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Mathbox for Glauco Siliprandi |
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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 2199 | . 2 ⊢ (𝜑 → ∀𝑥 𝐴 = 𝐵) |
4 | eqid 2778 | . . . 4 ⊢ 𝐶 = 𝐶 | |
5 | 4 | rgenw 3106 | . . 3 ⊢ ∀𝑥 ∈ 𝐴 𝐶 = 𝐶 |
6 | 5 | a1i 11 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐶 = 𝐶) |
7 | mpteq12f 4969 | . 2 ⊢ ((∀𝑥 𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐶) → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)) | |
8 | 3, 6, 7 | syl2anc 579 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1599 = wceq 1601 Ⅎwnf 1827 ∀wral 3090 ↦ cmpt 4967 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-12 2163 ax-ext 2754 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-ral 3095 df-opab 4951 df-mpt 4968 |
This theorem is referenced by: smfliminflem 41977 |
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