Step | Hyp | Ref
| Expression |
1 | | simpr 476 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ (SMblFn‘𝑆)) |
2 | | df-smblfn 39587 |
. . . . . . . . . 10
⊢ SMblFn =
(𝑠 ∈ SAlg ↦
{𝑓 ∈ (ℝ
↑pm ∪ 𝑠) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑠 ↾t dom 𝑓)}) |
3 | 2 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → SMblFn = (𝑠 ∈ SAlg ↦ {𝑓 ∈ (ℝ
↑pm ∪ 𝑠) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑠 ↾t dom 𝑓)})) |
4 | | unieq 4380 |
. . . . . . . . . . . . 13
⊢ (𝑠 = 𝑆 → ∪ 𝑠 = ∪
𝑆) |
5 | 4 | oveq2d 6565 |
. . . . . . . . . . . 12
⊢ (𝑠 = 𝑆 → (ℝ ↑pm
∪ 𝑠) = (ℝ ↑pm ∪ 𝑆)) |
6 | 5 | rabeqd 38304 |
. . . . . . . . . . 11
⊢ (𝑠 = 𝑆 → {𝑓 ∈ (ℝ ↑pm
∪ 𝑠) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑠 ↾t dom 𝑓)} = {𝑓 ∈ (ℝ ↑pm
∪ 𝑆) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑠 ↾t dom 𝑓)}) |
7 | | oveq1 6556 |
. . . . . . . . . . . . . 14
⊢ (𝑠 = 𝑆 → (𝑠 ↾t dom 𝑓) = (𝑆 ↾t dom 𝑓)) |
8 | 7 | eleq2d 2673 |
. . . . . . . . . . . . 13
⊢ (𝑠 = 𝑆 → ((◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑠 ↾t dom 𝑓) ↔ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝑓))) |
9 | 8 | ralbidv 2969 |
. . . . . . . . . . . 12
⊢ (𝑠 = 𝑆 → (∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑠 ↾t dom 𝑓) ↔ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝑓))) |
10 | 9 | rabbidv 3164 |
. . . . . . . . . . 11
⊢ (𝑠 = 𝑆 → {𝑓 ∈ (ℝ ↑pm
∪ 𝑆) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑠 ↾t dom 𝑓)} = {𝑓 ∈ (ℝ ↑pm
∪ 𝑆) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝑓)}) |
11 | 6, 10 | eqtrd 2644 |
. . . . . . . . . 10
⊢ (𝑠 = 𝑆 → {𝑓 ∈ (ℝ ↑pm
∪ 𝑠) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑠 ↾t dom 𝑓)} = {𝑓 ∈ (ℝ ↑pm
∪ 𝑆) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝑓)}) |
12 | 11 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑠 = 𝑆) → {𝑓 ∈ (ℝ ↑pm
∪ 𝑠) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑠 ↾t dom 𝑓)} = {𝑓 ∈ (ℝ ↑pm
∪ 𝑆) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝑓)}) |
13 | | issmflem.s |
. . . . . . . . 9
⊢ (𝜑 → 𝑆 ∈ SAlg) |
14 | | ovex 6577 |
. . . . . . . . . . 11
⊢ (ℝ
↑pm ∪ 𝑆) ∈ V |
15 | 14 | rabex 4740 |
. . . . . . . . . 10
⊢ {𝑓 ∈ (ℝ
↑pm ∪ 𝑆) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝑓)} ∈ V |
16 | 15 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → {𝑓 ∈ (ℝ ↑pm
∪ 𝑆) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝑓)} ∈ V) |
17 | 3, 12, 13, 16 | fvmptd 6197 |
. . . . . . . 8
⊢ (𝜑 → (SMblFn‘𝑆) = {𝑓 ∈ (ℝ ↑pm
∪ 𝑆) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝑓)}) |
18 | 17 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (SMblFn‘𝑆) = {𝑓 ∈ (ℝ ↑pm
∪ 𝑆) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝑓)}) |
19 | 1, 18 | eleqtrd 2690 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ {𝑓 ∈ (ℝ ↑pm
∪ 𝑆) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝑓)}) |
20 | | elrabi 3328 |
. . . . . 6
⊢ (𝐹 ∈ {𝑓 ∈ (ℝ ↑pm
∪ 𝑆) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝑓)} → 𝐹 ∈ (ℝ ↑pm
∪ 𝑆)) |
21 | 19, 20 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ (ℝ ↑pm
∪ 𝑆)) |
22 | | issmflem.d |
. . . . . . 7
⊢ 𝐷 = dom 𝐹 |
23 | | elpmi2 38413 |
. . . . . . 7
⊢ (𝐹 ∈ (ℝ
↑pm ∪ 𝑆) → dom 𝐹 ⊆ ∪ 𝑆) |
24 | 22, 23 | syl5eqss 3612 |
. . . . . 6
⊢ (𝐹 ∈ (ℝ
↑pm ∪ 𝑆) → 𝐷 ⊆ ∪ 𝑆) |
25 | 24 | adantl 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝐹 ∈ (ℝ ↑pm
∪ 𝑆)) → 𝐷 ⊆ ∪ 𝑆) |
26 | 21, 25 | syldan 486 |
. . . 4
⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 ⊆ ∪ 𝑆) |
27 | | elpmi 7762 |
. . . . . . 7
⊢ (𝐹 ∈ (ℝ
↑pm ∪ 𝑆) → (𝐹:dom 𝐹⟶ℝ ∧ dom 𝐹 ⊆ ∪ 𝑆)) |
28 | 21, 27 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (𝐹:dom 𝐹⟶ℝ ∧ dom 𝐹 ⊆ ∪ 𝑆)) |
29 | 28 | simpld 474 |
. . . . 5
⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:dom 𝐹⟶ℝ) |
30 | 22 | feq2i 5950 |
. . . . . 6
⊢ (𝐹:𝐷⟶ℝ ↔ 𝐹:dom 𝐹⟶ℝ) |
31 | 30 | a1i 11 |
. . . . 5
⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (𝐹:𝐷⟶ℝ ↔ 𝐹:dom 𝐹⟶ℝ)) |
32 | 29, 31 | mpbird 246 |
. . . 4
⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:𝐷⟶ℝ) |
33 | | cnveq 5218 |
. . . . . . . . . . . . . 14
⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹) |
34 | 33 | imaeq1d 5384 |
. . . . . . . . . . . . 13
⊢ (𝑓 = 𝐹 → (◡𝑓 “ (-∞(,)𝑎)) = (◡𝐹 “ (-∞(,)𝑎))) |
35 | | dmeq 5246 |
. . . . . . . . . . . . . 14
⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹) |
36 | 35 | oveq2d 6565 |
. . . . . . . . . . . . 13
⊢ (𝑓 = 𝐹 → (𝑆 ↾t dom 𝑓) = (𝑆 ↾t dom 𝐹)) |
37 | 34, 36 | eleq12d 2682 |
. . . . . . . . . . . 12
⊢ (𝑓 = 𝐹 → ((◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝑓) ↔ (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹))) |
38 | 37 | ralbidv 2969 |
. . . . . . . . . . 11
⊢ (𝑓 = 𝐹 → (∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝑓) ↔ ∀𝑎 ∈ ℝ (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹))) |
39 | 38 | elrab 3331 |
. . . . . . . . . 10
⊢ (𝐹 ∈ {𝑓 ∈ (ℝ ↑pm
∪ 𝑆) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝑓)} ↔ (𝐹 ∈ (ℝ ↑pm
∪ 𝑆) ∧ ∀𝑎 ∈ ℝ (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹))) |
40 | 39 | simprbi 479 |
. . . . . . . . 9
⊢ (𝐹 ∈ {𝑓 ∈ (ℝ ↑pm
∪ 𝑆) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝑓)} → ∀𝑎 ∈ ℝ (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹)) |
41 | 19, 40 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑎 ∈ ℝ (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹)) |
42 | 41 | adantr 480 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → ∀𝑎 ∈ ℝ (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹)) |
43 | | simpr 476 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ) |
44 | | rspa 2914 |
. . . . . . 7
⊢
((∀𝑎 ∈
ℝ (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹) ∧ 𝑎 ∈ ℝ) → (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹)) |
45 | 42, 43, 44 | syl2anc 691 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹)) |
46 | 32 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → 𝐹:𝐷⟶ℝ) |
47 | | simpl 472 |
. . . . . . . . . 10
⊢ ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → 𝐹:𝐷⟶ℝ) |
48 | | simpr 476 |
. . . . . . . . . . 11
⊢ ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ) |
49 | 48 | rexrd 9968 |
. . . . . . . . . 10
⊢ ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ*) |
50 | 47, 49 | preimaioomnf 39606 |
. . . . . . . . 9
⊢ ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → (◡𝐹 “ (-∞(,)𝑎)) = {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎}) |
51 | 50 | eqcomd 2616 |
. . . . . . . 8
⊢ ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} = (◡𝐹 “ (-∞(,)𝑎))) |
52 | 46, 43, 51 | syl2anc 691 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} = (◡𝐹 “ (-∞(,)𝑎))) |
53 | 22 | oveq2i 6560 |
. . . . . . . 8
⊢ (𝑆 ↾t 𝐷) = (𝑆 ↾t dom 𝐹) |
54 | 53 | a1i 11 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → (𝑆 ↾t 𝐷) = (𝑆 ↾t dom 𝐹)) |
55 | 52, 54 | eleq12d 2682 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → ({𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷) ↔ (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹))) |
56 | 45, 55 | mpbird 246 |
. . . . 5
⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)) |
57 | 56 | ralrimiva 2949 |
. . . 4
⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)) |
58 | 26, 32, 57 | 3jca 1235 |
. . 3
⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))) |
59 | 58 | ex 449 |
. 2
⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)))) |
60 | | reex 9906 |
. . . . . . . . 9
⊢ ℝ
∈ V |
61 | 60 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ)) → ℝ ∈
V) |
62 | 13 | uniexd 38310 |
. . . . . . . . 9
⊢ (𝜑 → ∪ 𝑆
∈ V) |
63 | 62 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ)) → ∪ 𝑆
∈ V) |
64 | | simprr 792 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ)) → 𝐹:𝐷⟶ℝ) |
65 | | fssxp 5973 |
. . . . . . . . . . . 12
⊢ (𝐹:𝐷⟶ℝ → 𝐹 ⊆ (𝐷 × ℝ)) |
66 | 65 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝐷 ⊆ ∪ 𝑆
∧ 𝐹:𝐷⟶ℝ) → 𝐹 ⊆ (𝐷 × ℝ)) |
67 | | xpss1 5151 |
. . . . . . . . . . . 12
⊢ (𝐷 ⊆ ∪ 𝑆
→ (𝐷 × ℝ)
⊆ (∪ 𝑆 × ℝ)) |
68 | 67 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝐷 ⊆ ∪ 𝑆
∧ 𝐹:𝐷⟶ℝ) → (𝐷 × ℝ) ⊆ (∪ 𝑆
× ℝ)) |
69 | 66, 68 | sstrd 3578 |
. . . . . . . . . 10
⊢ ((𝐷 ⊆ ∪ 𝑆
∧ 𝐹:𝐷⟶ℝ) → 𝐹 ⊆ (∪ 𝑆 ×
ℝ)) |
70 | 69 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ)) → 𝐹 ⊆ (∪ 𝑆 ×
ℝ)) |
71 | | dmss 5245 |
. . . . . . . . . . . 12
⊢ (𝐹 ⊆ (∪ 𝑆
× ℝ) → dom 𝐹 ⊆ dom (∪
𝑆 ×
ℝ)) |
72 | | dmxpss 5484 |
. . . . . . . . . . . . 13
⊢ dom
(∪ 𝑆 × ℝ) ⊆ ∪ 𝑆 |
73 | 72 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝐹 ⊆ (∪ 𝑆
× ℝ) → dom (∪ 𝑆 × ℝ) ⊆ ∪ 𝑆) |
74 | 71, 73 | sstrd 3578 |
. . . . . . . . . . 11
⊢ (𝐹 ⊆ (∪ 𝑆
× ℝ) → dom 𝐹 ⊆ ∪ 𝑆) |
75 | 74 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐹 ⊆ (∪ 𝑆 × ℝ)) → dom
𝐹 ⊆ ∪ 𝑆) |
76 | 22, 75 | syl5eqss 3612 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐹 ⊆ (∪ 𝑆 × ℝ)) → 𝐷 ⊆ ∪ 𝑆) |
77 | 70, 76 | syldan 486 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ)) → 𝐷 ⊆ ∪ 𝑆) |
78 | | elpm2r 7761 |
. . . . . . . 8
⊢
(((ℝ ∈ V ∧ ∪ 𝑆 ∈ V) ∧ (𝐹:𝐷⟶ℝ ∧ 𝐷 ⊆ ∪ 𝑆)) → 𝐹 ∈ (ℝ ↑pm
∪ 𝑆)) |
79 | 61, 63, 64, 77, 78 | syl22anc 1319 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ)) → 𝐹 ∈ (ℝ ↑pm
∪ 𝑆)) |
80 | 79 | 3adantr3 1215 |
. . . . . 6
⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))) → 𝐹 ∈ (ℝ ↑pm
∪ 𝑆)) |
81 | 22 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → 𝐷 = dom 𝐹) |
82 | 81 | oveq2d 6565 |
. . . . . . . . . . . 12
⊢ ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → (𝑆 ↾t 𝐷) = (𝑆 ↾t dom 𝐹)) |
83 | 51, 82 | eleq12d 2682 |
. . . . . . . . . . 11
⊢ ((𝐹:𝐷⟶ℝ ∧ 𝑎 ∈ ℝ) → ({𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷) ↔ (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹))) |
84 | 83 | ralbidva 2968 |
. . . . . . . . . 10
⊢ (𝐹:𝐷⟶ℝ → (∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷) ↔ ∀𝑎 ∈ ℝ (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹))) |
85 | 84 | biimpd 218 |
. . . . . . . . 9
⊢ (𝐹:𝐷⟶ℝ → (∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷) → ∀𝑎 ∈ ℝ (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹))) |
86 | 85 | imp 444 |
. . . . . . . 8
⊢ ((𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)) → ∀𝑎 ∈ ℝ (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹)) |
87 | 86 | adantl 481 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))) → ∀𝑎 ∈ ℝ (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹)) |
88 | 87 | 3adantr1 1213 |
. . . . . 6
⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))) → ∀𝑎 ∈ ℝ (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹)) |
89 | 80, 88 | jca 553 |
. . . . 5
⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))) → (𝐹 ∈ (ℝ ↑pm
∪ 𝑆) ∧ ∀𝑎 ∈ ℝ (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹))) |
90 | 89, 39 | sylibr 223 |
. . . 4
⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))) → 𝐹 ∈ {𝑓 ∈ (ℝ ↑pm
∪ 𝑆) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝑓)}) |
91 | 17 | eqcomd 2616 |
. . . . 5
⊢ (𝜑 → {𝑓 ∈ (ℝ ↑pm
∪ 𝑆) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝑓)} = (SMblFn‘𝑆)) |
92 | 91 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))) → {𝑓 ∈ (ℝ ↑pm
∪ 𝑆) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝑓)} = (SMblFn‘𝑆)) |
93 | 90, 92 | eleqtrd 2690 |
. . 3
⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))) → 𝐹 ∈ (SMblFn‘𝑆)) |
94 | 93 | ex 449 |
. 2
⊢ (𝜑 → ((𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)) → 𝐹 ∈ (SMblFn‘𝑆))) |
95 | 59, 94 | impbid 201 |
1
⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)))) |