Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 7169 | . 2 ⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ 𝑧 = 𝐶)} | |
2 | vex 3401 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | vex 3401 | . . . . 5 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | pm3.2i 474 | . . . 4 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V) |
5 | 4 | biantrur 534 | . . 3 ⊢ (𝑧 = 𝐶 ↔ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ 𝑧 = 𝐶)) |
6 | 5 | oprabbii 7229 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝐶} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ 𝑧 = 𝐶)} |
7 | 1, 6 | eqtr4i 2764 | 1 ⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝐶} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1542 ∈ wcel 2113 Vcvv 3397 {coprab 7165 ∈ cmpo 7166 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-12 2178 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1545 df-ex 1787 df-nf 1791 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-v 3399 df-oprab 7168 df-mpo 7169 |
This theorem is referenced by: 1st2val 7735 2nd2val 7736 |
Copyright terms: Public domain | W3C validator |