![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
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 2727 | . 2 ⊢ (𝜑 → 𝐶 = 𝐶) | |
4 | 1, 2, 3 | mpteq12df 5227 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 Ⅎwnf 1777 ↦ cmpt 5224 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-12 2163 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-opab 5204 df-mpt 5225 |
This theorem is referenced by: smfliminflem 46115 |
Copyright terms: Public domain | W3C validator |