𝑡𝑋 → ∃𝑣𝐺 𝑣𝑡))
2322ancld 577 . . . . . . . . . . . 12 ((((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ 𝑢𝐹) ∧ (𝑦𝐺 ∧ (𝑦𝑢𝑢𝑡))) → (𝑡𝑋 → (𝑡𝑋 ∧ ∃𝑣𝐺 𝑣𝑡)))
2423exp45 643 . . . . . . . . . . 11 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ 𝑢𝐹) → (𝑦𝐺 → (𝑦𝑢 → (𝑢𝑡 → (𝑡𝑋 → (𝑡𝑋 ∧ ∃𝑣𝐺 𝑣𝑡))))))
2524rexlimdv 3168 . . . . . . . . . 10 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ 𝑢𝐹) → (∃𝑦𝐺 𝑦𝑢 → (𝑢𝑡 → (𝑡𝑋 → (𝑡𝑋 ∧ ∃𝑣𝐺 𝑣𝑡)))))
2616, 25syld 47 . . . . . . . . 9 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ 𝑢𝐹) → (∀𝑥𝐹𝑦𝐺 𝑦𝑥 → (𝑢𝑡 → (𝑡𝑋 → (𝑡𝑋 ∧ ∃𝑣𝐺 𝑣𝑡)))))
2726impancom 455 . . . . . . . 8 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ ∀𝑥𝐹𝑦𝐺 𝑦𝑥) → (𝑢𝐹 → (𝑢𝑡 → (𝑡𝑋 → (𝑡𝑋 ∧ ∃𝑣𝐺 𝑣𝑡)))))
2827rexlimdv 3168 . . . . . . 7 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ ∀𝑥𝐹𝑦𝐺 𝑦𝑥) → (∃𝑢𝐹 𝑢𝑡 → (𝑡𝑋 → (𝑡𝑋 ∧ ∃𝑣𝐺 𝑣𝑡))))
2928com23 86 . . . . . 6 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ ∀𝑥𝐹𝑦𝐺 𝑦𝑥) → (𝑡𝑋 → (∃𝑢𝐹 𝑢𝑡 → (𝑡𝑋 ∧ ∃𝑣𝐺 𝑣𝑡))))
3029impd 446 . . . . 5 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ ∀𝑥𝐹𝑦𝐺 𝑦𝑥) → ((𝑡𝑋 ∧ ∃𝑢𝐹 𝑢𝑡) → (𝑡𝑋 ∧ ∃𝑣𝐺 𝑣𝑡)))
31 elfg 21876 . . . . . . 7 (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡𝑋 ∧ ∃𝑢𝐹 𝑢𝑡)))
3231adantr 472 . . . . . 6 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡𝑋 ∧ ∃𝑢𝐹 𝑢𝑡)))
3332adantr 472 . . . . 5 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ ∀𝑥𝐹𝑦𝐺 𝑦𝑥) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡𝑋 ∧ ∃𝑢𝐹 𝑢𝑡)))
34 elfg 21876 . . . . . . 7 (𝐺 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐺) ↔ (𝑡𝑋 ∧ ∃𝑣𝐺 𝑣𝑡)))
3534adantl 473 . . . . . 6 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (𝑡 ∈ (𝑋filGen𝐺) ↔ (𝑡𝑋 ∧ ∃𝑣𝐺 𝑣𝑡)))
3635adantr 472 . . . . 5 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ ∀𝑥𝐹𝑦𝐺 𝑦𝑥) → (𝑡 ∈ (𝑋filGen𝐺) ↔ (𝑡𝑋 ∧ ∃𝑣𝐺 𝑣𝑡)))
3730, 33, 363imtr4d 283 . . . 4 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ ∀𝑥𝐹𝑦𝐺 𝑦𝑥) → (𝑡 ∈ (𝑋filGen𝐹) → 𝑡 ∈ (𝑋filGen𝐺)))
3837ssrdv 3750 . . 3 (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) ∧ ∀𝑥𝐹𝑦𝐺 𝑦𝑥) → (𝑋filGen𝐹) ⊆ (𝑋filGen𝐺))
3938ex 449 . 2 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → (∀𝑥𝐹𝑦𝐺 𝑦𝑥 → (𝑋filGen𝐹) ⊆ (𝑋filGen𝐺)))
4012, 39impbid 202 1 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ((𝑋filGen𝐹) ⊆ (𝑋filGen𝐺) ↔ ∀𝑥𝐹𝑦𝐺 𝑦𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wcel 2139  wral 3050  wrex 3051  wss 3715  cfv 6049  (class class class)co 6813  fBascfbas 19936  filGencfg 19937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-fbas 19945  df-fg 19946
This theorem is referenced by: (None)
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