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Theorem fgss 23897
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 4065 . . . . 5 (𝐹𝐺 → (∃𝑥𝐹 𝑥𝑡 → ∃𝑥𝐺 𝑥𝑡))
21anim2d 612 . . . 4 (𝐹𝐺 → ((𝑡𝑋 ∧ ∃𝑥𝐹 𝑥𝑡) → (𝑡𝑋 ∧ ∃𝑥𝐺 𝑥𝑡)))
323ad2ant3 1134 . . 3 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋) ∧ 𝐹𝐺) → ((𝑡𝑋 ∧ ∃𝑥𝐹 𝑥𝑡) → (𝑡𝑋 ∧ ∃𝑥𝐺 𝑥𝑡)))
4 elfg 23895 . . . 4 (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡𝑋 ∧ ∃𝑥𝐹 𝑥𝑡)))
543ad2ant1 1132 . . 3 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋) ∧ 𝐹𝐺) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡𝑋 ∧ ∃𝑥𝐹 𝑥𝑡)))
6 elfg 23895 . . . 4 (𝐺 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐺) ↔ (𝑡𝑋 ∧ ∃𝑥𝐺 𝑥𝑡)))
763ad2ant2 1133 . . 3 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋) ∧ 𝐹𝐺) → (𝑡 ∈ (𝑋filGen𝐺) ↔ (𝑡𝑋 ∧ ∃𝑥𝐺 𝑥𝑡)))
83, 5, 73imtr4d 294 . 2 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋) ∧ 𝐹𝐺) → (𝑡 ∈ (𝑋filGen𝐹) → 𝑡 ∈ (𝑋filGen𝐺)))
98ssrdv 4001 1 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋) ∧ 𝐹𝐺) → (𝑋filGen𝐹) ⊆ (𝑋filGen𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086  wcel 2106  wrex 3068  wss 3963  cfv 6563  (class class class)co 7431  fBascfbas 21370  filGencfg 21371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-fg 21380
This theorem is referenced by:  fgabs  23903  fgtr  23914  fmss  23970
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