Metamath Proof Explorer
Description: Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015)
|
|
Ref |
Expression |
|
Hypotheses |
ersym.1 |
⊢ ( 𝜑 → 𝑅 Er 𝑋 ) |
|
|
ersym.2 |
⊢ ( 𝜑 → 𝐴 𝑅 𝐵 ) |
|
Assertion |
ercl |
⊢ ( 𝜑 → 𝐴 ∈ 𝑋 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
ersym.1 |
⊢ ( 𝜑 → 𝑅 Er 𝑋 ) |
2 |
|
ersym.2 |
⊢ ( 𝜑 → 𝐴 𝑅 𝐵 ) |
3 |
|
errel |
⊢ ( 𝑅 Er 𝑋 → Rel 𝑅 ) |
4 |
1 3
|
syl |
⊢ ( 𝜑 → Rel 𝑅 ) |
5 |
|
releldm |
⊢ ( ( Rel 𝑅 ∧ 𝐴 𝑅 𝐵 ) → 𝐴 ∈ dom 𝑅 ) |
6 |
4 2 5
|
syl2anc |
⊢ ( 𝜑 → 𝐴 ∈ dom 𝑅 ) |
7 |
|
erdm |
⊢ ( 𝑅 Er 𝑋 → dom 𝑅 = 𝑋 ) |
8 |
1 7
|
syl |
⊢ ( 𝜑 → dom 𝑅 = 𝑋 ) |
9 |
6 8
|
eleqtrd |
⊢ ( 𝜑 → 𝐴 ∈ 𝑋 ) |