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Mirrors > Home > MPE Home > Th. List > mptv | Structured version Visualization version GIF version |
Description: Function with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.) |
Ref | Expression |
---|---|
mptv | ⊢ (𝑥 ∈ V ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝐵} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mpt 5232 | . 2 ⊢ (𝑥 ∈ V ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 = 𝐵)} | |
2 | vex 3482 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | biantrur 530 | . . 3 ⊢ (𝑦 = 𝐵 ↔ (𝑥 ∈ V ∧ 𝑦 = 𝐵)) |
4 | 3 | opabbii 5215 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝐵} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 = 𝐵)} |
5 | 1, 4 | eqtr4i 2766 | 1 ⊢ (𝑥 ∈ V ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝐵} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 {copab 5210 ↦ cmpt 5231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-opab 5211 df-mpt 5232 |
This theorem is referenced by: df1st2 8122 df2nd2 8123 fsplit 8141 rankf 9832 |
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