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Mirrors > Home > MPE Home > Th. List > mpteq12df | Structured version Visualization version GIF version |
Description: An equality inference for the maps-to notation. Compare mpteq12dv 5200. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 11-Dec-2016.) (Proof shortened by SN, 11-Nov-2024.) |
Ref | Expression |
---|---|
mpteq12df.1 | ⊢ Ⅎ𝑥𝜑 |
mpteq12df.2 | ⊢ (𝜑 → 𝐴 = 𝐶) |
mpteq12df.3 | ⊢ (𝜑 → 𝐵 = 𝐷) |
Ref | Expression |
---|---|
mpteq12df | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpteq12df.1 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | mpteq12df.2 | . 2 ⊢ (𝜑 → 𝐴 = 𝐶) | |
3 | mpteq12df.3 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐷) | |
4 | 3 | adantr 482 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷) |
5 | 1, 2, 4 | mpteq12da 5194 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 Ⅎwnf 1786 ∈ wcel 2107 ↦ cmpt 5192 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-opab 5172 df-mpt 5193 |
This theorem is referenced by: esumrnmpt2 32731 exrecfnlem 35900 mpteq1df 43552 smflimmpt 45141 |
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