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| Mirrors > Home > MPE Home > Th. List > mpteq12dv | Structured version Visualization version GIF version | ||
| Description: An equality inference for the maps-to notation. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 16-Dec-2013.) Remove dependency on ax-10 2141, ax-12 2177. (Revised by SN and GG, 1-Dec-2023.) |
| Ref | Expression |
|---|---|
| mpteq12dv.1 | ⊢ (𝜑 → 𝐴 = 𝐶) |
| mpteq12dv.2 | ⊢ (𝜑 → 𝐵 = 𝐷) |
| Ref | Expression |
|---|---|
| mpteq12dv | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mpteq12dv.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐶) | |
| 2 | mpteq12dv.2 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐷) | |
| 3 | 2 | adantr 480 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷) |
| 4 | 1, 3 | mpteq12dva 5231 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) |
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