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Mirrors > Home > MPE Home > Th. List > mpt2v | Structured version Visualization version GIF version |
Description: Operation with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.) |
Ref | Expression |
---|---|
mpt2v | ⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝐶} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mpt2 6554 | . 2 ⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ 𝑧 = 𝐶)} | |
2 | vex 3176 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | vex 3176 | . . . . 5 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | pm3.2i 470 | . . . 4 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V) |
5 | 4 | biantrur 526 | . . 3 ⊢ (𝑧 = 𝐶 ↔ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ 𝑧 = 𝐶)) |
6 | 5 | oprabbii 6608 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝐶} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ 𝑧 = 𝐶)} |
7 | 1, 6 | eqtr4i 2635 | 1 ⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝐶} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 {coprab 6550 ↦ cmpt2 6551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-v 3175 df-oprab 6553 df-mpt2 6554 |
This theorem is referenced by: 1st2val 7085 2nd2val 7086 |
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