| Step | Hyp | Ref
| Expression |
| 1 | | bndth.4 |
. . . . 5
⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) |
| 2 | | bndth.1 |
. . . . . 6
⊢ 𝑋 = ∪
𝐽 |
| 3 | | bndth.2 |
. . . . . . . 8
⊢ 𝐾 = (topGen‘ran
(,)) |
| 4 | | retopon 24784 |
. . . . . . . 8
⊢
(topGen‘ran (,)) ∈ (TopOn‘ℝ) |
| 5 | 3, 4 | eqeltri 2837 |
. . . . . . 7
⊢ 𝐾 ∈
(TopOn‘ℝ) |
| 6 | 5 | toponunii 22922 |
. . . . . 6
⊢ ℝ =
∪ 𝐾 |
| 7 | 2, 6 | cnf 23254 |
. . . . 5
⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶ℝ) |
| 8 | 1, 7 | syl 17 |
. . . 4
⊢ (𝜑 → 𝐹:𝑋⟶ℝ) |
| 9 | 8 | frnd 6744 |
. . 3
⊢ (𝜑 → ran 𝐹 ⊆ ℝ) |
| 10 | | unieq 4918 |
. . . . . . 7
⊢ (𝑢 = ((,) “ ({-∞}
× ℝ)) → ∪ 𝑢 = ∪ ((,) “
({-∞} × ℝ))) |
| 11 | | imassrn 6089 |
. . . . . . . . . 10
⊢ ((,)
“ ({-∞} × ℝ)) ⊆ ran (,) |
| 12 | 11 | unissi 4916 |
. . . . . . . . 9
⊢ ∪ ((,) “ ({-∞} × ℝ)) ⊆ ∪ ran (,) |
| 13 | | unirnioo 13489 |
. . . . . . . . 9
⊢ ℝ =
∪ ran (,) |
| 14 | 12, 13 | sseqtrri 4033 |
. . . . . . . 8
⊢ ∪ ((,) “ ({-∞} × ℝ)) ⊆
ℝ |
| 15 | | id 22 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ → 𝑥 ∈
ℝ) |
| 16 | | ltp1 12107 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ → 𝑥 < (𝑥 + 1)) |
| 17 | | ressxr 11305 |
. . . . . . . . . . . . 13
⊢ ℝ
⊆ ℝ* |
| 18 | | peano2re 11434 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈
ℝ) |
| 19 | 17, 18 | sselid 3981 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈
ℝ*) |
| 20 | | elioomnf 13484 |
. . . . . . . . . . . 12
⊢ ((𝑥 + 1) ∈ ℝ*
→ (𝑥 ∈
(-∞(,)(𝑥 + 1)) ↔
(𝑥 ∈ ℝ ∧
𝑥 < (𝑥 + 1)))) |
| 21 | 19, 20 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ → (𝑥 ∈ (-∞(,)(𝑥 + 1)) ↔ (𝑥 ∈ ℝ ∧ 𝑥 < (𝑥 + 1)))) |
| 22 | 15, 16, 21 | mpbir2and 713 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℝ → 𝑥 ∈ (-∞(,)(𝑥 + 1))) |
| 23 | | df-ov 7434 |
. . . . . . . . . . 11
⊢
(-∞(,)(𝑥 + 1))
= ((,)‘〈-∞, (𝑥 + 1)〉) |
| 24 | | mnfxr 11318 |
. . . . . . . . . . . . . . 15
⊢ -∞
∈ ℝ* |
| 25 | 24 | elexi 3503 |
. . . . . . . . . . . . . 14
⊢ -∞
∈ V |
| 26 | 25 | snid 4662 |
. . . . . . . . . . . . 13
⊢ -∞
∈ {-∞} |
| 27 | | opelxpi 5722 |
. . . . . . . . . . . . 13
⊢
((-∞ ∈ {-∞} ∧ (𝑥 + 1) ∈ ℝ) → 〈-∞,
(𝑥 + 1)〉 ∈
({-∞} × ℝ)) |
| 28 | 26, 18, 27 | sylancr 587 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ →
〈-∞, (𝑥 +
1)〉 ∈ ({-∞} × ℝ)) |
| 29 | | ioof 13487 |
. . . . . . . . . . . . . 14
⊢
(,):(ℝ* × ℝ*)⟶𝒫
ℝ |
| 30 | | ffun 6739 |
. . . . . . . . . . . . . 14
⊢
((,):(ℝ* × ℝ*)⟶𝒫
ℝ → Fun (,)) |
| 31 | 29, 30 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢ Fun
(,) |
| 32 | | snssi 4808 |
. . . . . . . . . . . . . . . 16
⊢ (-∞
∈ ℝ* → {-∞} ⊆
ℝ*) |
| 33 | 24, 32 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢
{-∞} ⊆ ℝ* |
| 34 | | xpss12 5700 |
. . . . . . . . . . . . . . 15
⊢
(({-∞} ⊆ ℝ* ∧ ℝ ⊆
ℝ*) → ({-∞} × ℝ) ⊆
(ℝ* × ℝ*)) |
| 35 | 33, 17, 34 | mp2an 692 |
. . . . . . . . . . . . . 14
⊢
({-∞} × ℝ) ⊆ (ℝ* ×
ℝ*) |
| 36 | 29 | fdmi 6747 |
. . . . . . . . . . . . . 14
⊢ dom (,) =
(ℝ* × ℝ*) |
| 37 | 35, 36 | sseqtrri 4033 |
. . . . . . . . . . . . 13
⊢
({-∞} × ℝ) ⊆ dom (,) |
| 38 | | funfvima2 7251 |
. . . . . . . . . . . . 13
⊢ ((Fun (,)
∧ ({-∞} × ℝ) ⊆ dom (,)) → (〈-∞,
(𝑥 + 1)〉 ∈
({-∞} × ℝ) → ((,)‘〈-∞, (𝑥 + 1)〉) ∈ ((,) “
({-∞} × ℝ)))) |
| 39 | 31, 37, 38 | mp2an 692 |
. . . . . . . . . . . 12
⊢
(〈-∞, (𝑥
+ 1)〉 ∈ ({-∞} × ℝ) →
((,)‘〈-∞, (𝑥 + 1)〉) ∈ ((,) “ ({-∞}
× ℝ))) |
| 40 | 28, 39 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ →
((,)‘〈-∞, (𝑥 + 1)〉) ∈ ((,) “ ({-∞}
× ℝ))) |
| 41 | 23, 40 | eqeltrid 2845 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℝ →
(-∞(,)(𝑥 + 1)) ∈
((,) “ ({-∞} × ℝ))) |
| 42 | | elunii 4912 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ (-∞(,)(𝑥 + 1)) ∧ (-∞(,)(𝑥 + 1)) ∈ ((,) “
({-∞} × ℝ))) → 𝑥 ∈ ∪ ((,)
“ ({-∞} × ℝ))) |
| 43 | 22, 41, 42 | syl2anc 584 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ∪ ((,) “ ({-∞} ×
ℝ))) |
| 44 | 43 | ssriv 3987 |
. . . . . . . 8
⊢ ℝ
⊆ ∪ ((,) “ ({-∞} ×
ℝ)) |
| 45 | 14, 44 | eqssi 4000 |
. . . . . . 7
⊢ ∪ ((,) “ ({-∞} × ℝ)) =
ℝ |
| 46 | 10, 45 | eqtrdi 2793 |
. . . . . 6
⊢ (𝑢 = ((,) “ ({-∞}
× ℝ)) → ∪ 𝑢 = ℝ) |
| 47 | 46 | sseq2d 4016 |
. . . . 5
⊢ (𝑢 = ((,) “ ({-∞}
× ℝ)) → (ran 𝐹 ⊆ ∪ 𝑢 ↔ ran 𝐹 ⊆ ℝ)) |
| 48 | | pweq 4614 |
. . . . . . 7
⊢ (𝑢 = ((,) “ ({-∞}
× ℝ)) → 𝒫 𝑢 = 𝒫 ((,) “ ({-∞} ×
ℝ))) |
| 49 | 48 | ineq1d 4219 |
. . . . . 6
⊢ (𝑢 = ((,) “ ({-∞}
× ℝ)) → (𝒫 𝑢 ∩ Fin) = (𝒫 ((,) “
({-∞} × ℝ)) ∩ Fin)) |
| 50 | 49 | rexeqdv 3327 |
. . . . 5
⊢ (𝑢 = ((,) “ ({-∞}
× ℝ)) → (∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)ran 𝐹 ⊆ ∪ 𝑣 ↔ ∃𝑣 ∈ (𝒫 ((,) “
({-∞} × ℝ)) ∩ Fin)ran 𝐹 ⊆ ∪ 𝑣)) |
| 51 | 47, 50 | imbi12d 344 |
. . . 4
⊢ (𝑢 = ((,) “ ({-∞}
× ℝ)) → ((ran 𝐹 ⊆ ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)ran 𝐹 ⊆ ∪ 𝑣) ↔ (ran 𝐹 ⊆ ℝ → ∃𝑣 ∈ (𝒫 ((,) “
({-∞} × ℝ)) ∩ Fin)ran 𝐹 ⊆ ∪ 𝑣))) |
| 52 | | bndth.3 |
. . . . . 6
⊢ (𝜑 → 𝐽 ∈ Comp) |
| 53 | | rncmp 23404 |
. . . . . 6
⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐾 ↾t ran 𝐹) ∈ Comp) |
| 54 | 52, 1, 53 | syl2anc 584 |
. . . . 5
⊢ (𝜑 → (𝐾 ↾t ran 𝐹) ∈ Comp) |
| 55 | | retop 24782 |
. . . . . . 7
⊢
(topGen‘ran (,)) ∈ Top |
| 56 | 3, 55 | eqeltri 2837 |
. . . . . 6
⊢ 𝐾 ∈ Top |
| 57 | 6 | cmpsub 23408 |
. . . . . 6
⊢ ((𝐾 ∈ Top ∧ ran 𝐹 ⊆ ℝ) → ((𝐾 ↾t ran 𝐹) ∈ Comp ↔
∀𝑢 ∈ 𝒫
𝐾(ran 𝐹 ⊆ ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)ran 𝐹 ⊆ ∪ 𝑣))) |
| 58 | 56, 9, 57 | sylancr 587 |
. . . . 5
⊢ (𝜑 → ((𝐾 ↾t ran 𝐹) ∈ Comp ↔ ∀𝑢 ∈ 𝒫 𝐾(ran 𝐹 ⊆ ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)ran 𝐹 ⊆ ∪ 𝑣))) |
| 59 | 54, 58 | mpbid 232 |
. . . 4
⊢ (𝜑 → ∀𝑢 ∈ 𝒫 𝐾(ran 𝐹 ⊆ ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)ran 𝐹 ⊆ ∪ 𝑣)) |
| 60 | | retopbas 24781 |
. . . . . . . . 9
⊢ ran (,)
∈ TopBases |
| 61 | | bastg 22973 |
. . . . . . . . 9
⊢ (ran (,)
∈ TopBases → ran (,) ⊆ (topGen‘ran (,))) |
| 62 | 60, 61 | ax-mp 5 |
. . . . . . . 8
⊢ ran (,)
⊆ (topGen‘ran (,)) |
| 63 | 62, 3 | sseqtrri 4033 |
. . . . . . 7
⊢ ran (,)
⊆ 𝐾 |
| 64 | 11, 63 | sstri 3993 |
. . . . . 6
⊢ ((,)
“ ({-∞} × ℝ)) ⊆ 𝐾 |
| 65 | 56, 64 | elpwi2 5335 |
. . . . 5
⊢ ((,)
“ ({-∞} × ℝ)) ∈ 𝒫 𝐾 |
| 66 | 65 | a1i 11 |
. . . 4
⊢ (𝜑 → ((,) “ ({-∞}
× ℝ)) ∈ 𝒫 𝐾) |
| 67 | 51, 59, 66 | rspcdva 3623 |
. . 3
⊢ (𝜑 → (ran 𝐹 ⊆ ℝ → ∃𝑣 ∈ (𝒫 ((,) “
({-∞} × ℝ)) ∩ Fin)ran 𝐹 ⊆ ∪ 𝑣)) |
| 68 | 9, 67 | mpd 15 |
. 2
⊢ (𝜑 → ∃𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)ran 𝐹 ⊆ ∪ 𝑣) |
| 69 | | simpr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) → 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) |
| 70 | | elin 3967 |
. . . . . . 7
⊢ (𝑣 ∈ (𝒫 ((,) “
({-∞} × ℝ)) ∩ Fin) ↔ (𝑣 ∈ 𝒫 ((,) “ ({-∞}
× ℝ)) ∧ 𝑣
∈ Fin)) |
| 71 | 69, 70 | sylib 218 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) → (𝑣 ∈ 𝒫 ((,) “ ({-∞}
× ℝ)) ∧ 𝑣
∈ Fin)) |
| 72 | 71 | adantrr 717 |
. . . . 5
⊢ ((𝜑 ∧ (𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin) ∧ ran 𝐹 ⊆ ∪ 𝑣)) → (𝑣 ∈ 𝒫 ((,) “ ({-∞}
× ℝ)) ∧ 𝑣
∈ Fin)) |
| 73 | 72 | simprd 495 |
. . . 4
⊢ ((𝜑 ∧ (𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin) ∧ ran 𝐹 ⊆ ∪ 𝑣)) → 𝑣 ∈ Fin) |
| 74 | 71 | simpld 494 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) → 𝑣 ∈ 𝒫 ((,) “ ({-∞}
× ℝ))) |
| 75 | 74 | elpwid 4609 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) → 𝑣 ⊆ ((,) “ ({-∞} ×
ℝ))) |
| 76 | 33 | sseli 3979 |
. . . . . . . . . . . 12
⊢ (𝑢 ∈ {-∞} → 𝑢 ∈
ℝ*) |
| 77 | 76 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝑢 ∈ {-∞} ∧ 𝑤 ∈ ℝ) → 𝑢 ∈
ℝ*) |
| 78 | 17 | sseli 3979 |
. . . . . . . . . . . 12
⊢ (𝑤 ∈ ℝ → 𝑤 ∈
ℝ*) |
| 79 | 78 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝑢 ∈ {-∞} ∧ 𝑤 ∈ ℝ) → 𝑤 ∈
ℝ*) |
| 80 | | mnflt 13165 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 ∈ ℝ → -∞
< 𝑤) |
| 81 | | xrltnle 11328 |
. . . . . . . . . . . . . . . 16
⊢
((-∞ ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (-∞
< 𝑤 ↔ ¬ 𝑤 ≤
-∞)) |
| 82 | 24, 78, 81 | sylancr 587 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 ∈ ℝ → (-∞
< 𝑤 ↔ ¬ 𝑤 ≤
-∞)) |
| 83 | 80, 82 | mpbid 232 |
. . . . . . . . . . . . . 14
⊢ (𝑤 ∈ ℝ → ¬
𝑤 ≤
-∞) |
| 84 | 83 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝑢 ∈ {-∞} ∧ 𝑤 ∈ ℝ) → ¬
𝑤 ≤
-∞) |
| 85 | | elsni 4643 |
. . . . . . . . . . . . . . 15
⊢ (𝑢 ∈ {-∞} → 𝑢 = -∞) |
| 86 | 85 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝑢 ∈ {-∞} ∧ 𝑤 ∈ ℝ) → 𝑢 = -∞) |
| 87 | 86 | breq2d 5155 |
. . . . . . . . . . . . 13
⊢ ((𝑢 ∈ {-∞} ∧ 𝑤 ∈ ℝ) → (𝑤 ≤ 𝑢 ↔ 𝑤 ≤ -∞)) |
| 88 | 84, 87 | mtbird 325 |
. . . . . . . . . . . 12
⊢ ((𝑢 ∈ {-∞} ∧ 𝑤 ∈ ℝ) → ¬
𝑤 ≤ 𝑢) |
| 89 | | ioo0 13412 |
. . . . . . . . . . . . . 14
⊢ ((𝑢 ∈ ℝ*
∧ 𝑤 ∈
ℝ*) → ((𝑢(,)𝑤) = ∅ ↔ 𝑤 ≤ 𝑢)) |
| 90 | 76, 78, 89 | syl2an 596 |
. . . . . . . . . . . . 13
⊢ ((𝑢 ∈ {-∞} ∧ 𝑤 ∈ ℝ) → ((𝑢(,)𝑤) = ∅ ↔ 𝑤 ≤ 𝑢)) |
| 91 | 90 | necon3abid 2977 |
. . . . . . . . . . . 12
⊢ ((𝑢 ∈ {-∞} ∧ 𝑤 ∈ ℝ) → ((𝑢(,)𝑤) ≠ ∅ ↔ ¬ 𝑤 ≤ 𝑢)) |
| 92 | 88, 91 | mpbird 257 |
. . . . . . . . . . 11
⊢ ((𝑢 ∈ {-∞} ∧ 𝑤 ∈ ℝ) → (𝑢(,)𝑤) ≠ ∅) |
| 93 | | df-ioo 13391 |
. . . . . . . . . . . 12
⊢ (,) =
(𝑦 ∈
ℝ*, 𝑧
∈ ℝ* ↦ {𝑣 ∈ ℝ* ∣ (𝑦 < 𝑣 ∧ 𝑣 < 𝑧)}) |
| 94 | | idd 24 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℝ*
∧ 𝑤 ∈
ℝ*) → (𝑥 < 𝑤 → 𝑥 < 𝑤)) |
| 95 | | xrltle 13191 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℝ*
∧ 𝑤 ∈
ℝ*) → (𝑥 < 𝑤 → 𝑥 ≤ 𝑤)) |
| 96 | | idd 24 |
. . . . . . . . . . . 12
⊢ ((𝑢 ∈ ℝ*
∧ 𝑥 ∈
ℝ*) → (𝑢 < 𝑥 → 𝑢 < 𝑥)) |
| 97 | | xrltle 13191 |
. . . . . . . . . . . 12
⊢ ((𝑢 ∈ ℝ*
∧ 𝑥 ∈
ℝ*) → (𝑢 < 𝑥 → 𝑢 ≤ 𝑥)) |
| 98 | 93, 94, 95, 96, 97 | ixxub 13408 |
. . . . . . . . . . 11
⊢ ((𝑢 ∈ ℝ*
∧ 𝑤 ∈
ℝ* ∧ (𝑢(,)𝑤) ≠ ∅) → sup((𝑢(,)𝑤), ℝ*, < ) = 𝑤) |
| 99 | 77, 79, 92, 98 | syl3anc 1373 |
. . . . . . . . . 10
⊢ ((𝑢 ∈ {-∞} ∧ 𝑤 ∈ ℝ) →
sup((𝑢(,)𝑤), ℝ*, < ) = 𝑤) |
| 100 | | simpr 484 |
. . . . . . . . . 10
⊢ ((𝑢 ∈ {-∞} ∧ 𝑤 ∈ ℝ) → 𝑤 ∈
ℝ) |
| 101 | 99, 100 | eqeltrd 2841 |
. . . . . . . . 9
⊢ ((𝑢 ∈ {-∞} ∧ 𝑤 ∈ ℝ) →
sup((𝑢(,)𝑤), ℝ*, < ) ∈
ℝ) |
| 102 | 101 | rgen2 3199 |
. . . . . . . 8
⊢
∀𝑢 ∈
{-∞}∀𝑤 ∈
ℝ sup((𝑢(,)𝑤), ℝ*, < )
∈ ℝ |
| 103 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑧 = 〈𝑢, 𝑤〉 → ((,)‘𝑧) = ((,)‘〈𝑢, 𝑤〉)) |
| 104 | | df-ov 7434 |
. . . . . . . . . . . 12
⊢ (𝑢(,)𝑤) = ((,)‘〈𝑢, 𝑤〉) |
| 105 | 103, 104 | eqtr4di 2795 |
. . . . . . . . . . 11
⊢ (𝑧 = 〈𝑢, 𝑤〉 → ((,)‘𝑧) = (𝑢(,)𝑤)) |
| 106 | 105 | supeq1d 9486 |
. . . . . . . . . 10
⊢ (𝑧 = 〈𝑢, 𝑤〉 → sup(((,)‘𝑧), ℝ*, < ) =
sup((𝑢(,)𝑤), ℝ*, <
)) |
| 107 | 106 | eleq1d 2826 |
. . . . . . . . 9
⊢ (𝑧 = 〈𝑢, 𝑤〉 → (sup(((,)‘𝑧), ℝ*, < )
∈ ℝ ↔ sup((𝑢(,)𝑤), ℝ*, < ) ∈
ℝ)) |
| 108 | 107 | ralxp 5852 |
. . . . . . . 8
⊢
(∀𝑧 ∈
({-∞} × ℝ)sup(((,)‘𝑧), ℝ*, < ) ∈ ℝ
↔ ∀𝑢 ∈
{-∞}∀𝑤 ∈
ℝ sup((𝑢(,)𝑤), ℝ*, < )
∈ ℝ) |
| 109 | 102, 108 | mpbir 231 |
. . . . . . 7
⊢
∀𝑧 ∈
({-∞} × ℝ)sup(((,)‘𝑧), ℝ*, < ) ∈
ℝ |
| 110 | | ffn 6736 |
. . . . . . . . 9
⊢
((,):(ℝ* × ℝ*)⟶𝒫
ℝ → (,) Fn (ℝ* ×
ℝ*)) |
| 111 | 29, 110 | ax-mp 5 |
. . . . . . . 8
⊢ (,) Fn
(ℝ* × ℝ*) |
| 112 | | supeq1 9485 |
. . . . . . . . . 10
⊢ (𝑤 = ((,)‘𝑧) → sup(𝑤, ℝ*, < ) =
sup(((,)‘𝑧),
ℝ*, < )) |
| 113 | 112 | eleq1d 2826 |
. . . . . . . . 9
⊢ (𝑤 = ((,)‘𝑧) → (sup(𝑤, ℝ*, < ) ∈ ℝ
↔ sup(((,)‘𝑧),
ℝ*, < ) ∈ ℝ)) |
| 114 | 113 | ralima 7257 |
. . . . . . . 8
⊢ (((,) Fn
(ℝ* × ℝ*) ∧ ({-∞} ×
ℝ) ⊆ (ℝ* × ℝ*)) →
(∀𝑤 ∈ ((,)
“ ({-∞} × ℝ))sup(𝑤, ℝ*, < ) ∈ ℝ
↔ ∀𝑧 ∈
({-∞} × ℝ)sup(((,)‘𝑧), ℝ*, < ) ∈
ℝ)) |
| 115 | 111, 35, 114 | mp2an 692 |
. . . . . . 7
⊢
(∀𝑤 ∈
((,) “ ({-∞} × ℝ))sup(𝑤, ℝ*, < ) ∈ ℝ
↔ ∀𝑧 ∈
({-∞} × ℝ)sup(((,)‘𝑧), ℝ*, < ) ∈
ℝ) |
| 116 | 109, 115 | mpbir 231 |
. . . . . 6
⊢
∀𝑤 ∈
((,) “ ({-∞} × ℝ))sup(𝑤, ℝ*, < ) ∈
ℝ |
| 117 | | ssralv 4052 |
. . . . . 6
⊢ (𝑣 ⊆ ((,) “
({-∞} × ℝ)) → (∀𝑤 ∈ ((,) “ ({-∞} ×
ℝ))sup(𝑤,
ℝ*, < ) ∈ ℝ → ∀𝑤 ∈ 𝑣 sup(𝑤, ℝ*, < ) ∈
ℝ)) |
| 118 | 75, 116, 117 | mpisyl 21 |
. . . . 5
⊢ ((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) → ∀𝑤 ∈ 𝑣 sup(𝑤, ℝ*, < ) ∈
ℝ) |
| 119 | 118 | adantrr 717 |
. . . 4
⊢ ((𝜑 ∧ (𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin) ∧ ran 𝐹 ⊆ ∪ 𝑣)) → ∀𝑤 ∈ 𝑣 sup(𝑤, ℝ*, < ) ∈
ℝ) |
| 120 | | fimaxre3 12214 |
. . . 4
⊢ ((𝑣 ∈ Fin ∧ ∀𝑤 ∈ 𝑣 sup(𝑤, ℝ*, < ) ∈ ℝ)
→ ∃𝑥 ∈
ℝ ∀𝑤 ∈
𝑣 sup(𝑤, ℝ*, < ) ≤ 𝑥) |
| 121 | 73, 119, 120 | syl2anc 584 |
. . 3
⊢ ((𝜑 ∧ (𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin) ∧ ran 𝐹 ⊆ ∪ 𝑣)) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑣 sup(𝑤, ℝ*, < ) ≤ 𝑥) |
| 122 | | simplrr 778 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin) ∧ ran 𝐹 ⊆ ∪ 𝑣)) ∧ 𝑥 ∈ ℝ) → ran 𝐹 ⊆ ∪ 𝑣) |
| 123 | 122 | sselda 3983 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin) ∧ ran 𝐹 ⊆ ∪ 𝑣)) ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ran 𝐹) → 𝑧 ∈ ∪ 𝑣) |
| 124 | | eluni2 4911 |
. . . . . . . 8
⊢ (𝑧 ∈ ∪ 𝑣
↔ ∃𝑤 ∈
𝑣 𝑧 ∈ 𝑤) |
| 125 | | r19.29r 3116 |
. . . . . . . . . 10
⊢
((∃𝑤 ∈
𝑣 𝑧 ∈ 𝑤 ∧ ∀𝑤 ∈ 𝑣 sup(𝑤, ℝ*, < ) ≤ 𝑥) → ∃𝑤 ∈ 𝑣 (𝑧 ∈ 𝑤 ∧ sup(𝑤, ℝ*, < ) ≤ 𝑥)) |
| 126 | | sspwuni 5100 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((,)
“ ({-∞} × ℝ)) ⊆ 𝒫 ℝ ↔ ∪ ((,) “ ({-∞} × ℝ)) ⊆
ℝ) |
| 127 | 14, 126 | mpbir 231 |
. . . . . . . . . . . . . . . . . 18
⊢ ((,)
“ ({-∞} × ℝ)) ⊆ 𝒫
ℝ |
| 128 | 75 | 3ad2ant1 1134 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) ∧ (𝑥 ∈ ℝ ∧ 𝑤 ∈ 𝑣) ∧ (𝑧 ∈ 𝑤 ∧ sup(𝑤, ℝ*, < ) ≤ 𝑥)) → 𝑣 ⊆ ((,) “ ({-∞} ×
ℝ))) |
| 129 | | simp2r 1201 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) ∧ (𝑥 ∈ ℝ ∧ 𝑤 ∈ 𝑣) ∧ (𝑧 ∈ 𝑤 ∧ sup(𝑤, ℝ*, < ) ≤ 𝑥)) → 𝑤 ∈ 𝑣) |
| 130 | 128, 129 | sseldd 3984 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) ∧ (𝑥 ∈ ℝ ∧ 𝑤 ∈ 𝑣) ∧ (𝑧 ∈ 𝑤 ∧ sup(𝑤, ℝ*, < ) ≤ 𝑥)) → 𝑤 ∈ ((,) “ ({-∞} ×
ℝ))) |
| 131 | 127, 130 | sselid 3981 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) ∧ (𝑥 ∈ ℝ ∧ 𝑤 ∈ 𝑣) ∧ (𝑧 ∈ 𝑤 ∧ sup(𝑤, ℝ*, < ) ≤ 𝑥)) → 𝑤 ∈ 𝒫 ℝ) |
| 132 | 131 | elpwid 4609 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) ∧ (𝑥 ∈ ℝ ∧ 𝑤 ∈ 𝑣) ∧ (𝑧 ∈ 𝑤 ∧ sup(𝑤, ℝ*, < ) ≤ 𝑥)) → 𝑤 ⊆ ℝ) |
| 133 | | simp3l 1202 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) ∧ (𝑥 ∈ ℝ ∧ 𝑤 ∈ 𝑣) ∧ (𝑧 ∈ 𝑤 ∧ sup(𝑤, ℝ*, < ) ≤ 𝑥)) → 𝑧 ∈ 𝑤) |
| 134 | 132, 133 | sseldd 3984 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) ∧ (𝑥 ∈ ℝ ∧ 𝑤 ∈ 𝑣) ∧ (𝑧 ∈ 𝑤 ∧ sup(𝑤, ℝ*, < ) ≤ 𝑥)) → 𝑧 ∈ ℝ) |
| 135 | 118 | r19.21bi 3251 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) ∧ 𝑤 ∈ 𝑣) → sup(𝑤, ℝ*, < ) ∈
ℝ) |
| 136 | 135 | adantrl 716 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) ∧ (𝑥 ∈ ℝ ∧ 𝑤 ∈ 𝑣)) → sup(𝑤, ℝ*, < ) ∈
ℝ) |
| 137 | 136 | 3adant3 1133 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) ∧ (𝑥 ∈ ℝ ∧ 𝑤 ∈ 𝑣) ∧ (𝑧 ∈ 𝑤 ∧ sup(𝑤, ℝ*, < ) ≤ 𝑥)) → sup(𝑤, ℝ*, < ) ∈
ℝ) |
| 138 | | simp2l 1200 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) ∧ (𝑥 ∈ ℝ ∧ 𝑤 ∈ 𝑣) ∧ (𝑧 ∈ 𝑤 ∧ sup(𝑤, ℝ*, < ) ≤ 𝑥)) → 𝑥 ∈ ℝ) |
| 139 | 132, 17 | sstrdi 3996 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) ∧ (𝑥 ∈ ℝ ∧ 𝑤 ∈ 𝑣) ∧ (𝑧 ∈ 𝑤 ∧ sup(𝑤, ℝ*, < ) ≤ 𝑥)) → 𝑤 ⊆
ℝ*) |
| 140 | | supxrub 13366 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑤 ⊆ ℝ*
∧ 𝑧 ∈ 𝑤) → 𝑧 ≤ sup(𝑤, ℝ*, <
)) |
| 141 | 139, 133,
140 | syl2anc 584 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) ∧ (𝑥 ∈ ℝ ∧ 𝑤 ∈ 𝑣) ∧ (𝑧 ∈ 𝑤 ∧ sup(𝑤, ℝ*, < ) ≤ 𝑥)) → 𝑧 ≤ sup(𝑤, ℝ*, <
)) |
| 142 | | simp3r 1203 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) ∧ (𝑥 ∈ ℝ ∧ 𝑤 ∈ 𝑣) ∧ (𝑧 ∈ 𝑤 ∧ sup(𝑤, ℝ*, < ) ≤ 𝑥)) → sup(𝑤, ℝ*, < ) ≤ 𝑥) |
| 143 | 134, 137,
138, 141, 142 | letrd 11418 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) ∧ (𝑥 ∈ ℝ ∧ 𝑤 ∈ 𝑣) ∧ (𝑧 ∈ 𝑤 ∧ sup(𝑤, ℝ*, < ) ≤ 𝑥)) → 𝑧 ≤ 𝑥) |
| 144 | 143 | 3expia 1122 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) ∧ (𝑥 ∈ ℝ ∧ 𝑤 ∈ 𝑣)) → ((𝑧 ∈ 𝑤 ∧ sup(𝑤, ℝ*, < ) ≤ 𝑥) → 𝑧 ≤ 𝑥)) |
| 145 | 144 | anassrs 467 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) ∧ 𝑥 ∈ ℝ) ∧ 𝑤 ∈ 𝑣) → ((𝑧 ∈ 𝑤 ∧ sup(𝑤, ℝ*, < ) ≤ 𝑥) → 𝑧 ≤ 𝑥)) |
| 146 | 145 | rexlimdva 3155 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin)) ∧ 𝑥 ∈ ℝ) → (∃𝑤 ∈ 𝑣 (𝑧 ∈ 𝑤 ∧ sup(𝑤, ℝ*, < ) ≤ 𝑥) → 𝑧 ≤ 𝑥)) |
| 147 | 146 | adantlrr 721 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin) ∧ ran 𝐹 ⊆ ∪ 𝑣)) ∧ 𝑥 ∈ ℝ) → (∃𝑤 ∈ 𝑣 (𝑧 ∈ 𝑤 ∧ sup(𝑤, ℝ*, < ) ≤ 𝑥) → 𝑧 ≤ 𝑥)) |
| 148 | 125, 147 | syl5 34 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin) ∧ ran 𝐹 ⊆ ∪ 𝑣)) ∧ 𝑥 ∈ ℝ) → ((∃𝑤 ∈ 𝑣 𝑧 ∈ 𝑤 ∧ ∀𝑤 ∈ 𝑣 sup(𝑤, ℝ*, < ) ≤ 𝑥) → 𝑧 ≤ 𝑥)) |
| 149 | 148 | expdimp 452 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin) ∧ ran 𝐹 ⊆ ∪ 𝑣)) ∧ 𝑥 ∈ ℝ) ∧ ∃𝑤 ∈ 𝑣 𝑧 ∈ 𝑤) → (∀𝑤 ∈ 𝑣 sup(𝑤, ℝ*, < ) ≤ 𝑥 → 𝑧 ≤ 𝑥)) |
| 150 | 124, 149 | sylan2b 594 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin) ∧ ran 𝐹 ⊆ ∪ 𝑣)) ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ∪ 𝑣) → (∀𝑤 ∈ 𝑣 sup(𝑤, ℝ*, < ) ≤ 𝑥 → 𝑧 ≤ 𝑥)) |
| 151 | 123, 150 | syldan 591 |
. . . . . 6
⊢ ((((𝜑 ∧ (𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin) ∧ ran 𝐹 ⊆ ∪ 𝑣)) ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ran 𝐹) → (∀𝑤 ∈ 𝑣 sup(𝑤, ℝ*, < ) ≤ 𝑥 → 𝑧 ≤ 𝑥)) |
| 152 | 151 | ralrimdva 3154 |
. . . . 5
⊢ (((𝜑 ∧ (𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin) ∧ ran 𝐹 ⊆ ∪ 𝑣)) ∧ 𝑥 ∈ ℝ) → (∀𝑤 ∈ 𝑣 sup(𝑤, ℝ*, < ) ≤ 𝑥 → ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑥)) |
| 153 | 8 | ffnd 6737 |
. . . . . . 7
⊢ (𝜑 → 𝐹 Fn 𝑋) |
| 154 | 153 | ad2antrr 726 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin) ∧ ran 𝐹 ⊆ ∪ 𝑣)) ∧ 𝑥 ∈ ℝ) → 𝐹 Fn 𝑋) |
| 155 | | breq1 5146 |
. . . . . . 7
⊢ (𝑧 = (𝐹‘𝑦) → (𝑧 ≤ 𝑥 ↔ (𝐹‘𝑦) ≤ 𝑥)) |
| 156 | 155 | ralrn 7108 |
. . . . . 6
⊢ (𝐹 Fn 𝑋 → (∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑥 ↔ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ 𝑥)) |
| 157 | 154, 156 | syl 17 |
. . . . 5
⊢ (((𝜑 ∧ (𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin) ∧ ran 𝐹 ⊆ ∪ 𝑣)) ∧ 𝑥 ∈ ℝ) → (∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑥 ↔ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ 𝑥)) |
| 158 | 152, 157 | sylibd 239 |
. . . 4
⊢ (((𝜑 ∧ (𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin) ∧ ran 𝐹 ⊆ ∪ 𝑣)) ∧ 𝑥 ∈ ℝ) → (∀𝑤 ∈ 𝑣 sup(𝑤, ℝ*, < ) ≤ 𝑥 → ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ 𝑥)) |
| 159 | 158 | reximdva 3168 |
. . 3
⊢ ((𝜑 ∧ (𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin) ∧ ran 𝐹 ⊆ ∪ 𝑣)) → (∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑣 sup(𝑤, ℝ*, < ) ≤ 𝑥 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ 𝑥)) |
| 160 | 121, 159 | mpd 15 |
. 2
⊢ ((𝜑 ∧ (𝑣 ∈ (𝒫 ((,) “ ({-∞}
× ℝ)) ∩ Fin) ∧ ran 𝐹 ⊆ ∪ 𝑣)) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ 𝑥) |
| 161 | 68, 160 | rexlimddv 3161 |
1
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ 𝑥) |