Theorem cfilss 25323

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) ∧ 𝐹 ⊆ 𝐺)) → 𝐺 ∈ (CauFil‘𝐷))

Proof of Theorem cfilss

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprl 770	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) ∧ (𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺)) → 𝐺 ∈ (Fil‘𝑋))
2		simprr 772	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) ∧ (𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺)) → 𝐹 ⊆ 𝐺)
3		iscfil 25318	. . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (CauFil‘𝐷) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐹 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))))
4	3	simplbda 499	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) → ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐹 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))
5	4	adantr 480	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) ∧ (𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺)) → ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐹 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))
6		ssrexv 4078	. . . 4 ⊢ (𝐹 ⊆ 𝐺 → (∃𝑦 ∈ 𝐹 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥) → ∃𝑦 ∈ 𝐺 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
7	6	ralimdv 3175	. . 3 ⊢ (𝐹 ⊆ 𝐺 → (∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐹 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥) → ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐺 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
8	2, 5, 7	sylc 65	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) ∧ (𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺)) → ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐺 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))
9		iscfil 25318	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐺 ∈ (CauFil‘𝐷) ↔ (𝐺 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐺 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))))
10	9	ad2antrr 725	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) ∧ (𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺)) → (𝐺 ∈ (CauFil‘𝐷) ↔ (𝐺 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐺 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))))
11	1, 8, 10	mpbir2and 712	1 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) ∧ (𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺)) → 𝐺 ∈ (CauFil‘𝐷))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076   ⊆ wss 3976   × cxp 5698   “ cima 5703  ‘cfv 6573  (class class class)co 7448  0cc0 11184  ℝ⁺crp 13057  [,)cico 13409  ∞Metcxmet 21372  Filcfil 23874  CauFilccfil 25305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-map 8886  df-xr 11328  df-xmet 21380  df-cfil 25308

This theorem is referenced by: (None)