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Theorem cfilss 25305
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 770 . 2 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) ∧ (𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺)) → 𝐺 ∈ (Fil‘𝑋))
2 simprr 772 . . 3 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) ∧ (𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺)) → 𝐹𝐺)
3 iscfil 25300 . . . . 5 (𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (CauFil‘𝐷) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐹 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))))
43simplbda 499 . . . 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 4052 . . . 4 (𝐹𝐺 → (∃𝑦𝐹 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥) → ∃𝑦𝐺 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
76ralimdv 3168 . . 3 (𝐹𝐺 → (∀𝑥 ∈ ℝ+𝑦𝐹 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥) → ∀𝑥 ∈ ℝ+𝑦𝐺 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
82, 5, 7sylc 65 . 2 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) ∧ (𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺)) → ∀𝑥 ∈ ℝ+𝑦𝐺 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))
9 iscfil 25300 . . 3 (𝐷 ∈ (∞Met‘𝑋) → (𝐺 ∈ (CauFil‘𝐷) ↔ (𝐺 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐺 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))))
109ad2antrr 726 . 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 713 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 206  wa 395  wcel 2107  wral 3060  wrex 3069  wss 3950   × cxp 5682  cima 5687  cfv 6560  (class class class)co 7432  0cc0 11156  +crp 13035  [,)cico 13390  ∞Metcxmet 21350  Filcfil 23854  CauFilccfil 25287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-map 8869  df-xr 11300  df-xmet 21358  df-cfil 25290
This theorem is referenced by: (None)
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