Step | Hyp | Ref
| Expression |
1 | | simplll 798 |
. . . 4
⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) → 𝑈 ∈ (UnifOn‘𝑋)) |
2 | | simplr 792 |
. . . 4
⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) → 𝑣 ∈ 𝑈) |
3 | | ustexsym 22019 |
. . . 4
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑣)) |
4 | 1, 2, 3 | syl2anc 693 |
. . 3
⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑣)) |
5 | | simprl 794 |
. . . . . 6
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑣)) → ◡𝑤 = 𝑤) |
6 | | coss1 5277 |
. . . . . . . . 9
⊢ (𝑤 ⊆ 𝑣 → (𝑤 ∘ 𝑤) ⊆ (𝑣 ∘ 𝑤)) |
7 | | coss2 5278 |
. . . . . . . . 9
⊢ (𝑤 ⊆ 𝑣 → (𝑣 ∘ 𝑤) ⊆ (𝑣 ∘ 𝑣)) |
8 | 6, 7 | sstrd 3613 |
. . . . . . . 8
⊢ (𝑤 ⊆ 𝑣 → (𝑤 ∘ 𝑤) ⊆ (𝑣 ∘ 𝑣)) |
9 | 8 | ad2antll 765 |
. . . . . . 7
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑣)) → (𝑤 ∘ 𝑤) ⊆ (𝑣 ∘ 𝑣)) |
10 | | simpllr 799 |
. . . . . . 7
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑣)) → (𝑣 ∘ 𝑣) ⊆ 𝑉) |
11 | 9, 10 | sstrd 3613 |
. . . . . 6
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑣)) → (𝑤 ∘ 𝑤) ⊆ 𝑉) |
12 | 5, 11 | jca 554 |
. . . . 5
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑣)) → (◡𝑤 = 𝑤 ∧ (𝑤 ∘ 𝑤) ⊆ 𝑉)) |
13 | 12 | ex 450 |
. . . 4
⊢
(((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) ∧ 𝑤 ∈ 𝑈) → ((◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑣) → (◡𝑤 = 𝑤 ∧ (𝑤 ∘ 𝑤) ⊆ 𝑉))) |
14 | 13 | reximdva 3017 |
. . 3
⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) → (∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑣) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ (𝑤 ∘ 𝑤) ⊆ 𝑉))) |
15 | 4, 14 | mpd 15 |
. 2
⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ (𝑤 ∘ 𝑤) ⊆ 𝑉)) |
16 | | ustexhalf 22014 |
. 2
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ∃𝑣 ∈ 𝑈 (𝑣 ∘ 𝑣) ⊆ 𝑉) |
17 | 15, 16 | r19.29a 3078 |
1
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ (𝑤 ∘ 𝑤) ⊆ 𝑉)) |