| Step | Hyp | Ref
| Expression |
| 1 | | issmfle.s |
. . . . . . 7
⊢ (𝜑 → 𝑆 ∈ SAlg) |
| 2 | 1 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝑆 ∈ SAlg) |
| 3 | | simpr 484 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ (SMblFn‘𝑆)) |
| 4 | | issmfle.d |
. . . . . 6
⊢ 𝐷 = dom 𝐹 |
| 5 | 2, 3, 4 | smfdmss 46748 |
. . . . 5
⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 ⊆ ∪ 𝑆) |
| 6 | 2, 3, 4 | smff 46747 |
. . . . 5
⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:𝐷⟶ℝ) |
| 7 | | nfv 1914 |
. . . . . . 7
⊢
Ⅎ𝑏𝜑 |
| 8 | | nfv 1914 |
. . . . . . 7
⊢
Ⅎ𝑏 𝐹 ∈ (SMblFn‘𝑆) |
| 9 | 7, 8 | nfan 1899 |
. . . . . 6
⊢
Ⅎ𝑏(𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) |
| 10 | | nfv 1914 |
. . . . . . . . . 10
⊢
Ⅎ𝑦𝜑 |
| 11 | | nfv 1914 |
. . . . . . . . . 10
⊢
Ⅎ𝑦 𝐹 ∈ (SMblFn‘𝑆) |
| 12 | 10, 11 | nfan 1899 |
. . . . . . . . 9
⊢
Ⅎ𝑦(𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) |
| 13 | | nfv 1914 |
. . . . . . . . 9
⊢
Ⅎ𝑦 𝑏 ∈ ℝ |
| 14 | 12, 13 | nfan 1899 |
. . . . . . . 8
⊢
Ⅎ𝑦((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) |
| 15 | | nfv 1914 |
. . . . . . . 8
⊢
Ⅎ𝑐((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) |
| 16 | 1 | uniexd 7762 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ∪ 𝑆
∈ V) |
| 17 | 16 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → ∪ 𝑆
∈ V) |
| 18 | | simpr 484 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → 𝐷 ⊆ ∪ 𝑆) |
| 19 | 17, 18 | ssexd 5324 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → 𝐷 ∈ V) |
| 20 | 5, 19 | syldan 591 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 ∈ V) |
| 21 | | eqid 2737 |
. . . . . . . . . 10
⊢ (𝑆 ↾t 𝐷) = (𝑆 ↾t 𝐷) |
| 22 | 2, 20, 21 | subsalsal 46374 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (𝑆 ↾t 𝐷) ∈ SAlg) |
| 23 | 22 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → (𝑆 ↾t 𝐷) ∈ SAlg) |
| 24 | 6 | frexr 45396 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:𝐷⟶ℝ*) |
| 25 | 24 | adantr 480 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → 𝐹:𝐷⟶ℝ*) |
| 26 | 25 | ffvelcdmda 7104 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑦 ∈ 𝐷) → (𝐹‘𝑦) ∈
ℝ*) |
| 27 | 2 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝑆 ∈ SAlg) |
| 28 | 3 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝐹 ∈ (SMblFn‘𝑆)) |
| 29 | | simpr 484 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝑐 ∈ ℝ) |
| 30 | 27, 28, 4, 29 | smfpreimalt 46746 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑐} ∈ (𝑆 ↾t 𝐷)) |
| 31 | 30 | adantlr 715 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑐 ∈ ℝ) → {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑐} ∈ (𝑆 ↾t 𝐷)) |
| 32 | | simpr 484 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → 𝑏 ∈ ℝ) |
| 33 | 14, 15, 23, 26, 31, 32 | salpreimaltle 46741 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)) |
| 34 | 33 | ex 412 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (𝑏 ∈ ℝ → {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))) |
| 35 | 9, 34 | ralrimi 3257 |
. . . . 5
⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)) |
| 36 | 5, 6, 35 | 3jca 1129 |
. . . 4
⊢ ((𝜑 ∧ 𝐹 ∈ (SMblFn‘𝑆)) → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))) |
| 37 | 36 | ex 412 |
. . 3
⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)))) |
| 38 | | nfv 1914 |
. . . . . . 7
⊢
Ⅎ𝑦 𝐷 ⊆ ∪ 𝑆 |
| 39 | | nfv 1914 |
. . . . . . 7
⊢
Ⅎ𝑦 𝐹:𝐷⟶ℝ |
| 40 | | nfcv 2905 |
. . . . . . . 8
⊢
Ⅎ𝑦ℝ |
| 41 | | nfrab1 3457 |
. . . . . . . . 9
⊢
Ⅎ𝑦{𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} |
| 42 | | nfcv 2905 |
. . . . . . . . 9
⊢
Ⅎ𝑦(𝑆 ↾t 𝐷) |
| 43 | 41, 42 | nfel 2920 |
. . . . . . . 8
⊢
Ⅎ𝑦{𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷) |
| 44 | 40, 43 | nfralw 3311 |
. . . . . . 7
⊢
Ⅎ𝑦∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷) |
| 45 | 38, 39, 44 | nf3an 1901 |
. . . . . 6
⊢
Ⅎ𝑦(𝐷 ⊆ ∪ 𝑆
∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)) |
| 46 | 10, 45 | nfan 1899 |
. . . . 5
⊢
Ⅎ𝑦(𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))) |
| 47 | | nfv 1914 |
. . . . . . 7
⊢
Ⅎ𝑏 𝐷 ⊆ ∪ 𝑆 |
| 48 | | nfv 1914 |
. . . . . . 7
⊢
Ⅎ𝑏 𝐹:𝐷⟶ℝ |
| 49 | | nfra1 3284 |
. . . . . . 7
⊢
Ⅎ𝑏∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷) |
| 50 | 47, 48, 49 | nf3an 1901 |
. . . . . 6
⊢
Ⅎ𝑏(𝐷 ⊆ ∪ 𝑆
∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)) |
| 51 | 7, 50 | nfan 1899 |
. . . . 5
⊢
Ⅎ𝑏(𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))) |
| 52 | 1 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))) → 𝑆 ∈ SAlg) |
| 53 | | simpr1 1195 |
. . . . 5
⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))) → 𝐷 ⊆ ∪ 𝑆) |
| 54 | | simpr2 1196 |
. . . . 5
⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))) → 𝐹:𝐷⟶ℝ) |
| 55 | | rspa 3248 |
. . . . . . 7
⊢
((∀𝑏 ∈
ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷) ∧ 𝑏 ∈ ℝ) → {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)) |
| 56 | 55 | 3ad2antl3 1188 |
. . . . . 6
⊢ (((𝐷 ⊆ ∪ 𝑆
∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)) ∧ 𝑏 ∈ ℝ) → {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)) |
| 57 | 56 | adantll 714 |
. . . . 5
⊢ (((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))) ∧ 𝑏 ∈ ℝ) → {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)) |
| 58 | 46, 51, 52, 4, 53, 54, 57 | issmflelem 46759 |
. . . 4
⊢ ((𝜑 ∧ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷))) → 𝐹 ∈ (SMblFn‘𝑆)) |
| 59 | 58 | ex 412 |
. . 3
⊢ (𝜑 → ((𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)) → 𝐹 ∈ (SMblFn‘𝑆))) |
| 60 | 37, 59 | impbid 212 |
. 2
⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)))) |
| 61 | | breq2 5147 |
. . . . . . . 8
⊢ (𝑏 = 𝑎 → ((𝐹‘𝑦) ≤ 𝑏 ↔ (𝐹‘𝑦) ≤ 𝑎)) |
| 62 | 61 | rabbidv 3444 |
. . . . . . 7
⊢ (𝑏 = 𝑎 → {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} = {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑎}) |
| 63 | | fveq2 6906 |
. . . . . . . . . 10
⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥)) |
| 64 | 63 | breq1d 5153 |
. . . . . . . . 9
⊢ (𝑦 = 𝑥 → ((𝐹‘𝑦) ≤ 𝑎 ↔ (𝐹‘𝑥) ≤ 𝑎)) |
| 65 | 64 | cbvrabv 3447 |
. . . . . . . 8
⊢ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑎} = {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎} |
| 66 | 65 | a1i 11 |
. . . . . . 7
⊢ (𝑏 = 𝑎 → {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑎} = {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎}) |
| 67 | 62, 66 | eqtrd 2777 |
. . . . . 6
⊢ (𝑏 = 𝑎 → {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} = {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎}) |
| 68 | 67 | eleq1d 2826 |
. . . . 5
⊢ (𝑏 = 𝑎 → ({𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷) ↔ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎} ∈ (𝑆 ↾t 𝐷))) |
| 69 | 68 | cbvralvw 3237 |
. . . 4
⊢
(∀𝑏 ∈
ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷) ↔ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎} ∈ (𝑆 ↾t 𝐷)) |
| 70 | 69 | 3anbi3i 1160 |
. . 3
⊢ ((𝐷 ⊆ ∪ 𝑆
∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎} ∈ (𝑆 ↾t 𝐷))) |
| 71 | 70 | a1i 11 |
. 2
⊢ (𝜑 → ((𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) ≤ 𝑏} ∈ (𝑆 ↾t 𝐷)) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎} ∈ (𝑆 ↾t 𝐷)))) |
| 72 | 60, 71 | bitrd 279 |
1
⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎} ∈ (𝑆 ↾t 𝐷)))) |