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| Mirrors > Home > MPE Home > Th. List > Mathboxes > mpteq1df | Structured version Visualization version GIF version | ||
| Description: An equality theorem for the maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.) (Proof shortened by SN, 11-Nov-2024.) | 
| Ref | Expression | 
|---|---|
| mpteq1df.1 | ⊢ Ⅎ𝑥𝜑 | 
| mpteq1df.2 | ⊢ (𝜑 → 𝐴 = 𝐵) | 
| Ref | Expression | 
|---|---|
| mpteq1df | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | mpteq1df.1 | . 2 ⊢ Ⅎ𝑥𝜑 | |
| 2 | mpteq1df.2 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 3 | eqidd 2738 | . 2 ⊢ (𝜑 → 𝐶 = 𝐶) | |
| 4 | 1, 2, 3 | mpteq12df 5228 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 Ⅎwnf 1783 ↦ cmpt 5225 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-opab 5206 df-mpt 5226 | 
| This theorem is referenced by: smfliminflem 46845 | 
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