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Mirrors > Home > MPE Home > Th. List > cfilfval | Structured version Visualization version GIF version |
Description: The set of Cauchy filters on a metric space. (Contributed by Mario Carneiro, 13-Oct-2015.) |
Ref | Expression |
---|---|
cfilfval | ⊢ (𝐷 ∈ (∞Met‘𝑋) → (CauFil‘𝐷) = {𝑓 ∈ (Fil‘𝑋) ∣ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvssunirn 6522 | . . . 4 ⊢ (∞Met‘𝑋) ⊆ ∪ ran ∞Met | |
2 | 1 | sseli 3848 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ ∪ ran ∞Met) |
3 | dmeq 5616 | . . . . . . 7 ⊢ (𝑑 = 𝐷 → dom 𝑑 = dom 𝐷) | |
4 | 3 | dmeqd 5618 | . . . . . 6 ⊢ (𝑑 = 𝐷 → dom dom 𝑑 = dom dom 𝐷) |
5 | 4 | fveq2d 6497 | . . . . 5 ⊢ (𝑑 = 𝐷 → (Fil‘dom dom 𝑑) = (Fil‘dom dom 𝐷)) |
6 | imaeq1 5759 | . . . . . . . 8 ⊢ (𝑑 = 𝐷 → (𝑑 “ (𝑦 × 𝑦)) = (𝐷 “ (𝑦 × 𝑦))) | |
7 | 6 | sseq1d 3882 | . . . . . . 7 ⊢ (𝑑 = 𝐷 → ((𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥) ↔ (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) |
8 | 7 | rexbidv 3236 | . . . . . 6 ⊢ (𝑑 = 𝐷 → (∃𝑦 ∈ 𝑓 (𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥) ↔ ∃𝑦 ∈ 𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) |
9 | 8 | ralbidv 3141 | . . . . 5 ⊢ (𝑑 = 𝐷 → (∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑓 (𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))) |
10 | 5, 9 | rabeqbidv 3402 | . . . 4 ⊢ (𝑑 = 𝐷 → {𝑓 ∈ (Fil‘dom dom 𝑑) ∣ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑓 (𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)} = {𝑓 ∈ (Fil‘dom dom 𝐷) ∣ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)}) |
11 | df-cfil 23555 | . . . 4 ⊢ CauFil = (𝑑 ∈ ∪ ran ∞Met ↦ {𝑓 ∈ (Fil‘dom dom 𝑑) ∣ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑓 (𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)}) | |
12 | fvex 6506 | . . . . 5 ⊢ (Fil‘dom dom 𝐷) ∈ V | |
13 | 12 | rabex 5085 | . . . 4 ⊢ {𝑓 ∈ (Fil‘dom dom 𝐷) ∣ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)} ∈ V |
14 | 10, 11, 13 | fvmpt 6589 | . . 3 ⊢ (𝐷 ∈ ∪ ran ∞Met → (CauFil‘𝐷) = {𝑓 ∈ (Fil‘dom dom 𝐷) ∣ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)}) |
15 | 2, 14 | syl 17 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (CauFil‘𝐷) = {𝑓 ∈ (Fil‘dom dom 𝐷) ∣ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)}) |
16 | xmetdmdm 22642 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = dom dom 𝐷) | |
17 | 16 | fveq2d 6497 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (Fil‘𝑋) = (Fil‘dom dom 𝐷)) |
18 | 17 | rabeqdv 3401 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → {𝑓 ∈ (Fil‘𝑋) ∣ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)} = {𝑓 ∈ (Fil‘dom dom 𝐷) ∣ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)}) |
19 | 15, 18 | eqtr4d 2811 | 1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (CauFil‘𝐷) = {𝑓 ∈ (Fil‘𝑋) ∣ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 ∀wral 3082 ∃wrex 3083 {crab 3086 ⊆ wss 3823 ∪ cuni 4706 × cxp 5399 dom cdm 5401 ran crn 5402 “ cima 5404 ‘cfv 6182 (class class class)co 6970 0cc0 10329 ℝ+crp 12198 [,)cico 12550 ∞Metcxmet 20226 Filcfil 22151 CauFilccfil 23552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10385 ax-resscn 10386 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-sbc 3676 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5306 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-fv 6190 df-ov 6973 df-oprab 6974 df-mpo 6975 df-map 8202 df-xr 10472 df-xmet 20234 df-cfil 23555 |
This theorem is referenced by: iscfil 23565 |
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