Description: Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by Mario Carneiro, 19-Feb-2013) Add disjoint variable condition to avoid auxiliary axioms . See cbvmptvg for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 17-Nov-2024)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | cbvmptv.1 | ⊢ ( 𝑥 = 𝑦 → 𝐵 = 𝐶 ) | |
| Assertion | cbvmptv | ⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑦 ∈ 𝐴 ↦ 𝐶 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cbvmptv.1 | ⊢ ( 𝑥 = 𝑦 → 𝐵 = 𝐶 ) | |
| 2 | eleq1w | ⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 ) ) | |
| 3 | 1 | eqeq2d | ⊢ ( 𝑥 = 𝑦 → ( 𝑧 = 𝐵 ↔ 𝑧 = 𝐶 ) ) |
| 4 | 2 3 | anbi12d | ⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵 ) ↔ ( 𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶 ) ) ) |
| 5 | 4 | cbvopab1v | ⊢ { ⟨ 𝑥 , 𝑧 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵 ) } = { ⟨ 𝑦 , 𝑧 ⟩ ∣ ( 𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶 ) } |
| 6 | df-mpt | ⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = { ⟨ 𝑥 , 𝑧 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵 ) } | |
| 7 | df-mpt | ⊢ ( 𝑦 ∈ 𝐴 ↦ 𝐶 ) = { ⟨ 𝑦 , 𝑧 ⟩ ∣ ( 𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶 ) } | |
| 8 | 5 6 7 | 3eqtr4i | ⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑦 ∈ 𝐴 ↦ 𝐶 ) |