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Mirrors > Home > MPE Home > Th. List > mpov | Structured version Visualization version GIF version |
Description: Operation with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.) |
Ref | Expression |
---|---|
mpov | ⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝐶} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mpo 7161 | . 2 ⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ 𝑧 = 𝐶)} | |
2 | vex 3497 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | vex 3497 | . . . . 5 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | pm3.2i 473 | . . . 4 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V) |
5 | 4 | biantrur 533 | . . 3 ⊢ (𝑧 = 𝐶 ↔ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ 𝑧 = 𝐶)) |
6 | 5 | oprabbii 7221 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝐶} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ 𝑧 = 𝐶)} |
7 | 1, 6 | eqtr4i 2847 | 1 ⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝐶} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 {coprab 7157 ∈ cmpo 7158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-v 3496 df-oprab 7160 df-mpo 7161 |
This theorem is referenced by: 1st2val 7717 2nd2val 7718 |
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