Description: The equivalence class of a cartesian product is the whole set. (Contributed by Thierry Arnoux, 15-Jan-2024)
Ref | Expression | ||
---|---|---|---|
Assertion | ecxpid | ⊢ ( 𝑋 ∈ 𝐴 → [ 𝑋 ] ( 𝐴 × 𝐴 ) = 𝐴 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex | ⊢ 𝑥 ∈ V | |
2 | elecg | ⊢ ( ( 𝑥 ∈ V ∧ 𝑋 ∈ 𝐴 ) → ( 𝑥 ∈ [ 𝑋 ] ( 𝐴 × 𝐴 ) ↔ 𝑋 ( 𝐴 × 𝐴 ) 𝑥 ) ) | |
3 | 1 2 | mpan | ⊢ ( 𝑋 ∈ 𝐴 → ( 𝑥 ∈ [ 𝑋 ] ( 𝐴 × 𝐴 ) ↔ 𝑋 ( 𝐴 × 𝐴 ) 𝑥 ) ) |
4 | brxp | ⊢ ( 𝑋 ( 𝐴 × 𝐴 ) 𝑥 ↔ ( 𝑋 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴 ) ) | |
5 | 4 | baib | ⊢ ( 𝑋 ∈ 𝐴 → ( 𝑋 ( 𝐴 × 𝐴 ) 𝑥 ↔ 𝑥 ∈ 𝐴 ) ) |
6 | 3 5 | bitrd | ⊢ ( 𝑋 ∈ 𝐴 → ( 𝑥 ∈ [ 𝑋 ] ( 𝐴 × 𝐴 ) ↔ 𝑥 ∈ 𝐴 ) ) |
7 | 6 | eqrdv | ⊢ ( 𝑋 ∈ 𝐴 → [ 𝑋 ] ( 𝐴 × 𝐴 ) = 𝐴 ) |