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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 2211 | . 2 ⊢ (𝜑 → ∀𝑥 𝐴 = 𝐶) |
4 | mpteq12da.3 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷) | |
5 | 1, 4 | ralrimia 41767 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) |
6 | mpteq12f 5113 | . 2 ⊢ ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) | |
7 | 3, 5, 6 | syl2anc 587 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∀wal 1536 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 ∀wral 3106 ↦ cmpt 5110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ral 3111 df-opab 5093 df-mpt 5111 |
This theorem is referenced by: smflimmpt 43441 smflimsupmpt 43460 smfliminfmpt 43463 |
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