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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mpteq12da | Structured version Visualization version GIF version |
Description: An equality inference for the maps to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
mpteq12da.1 | ⊢ Ⅎ𝑥𝜑 |
mpteq12da.2 | ⊢ (𝜑 → 𝐴 = 𝐶) |
mpteq12da.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷) |
Ref | Expression |
---|---|
mpteq12da | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpteq12da.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | mpteq12da.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐶) | |
3 | 1, 2 | alrimi 2238 | . 2 ⊢ (𝜑 → ∀𝑥 𝐴 = 𝐶) |
4 | mpteq12da.3 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷) | |
5 | 4 | ex 397 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 = 𝐷)) |
6 | 1, 5 | ralrimi 3106 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) |
7 | mpteq12f 4866 | . 2 ⊢ ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) | |
8 | 3, 6, 7 | syl2anc 573 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∀wal 1629 = wceq 1631 Ⅎwnf 1856 ∈ wcel 2145 ∀wral 3061 ↦ cmpt 4864 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-ext 2751 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-ral 3066 df-opab 4848 df-mpt 4865 |
This theorem is referenced by: smflimmpt 41531 smflimsupmpt 41550 smfliminfmpt 41553 |
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