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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 2741 | . 2 ⊢ (𝜑 → 𝐶 = 𝐶) | |
4 | 1, 2, 3 | mpteq12df 5165 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 Ⅎwnf 1790 ↦ cmpt 5162 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-12 2175 ax-ext 2711 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1787 df-nf 1791 df-sb 2072 df-clab 2718 df-cleq 2732 df-clel 2818 df-opab 5142 df-mpt 5163 |
This theorem is referenced by: smfliminflem 44329 |
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