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Theorem cfilss 22876
 Description: A filter finer than a Cauchy filter is Cauchy. (Contributed by Mario Carneiro, 13-Oct-2015.)
Assertion
Ref Expression
cfilss (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) ∧ (𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺)) → 𝐺 ∈ (CauFil‘𝐷))

Proof of Theorem cfilss
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simprl 790 . 2 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) ∧ (𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺)) → 𝐺 ∈ (Fil‘𝑋))
2 simprr 792 . . 3 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) ∧ (𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺)) → 𝐹𝐺)
3 iscfil 22871 . . . . 5 (𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (CauFil‘𝐷) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐹 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))))
43simplbda 652 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) → ∀𝑥 ∈ ℝ+𝑦𝐹 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))
54adantr 480 . . 3 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) ∧ (𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺)) → ∀𝑥 ∈ ℝ+𝑦𝐹 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))
6 ssrexv 3630 . . . 4 (𝐹𝐺 → (∃𝑦𝐹 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥) → ∃𝑦𝐺 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
76ralimdv 2946 . . 3 (𝐹𝐺 → (∀𝑥 ∈ ℝ+𝑦𝐹 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥) → ∀𝑥 ∈ ℝ+𝑦𝐺 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
82, 5, 7sylc 63 . 2 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) ∧ (𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺)) → ∀𝑥 ∈ ℝ+𝑦𝐺 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))
9 iscfil 22871 . . 3 (𝐷 ∈ (∞Met‘𝑋) → (𝐺 ∈ (CauFil‘𝐷) ↔ (𝐺 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐺 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))))
109ad2antrr 758 . 2 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) ∧ (𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺)) → (𝐺 ∈ (CauFil‘𝐷) ↔ (𝐺 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐺 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))))
111, 8, 10mpbir2and 959 1 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) ∧ (𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺)) → 𝐺 ∈ (CauFil‘𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540   × cxp 5036   “ cima 5041  ‘cfv 5804  (class class class)co 6549  0cc0 9815  ℝ+crp 11708  [,)cico 12048  ∞Metcxmt 19552  Filcfil 21459  CauFilccfil 22858 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-xr 9957  df-xmet 19560  df-cfil 22861 This theorem is referenced by: (None)
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