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Theorem fgss 23376
Description: A bigger base generates a bigger filter. (Contributed by NM, 5-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
fgss ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋) ∧ 𝐹𝐺) → (𝑋filGen𝐹) ⊆ (𝑋filGen𝐺))

Proof of Theorem fgss
Dummy variables 𝑥 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrexv 4051 . . . . 5 (𝐹𝐺 → (∃𝑥𝐹 𝑥𝑡 → ∃𝑥𝐺 𝑥𝑡))
21anim2d 612 . . . 4 (𝐹𝐺 → ((𝑡𝑋 ∧ ∃𝑥𝐹 𝑥𝑡) → (𝑡𝑋 ∧ ∃𝑥𝐺 𝑥𝑡)))
323ad2ant3 1135 . . 3 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋) ∧ 𝐹𝐺) → ((𝑡𝑋 ∧ ∃𝑥𝐹 𝑥𝑡) → (𝑡𝑋 ∧ ∃𝑥𝐺 𝑥𝑡)))
4 elfg 23374 . . . 4 (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡𝑋 ∧ ∃𝑥𝐹 𝑥𝑡)))
543ad2ant1 1133 . . 3 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋) ∧ 𝐹𝐺) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡𝑋 ∧ ∃𝑥𝐹 𝑥𝑡)))
6 elfg 23374 . . . 4 (𝐺 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐺) ↔ (𝑡𝑋 ∧ ∃𝑥𝐺 𝑥𝑡)))
763ad2ant2 1134 . . 3 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋) ∧ 𝐹𝐺) → (𝑡 ∈ (𝑋filGen𝐺) ↔ (𝑡𝑋 ∧ ∃𝑥𝐺 𝑥𝑡)))
83, 5, 73imtr4d 293 . 2 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋) ∧ 𝐹𝐺) → (𝑡 ∈ (𝑋filGen𝐹) → 𝑡 ∈ (𝑋filGen𝐺)))
98ssrdv 3988 1 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋) ∧ 𝐹𝐺) → (𝑋filGen𝐹) ⊆ (𝑋filGen𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087  wcel 2106  wrex 3070  wss 3948  cfv 6543  (class class class)co 7408  fBascfbas 20931  filGencfg 20932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-fg 20941
This theorem is referenced by:  fgabs  23382  fgtr  23393  fmss  23449
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