Theorem cfilss 25333

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 780	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) ∧ (𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺)) → 𝐺 ∈ (Fil‘𝑋))
2		simprr 782	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) ∧ (𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺)) → 𝐹 ⊆ 𝐺)
3		iscfil 25328	. . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (CauFil‘𝐷) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐹 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))))
4	3	simplbda 503	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) → ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐹 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))
5	4	adantr 484	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) ∧ (𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺)) → ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐹 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))
6		ssrexv 4007	. . . 4 ⊢ (𝐹 ⊆ 𝐺 → (∃𝑦 ∈ 𝐹 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥) → ∃𝑦 ∈ 𝐺 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))
7	6	ralimdv 3177	. . 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 25328	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐺 ∈ (CauFil‘𝐷) ↔ (𝐺 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐺 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))))
10	9	ad2antrr 736	. 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 723	1 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) ∧ (𝐺 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺)) → 𝐺 ∈ (CauFil‘𝐷))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∈ wcel 2143  ∀wral 3077  ∃wrex 3087   ⊆ wss 3905   × cxp 5646   “ cima 5651  ‘cfv 6522  (class class class)co 7397  0cc0 11074  ℝ⁺crp 12994  [,)cico 13352  ∞Metcxmet 21410  Filcfil 23906  CauFilccfil 25315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719  ax-cnex 11130  ax-resscn 11131

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-sbc 3746  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-fv 6530  df-ov 7400  df-oprab 7401  df-mpo 7402  df-map 8811  df-xr 11221  df-xmet 21418  df-cfil 25318

This theorem is referenced by: (None)