Description: Slightly more general equality inference for the maps-to notation. (Contributed by FL, 14-Sep-2013) (Revised by Mario Carneiro, 16-Dec-2013)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | mpteq2da.1 | ⊢ Ⅎ 𝑥 𝜑 | |
| mpteq2da.2 | ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = 𝐶 ) | ||
| Assertion | mpteq2da | ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mpteq2da.1 | ⊢ Ⅎ 𝑥 𝜑 | |
| 2 | mpteq2da.2 | ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = 𝐶 ) | |
| 3 | eqid | ⊢ 𝐴 = 𝐴 | |
| 4 | 3 | ax-gen | ⊢ ∀ 𝑥 𝐴 = 𝐴 |
| 5 | 2 | ex | ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → 𝐵 = 𝐶 ) ) |
| 6 | 1 5 | ralrimi | ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝐵 = 𝐶 ) |
| 7 | mpteq12f | ⊢ ( ( ∀ 𝑥 𝐴 = 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝐵 = 𝐶 ) → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ) | |
| 8 | 4 6 7 | sylancr | ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ) |