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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 6655 | . 2 ⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ 𝑧 = 𝐶)} | |
2 | vex 3203 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | vex 3203 | . . . . 5 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | pm3.2i 471 | . . . 4 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V) |
5 | 4 | biantrur 527 | . . 3 ⊢ (𝑧 = 𝐶 ↔ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ 𝑧 = 𝐶)) |
6 | 5 | oprabbii 6710 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝐶} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ V ∧ 𝑦 ∈ V) ∧ 𝑧 = 𝐶)} |
7 | 1, 6 | eqtr4i 2647 | 1 ⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝐶} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 {coprab 6651 ↦ cmpt2 6652 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-v 3202 df-oprab 6654 df-mpt2 6655 |
This theorem is referenced by: 1st2val 7194 2nd2val 7195 |
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