Step |
Hyp |
Ref |
Expression |
1 |
|
psref.1 |
⊢ 𝑋 = dom 𝑅 |
2 |
|
psdmrn |
⊢ ( 𝑅 ∈ PosetRel → ( dom 𝑅 = ∪ ∪ 𝑅 ∧ ran 𝑅 = ∪ ∪ 𝑅 ) ) |
3 |
2
|
simpld |
⊢ ( 𝑅 ∈ PosetRel → dom 𝑅 = ∪ ∪ 𝑅 ) |
4 |
1 3
|
syl5eq |
⊢ ( 𝑅 ∈ PosetRel → 𝑋 = ∪ ∪ 𝑅 ) |
5 |
4
|
eleq2d |
⊢ ( 𝑅 ∈ PosetRel → ( 𝐴 ∈ 𝑋 ↔ 𝐴 ∈ ∪ ∪ 𝑅 ) ) |
6 |
|
pslem |
⊢ ( 𝑅 ∈ PosetRel → ( ( ( 𝐴 𝑅 𝐴 ∧ 𝐴 𝑅 𝐴 ) → 𝐴 𝑅 𝐴 ) ∧ ( 𝐴 ∈ ∪ ∪ 𝑅 → 𝐴 𝑅 𝐴 ) ∧ ( ( 𝐴 𝑅 𝐴 ∧ 𝐴 𝑅 𝐴 ) → 𝐴 = 𝐴 ) ) ) |
7 |
6
|
simp2d |
⊢ ( 𝑅 ∈ PosetRel → ( 𝐴 ∈ ∪ ∪ 𝑅 → 𝐴 𝑅 𝐴 ) ) |
8 |
5 7
|
sylbid |
⊢ ( 𝑅 ∈ PosetRel → ( 𝐴 ∈ 𝑋 → 𝐴 𝑅 𝐴 ) ) |
9 |
8
|
imp |
⊢ ( ( 𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑋 ) → 𝐴 𝑅 𝐴 ) |