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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.) |
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 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | mpteq1df.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
3 | 1, 2 | alrimi 2212 | . 2 ⊢ (𝜑 → ∀𝑥 𝐴 = 𝐵) |
4 | eqid 2824 | . . 3 ⊢ 𝐶 = 𝐶 | |
5 | 4 | rgenw 3153 | . 2 ⊢ ∀𝑥 ∈ 𝐴 𝐶 = 𝐶 |
6 | mpteq12f 5152 | . 2 ⊢ ((∀𝑥 𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐶) → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)) | |
7 | 3, 5, 6 | sylancl 588 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1534 = wceq 1536 Ⅎwnf 1783 ∀wral 3141 ↦ cmpt 5149 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-ral 3146 df-opab 5132 df-mpt 5150 |
This theorem is referenced by: smfliminflem 43111 |
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