Step | Hyp | Ref
| Expression |
1 | | dirge.1 |
. . . . . . 7
⊢ 𝑋 = dom 𝑅 |
2 | | dirdm 18549 |
. . . . . . 7
⊢ (𝑅 ∈ DirRel → dom 𝑅 = ∪
∪ 𝑅) |
3 | 1, 2 | eqtrid 2784 |
. . . . . 6
⊢ (𝑅 ∈ DirRel → 𝑋 = ∪
∪ 𝑅) |
4 | 3 | eleq2d 2819 |
. . . . 5
⊢ (𝑅 ∈ DirRel → (𝐴 ∈ 𝑋 ↔ 𝐴 ∈ ∪ ∪ 𝑅)) |
5 | 3 | eleq2d 2819 |
. . . . 5
⊢ (𝑅 ∈ DirRel → (𝐵 ∈ 𝑋 ↔ 𝐵 ∈ ∪ ∪ 𝑅)) |
6 | 4, 5 | anbi12d 631 |
. . . 4
⊢ (𝑅 ∈ DirRel → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ↔ (𝐴 ∈ ∪ ∪ 𝑅
∧ 𝐵 ∈ ∪ ∪ 𝑅))) |
7 | | eqid 2732 |
. . . . . . . . 9
⊢ ∪ ∪ 𝑅 = ∪ ∪ 𝑅 |
8 | 7 | isdir 18547 |
. . . . . . . 8
⊢ (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel
𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅
× ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅))))) |
9 | 8 | ibi 266 |
. . . . . . 7
⊢ (𝑅 ∈ DirRel → ((Rel
𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅
× ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅)))) |
10 | 9 | simprrd 772 |
. . . . . 6
⊢ (𝑅 ∈ DirRel → (∪ ∪ 𝑅 × ∪ ∪ 𝑅)
⊆ (◡𝑅 ∘ 𝑅)) |
11 | | codir 6118 |
. . . . . 6
⊢ ((∪ ∪ 𝑅 × ∪ ∪ 𝑅)
⊆ (◡𝑅 ∘ 𝑅) ↔ ∀𝑦 ∈ ∪ ∪ 𝑅∀𝑧 ∈ ∪ ∪ 𝑅∃𝑥(𝑦𝑅𝑥 ∧ 𝑧𝑅𝑥)) |
12 | 10, 11 | sylib 217 |
. . . . 5
⊢ (𝑅 ∈ DirRel →
∀𝑦 ∈ ∪ ∪ 𝑅∀𝑧 ∈ ∪ ∪ 𝑅∃𝑥(𝑦𝑅𝑥 ∧ 𝑧𝑅𝑥)) |
13 | | breq1 5150 |
. . . . . . . 8
⊢ (𝑦 = 𝐴 → (𝑦𝑅𝑥 ↔ 𝐴𝑅𝑥)) |
14 | 13 | anbi1d 630 |
. . . . . . 7
⊢ (𝑦 = 𝐴 → ((𝑦𝑅𝑥 ∧ 𝑧𝑅𝑥) ↔ (𝐴𝑅𝑥 ∧ 𝑧𝑅𝑥))) |
15 | 14 | exbidv 1924 |
. . . . . 6
⊢ (𝑦 = 𝐴 → (∃𝑥(𝑦𝑅𝑥 ∧ 𝑧𝑅𝑥) ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝑧𝑅𝑥))) |
16 | | breq1 5150 |
. . . . . . . 8
⊢ (𝑧 = 𝐵 → (𝑧𝑅𝑥 ↔ 𝐵𝑅𝑥)) |
17 | 16 | anbi2d 629 |
. . . . . . 7
⊢ (𝑧 = 𝐵 → ((𝐴𝑅𝑥 ∧ 𝑧𝑅𝑥) ↔ (𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥))) |
18 | 17 | exbidv 1924 |
. . . . . 6
⊢ (𝑧 = 𝐵 → (∃𝑥(𝐴𝑅𝑥 ∧ 𝑧𝑅𝑥) ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥))) |
19 | 15, 18 | rspc2v 3621 |
. . . . 5
⊢ ((𝐴 ∈ ∪ ∪ 𝑅 ∧ 𝐵 ∈ ∪ ∪ 𝑅)
→ (∀𝑦 ∈
∪ ∪ 𝑅∀𝑧 ∈ ∪ ∪ 𝑅∃𝑥(𝑦𝑅𝑥 ∧ 𝑧𝑅𝑥) → ∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥))) |
20 | 12, 19 | syl5com 31 |
. . . 4
⊢ (𝑅 ∈ DirRel → ((𝐴 ∈ ∪ ∪ 𝑅 ∧ 𝐵 ∈ ∪ ∪ 𝑅)
→ ∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥))) |
21 | 6, 20 | sylbid 239 |
. . 3
⊢ (𝑅 ∈ DirRel → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥))) |
22 | | reldir 18548 |
. . . . . . . . . 10
⊢ (𝑅 ∈ DirRel → Rel 𝑅) |
23 | | relelrn 5942 |
. . . . . . . . . 10
⊢ ((Rel
𝑅 ∧ 𝐴𝑅𝑥) → 𝑥 ∈ ran 𝑅) |
24 | 22, 23 | sylan 580 |
. . . . . . . . 9
⊢ ((𝑅 ∈ DirRel ∧ 𝐴𝑅𝑥) → 𝑥 ∈ ran 𝑅) |
25 | 24 | ex 413 |
. . . . . . . 8
⊢ (𝑅 ∈ DirRel → (𝐴𝑅𝑥 → 𝑥 ∈ ran 𝑅)) |
26 | | ssun2 4172 |
. . . . . . . . . . 11
⊢ ran 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅) |
27 | | dmrnssfld 5967 |
. . . . . . . . . . 11
⊢ (dom
𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅 |
28 | 26, 27 | sstri 3990 |
. . . . . . . . . 10
⊢ ran 𝑅 ⊆ ∪ ∪ 𝑅 |
29 | 28, 3 | sseqtrrid 4034 |
. . . . . . . . 9
⊢ (𝑅 ∈ DirRel → ran 𝑅 ⊆ 𝑋) |
30 | 29 | sseld 3980 |
. . . . . . . 8
⊢ (𝑅 ∈ DirRel → (𝑥 ∈ ran 𝑅 → 𝑥 ∈ 𝑋)) |
31 | 25, 30 | syld 47 |
. . . . . . 7
⊢ (𝑅 ∈ DirRel → (𝐴𝑅𝑥 → 𝑥 ∈ 𝑋)) |
32 | 31 | adantrd 492 |
. . . . . 6
⊢ (𝑅 ∈ DirRel → ((𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) → 𝑥 ∈ 𝑋)) |
33 | 32 | ancrd 552 |
. . . . 5
⊢ (𝑅 ∈ DirRel → ((𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) → (𝑥 ∈ 𝑋 ∧ (𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥)))) |
34 | 33 | eximdv 1920 |
. . . 4
⊢ (𝑅 ∈ DirRel →
(∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) → ∃𝑥(𝑥 ∈ 𝑋 ∧ (𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥)))) |
35 | | df-rex 3071 |
. . . 4
⊢
(∃𝑥 ∈
𝑋 (𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ↔ ∃𝑥(𝑥 ∈ 𝑋 ∧ (𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥))) |
36 | 34, 35 | syl6ibr 251 |
. . 3
⊢ (𝑅 ∈ DirRel →
(∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) → ∃𝑥 ∈ 𝑋 (𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥))) |
37 | 21, 36 | syld 47 |
. 2
⊢ (𝑅 ∈ DirRel → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ∃𝑥 ∈ 𝑋 (𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥))) |
38 | 37 | 3impib 1116 |
1
⊢ ((𝑅 ∈ DirRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ∃𝑥 ∈ 𝑋 (𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥)) |