| Step | Hyp | Ref
| Expression |
| 1 | | metust.1 |
. . . . . . 7
⊢ 𝐹 = ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) |
| 2 | 1 | metustel 24563 |
. . . . . 6
⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝑥 ∈ 𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝑥 = (◡𝐷 “ (0[,)𝑎)))) |
| 3 | | simpr 484 |
. . . . . . . . 9
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 = (◡𝐷 “ (0[,)𝑎))) → 𝑥 = (◡𝐷 “ (0[,)𝑎))) |
| 4 | | cnvimass 6100 |
. . . . . . . . . 10
⊢ (◡𝐷 “ (0[,)𝑎)) ⊆ dom 𝐷 |
| 5 | | psmetf 24316 |
. . . . . . . . . . . 12
⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) |
| 6 | 5 | fdmd 6746 |
. . . . . . . . . . 11
⊢ (𝐷 ∈ (PsMet‘𝑋) → dom 𝐷 = (𝑋 × 𝑋)) |
| 7 | 6 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 = (◡𝐷 “ (0[,)𝑎))) → dom 𝐷 = (𝑋 × 𝑋)) |
| 8 | 4, 7 | sseqtrid 4026 |
. . . . . . . . 9
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 = (◡𝐷 “ (0[,)𝑎))) → (◡𝐷 “ (0[,)𝑎)) ⊆ (𝑋 × 𝑋)) |
| 9 | 3, 8 | eqsstrd 4018 |
. . . . . . . 8
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 = (◡𝐷 “ (0[,)𝑎))) → 𝑥 ⊆ (𝑋 × 𝑋)) |
| 10 | 9 | ex 412 |
. . . . . . 7
⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝑥 = (◡𝐷 “ (0[,)𝑎)) → 𝑥 ⊆ (𝑋 × 𝑋))) |
| 11 | 10 | rexlimdvw 3160 |
. . . . . 6
⊢ (𝐷 ∈ (PsMet‘𝑋) → (∃𝑎 ∈ ℝ+
𝑥 = (◡𝐷 “ (0[,)𝑎)) → 𝑥 ⊆ (𝑋 × 𝑋))) |
| 12 | 2, 11 | sylbid 240 |
. . . . 5
⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝑥 ∈ 𝐹 → 𝑥 ⊆ (𝑋 × 𝑋))) |
| 13 | 12 | ralrimiv 3145 |
. . . 4
⊢ (𝐷 ∈ (PsMet‘𝑋) → ∀𝑥 ∈ 𝐹 𝑥 ⊆ (𝑋 × 𝑋)) |
| 14 | | pwssb 5101 |
. . . 4
⊢ (𝐹 ⊆ 𝒫 (𝑋 × 𝑋) ↔ ∀𝑥 ∈ 𝐹 𝑥 ⊆ (𝑋 × 𝑋)) |
| 15 | 13, 14 | sylibr 234 |
. . 3
⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐹 ⊆ 𝒫 (𝑋 × 𝑋)) |
| 16 | 15 | adantl 481 |
. 2
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝐹 ⊆ 𝒫 (𝑋 × 𝑋)) |
| 17 | | cnvexg 7946 |
. . . . . . 7
⊢ (𝐷 ∈ (PsMet‘𝑋) → ◡𝐷 ∈ V) |
| 18 | | imaexg 7935 |
. . . . . . 7
⊢ (◡𝐷 ∈ V → (◡𝐷 “ (0[,)1)) ∈ V) |
| 19 | | elisset 2823 |
. . . . . . 7
⊢ ((◡𝐷 “ (0[,)1)) ∈ V →
∃𝑥 𝑥 = (◡𝐷 “ (0[,)1))) |
| 20 | | 1rp 13038 |
. . . . . . . . 9
⊢ 1 ∈
ℝ+ |
| 21 | | oveq2 7439 |
. . . . . . . . . . 11
⊢ (𝑎 = 1 → (0[,)𝑎) = (0[,)1)) |
| 22 | 21 | imaeq2d 6078 |
. . . . . . . . . 10
⊢ (𝑎 = 1 → (◡𝐷 “ (0[,)𝑎)) = (◡𝐷 “ (0[,)1))) |
| 23 | 22 | rspceeqv 3645 |
. . . . . . . . 9
⊢ ((1
∈ ℝ+ ∧ 𝑥 = (◡𝐷 “ (0[,)1))) → ∃𝑎 ∈ ℝ+
𝑥 = (◡𝐷 “ (0[,)𝑎))) |
| 24 | 20, 23 | mpan 690 |
. . . . . . . 8
⊢ (𝑥 = (◡𝐷 “ (0[,)1)) → ∃𝑎 ∈ ℝ+
𝑥 = (◡𝐷 “ (0[,)𝑎))) |
| 25 | 24 | eximi 1835 |
. . . . . . 7
⊢
(∃𝑥 𝑥 = (◡𝐷 “ (0[,)1)) → ∃𝑥∃𝑎 ∈ ℝ+ 𝑥 = (◡𝐷 “ (0[,)𝑎))) |
| 26 | 17, 18, 19, 25 | 4syl 19 |
. . . . . 6
⊢ (𝐷 ∈ (PsMet‘𝑋) → ∃𝑥∃𝑎 ∈ ℝ+ 𝑥 = (◡𝐷 “ (0[,)𝑎))) |
| 27 | 2 | exbidv 1921 |
. . . . . 6
⊢ (𝐷 ∈ (PsMet‘𝑋) → (∃𝑥 𝑥 ∈ 𝐹 ↔ ∃𝑥∃𝑎 ∈ ℝ+ 𝑥 = (◡𝐷 “ (0[,)𝑎)))) |
| 28 | 26, 27 | mpbird 257 |
. . . . 5
⊢ (𝐷 ∈ (PsMet‘𝑋) → ∃𝑥 𝑥 ∈ 𝐹) |
| 29 | 28 | adantl 481 |
. . . 4
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ∃𝑥 𝑥 ∈ 𝐹) |
| 30 | | n0 4353 |
. . . 4
⊢ (𝐹 ≠ ∅ ↔
∃𝑥 𝑥 ∈ 𝐹) |
| 31 | 29, 30 | sylibr 234 |
. . 3
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝐹 ≠ ∅) |
| 32 | 1 | metustid 24567 |
. . . . . . 7
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝐹) → ( I ↾ 𝑋) ⊆ 𝑥) |
| 33 | 32 | adantll 714 |
. . . . . 6
⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑥 ∈ 𝐹) → ( I ↾ 𝑋) ⊆ 𝑥) |
| 34 | | n0 4353 |
. . . . . . . . . 10
⊢ (𝑋 ≠ ∅ ↔
∃𝑝 𝑝 ∈ 𝑋) |
| 35 | 34 | biimpi 216 |
. . . . . . . . 9
⊢ (𝑋 ≠ ∅ →
∃𝑝 𝑝 ∈ 𝑋) |
| 36 | 35 | adantr 480 |
. . . . . . . 8
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ∃𝑝 𝑝 ∈ 𝑋) |
| 37 | | opelidres 6009 |
. . . . . . . . . . 11
⊢ (𝑝 ∈ 𝑋 → (〈𝑝, 𝑝〉 ∈ ( I ↾ 𝑋) ↔ 𝑝 ∈ 𝑋)) |
| 38 | 37 | ibir 268 |
. . . . . . . . . 10
⊢ (𝑝 ∈ 𝑋 → 〈𝑝, 𝑝〉 ∈ ( I ↾ 𝑋)) |
| 39 | 38 | ne0d 4342 |
. . . . . . . . 9
⊢ (𝑝 ∈ 𝑋 → ( I ↾ 𝑋) ≠ ∅) |
| 40 | 39 | exlimiv 1930 |
. . . . . . . 8
⊢
(∃𝑝 𝑝 ∈ 𝑋 → ( I ↾ 𝑋) ≠ ∅) |
| 41 | 36, 40 | syl 17 |
. . . . . . 7
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ( I ↾ 𝑋) ≠ ∅) |
| 42 | 41 | adantr 480 |
. . . . . 6
⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑥 ∈ 𝐹) → ( I ↾ 𝑋) ≠ ∅) |
| 43 | | ssn0 4404 |
. . . . . 6
⊢ ((( I
↾ 𝑋) ⊆ 𝑥 ∧ ( I ↾ 𝑋) ≠ ∅) → 𝑥 ≠ ∅) |
| 44 | 33, 42, 43 | syl2anc 584 |
. . . . 5
⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑥 ∈ 𝐹) → 𝑥 ≠ ∅) |
| 45 | 44 | nelrdva 3711 |
. . . 4
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ¬ ∅ ∈
𝐹) |
| 46 | | df-nel 3047 |
. . . 4
⊢ (∅
∉ 𝐹 ↔ ¬
∅ ∈ 𝐹) |
| 47 | 45, 46 | sylibr 234 |
. . 3
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ∅ ∉ 𝐹) |
| 48 | | dfss2 3969 |
. . . . . . . . 9
⊢ (𝑥 ⊆ 𝑦 ↔ (𝑥 ∩ 𝑦) = 𝑥) |
| 49 | 48 | biimpi 216 |
. . . . . . . 8
⊢ (𝑥 ⊆ 𝑦 → (𝑥 ∩ 𝑦) = 𝑥) |
| 50 | 49 | adantl 481 |
. . . . . . 7
⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) ∧ 𝑥 ⊆ 𝑦) → (𝑥 ∩ 𝑦) = 𝑥) |
| 51 | | simplrl 777 |
. . . . . . 7
⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) ∧ 𝑥 ⊆ 𝑦) → 𝑥 ∈ 𝐹) |
| 52 | 50, 51 | eqeltrd 2841 |
. . . . . 6
⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) ∧ 𝑥 ⊆ 𝑦) → (𝑥 ∩ 𝑦) ∈ 𝐹) |
| 53 | | sseqin2 4223 |
. . . . . . . . 9
⊢ (𝑦 ⊆ 𝑥 ↔ (𝑥 ∩ 𝑦) = 𝑦) |
| 54 | 53 | biimpi 216 |
. . . . . . . 8
⊢ (𝑦 ⊆ 𝑥 → (𝑥 ∩ 𝑦) = 𝑦) |
| 55 | 54 | adantl 481 |
. . . . . . 7
⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) ∧ 𝑦 ⊆ 𝑥) → (𝑥 ∩ 𝑦) = 𝑦) |
| 56 | | simplrr 778 |
. . . . . . 7
⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ 𝐹) |
| 57 | 55, 56 | eqeltrd 2841 |
. . . . . 6
⊢ ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) ∧ 𝑦 ⊆ 𝑥) → (𝑥 ∩ 𝑦) ∈ 𝐹) |
| 58 | | simplr 769 |
. . . . . . 7
⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → 𝐷 ∈ (PsMet‘𝑋)) |
| 59 | | simprl 771 |
. . . . . . 7
⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → 𝑥 ∈ 𝐹) |
| 60 | | simprr 773 |
. . . . . . 7
⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → 𝑦 ∈ 𝐹) |
| 61 | 1 | metustto 24566 |
. . . . . . 7
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)) |
| 62 | 58, 59, 60, 61 | syl3anc 1373 |
. . . . . 6
⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)) |
| 63 | 52, 57, 62 | mpjaodan 961 |
. . . . 5
⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑥 ∩ 𝑦) ∈ 𝐹) |
| 64 | | ssidd 4007 |
. . . . 5
⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)) |
| 65 | | sseq1 4009 |
. . . . . 6
⊢ (𝑧 = (𝑥 ∩ 𝑦) → (𝑧 ⊆ (𝑥 ∩ 𝑦) ↔ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦))) |
| 66 | 65 | rspcev 3622 |
. . . . 5
⊢ (((𝑥 ∩ 𝑦) ∈ 𝐹 ∧ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)) → ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦)) |
| 67 | 63, 64, 66 | syl2anc 584 |
. . . 4
⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦)) |
| 68 | 67 | ralrimivva 3202 |
. . 3
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦)) |
| 69 | 31, 47, 68 | 3jca 1129 |
. 2
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦))) |
| 70 | | elfvex 6944 |
. . . . 5
⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V) |
| 71 | 70 | adantl 481 |
. . . 4
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝑋 ∈ V) |
| 72 | 71, 71 | xpexd 7771 |
. . 3
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑋 × 𝑋) ∈ V) |
| 73 | | isfbas2 23843 |
. . 3
⊢ ((𝑋 × 𝑋) ∈ V → (𝐹 ∈ (fBas‘(𝑋 × 𝑋)) ↔ (𝐹 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦))))) |
| 74 | 72, 73 | syl 17 |
. 2
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝐹 ∈ (fBas‘(𝑋 × 𝑋)) ↔ (𝐹 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦))))) |
| 75 | 16, 69, 74 | mpbir2and 713 |
1
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝐹 ∈ (fBas‘(𝑋 × 𝑋))) |