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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 5226 | . 2 ⊢ (𝑥 ∈ V ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 = 𝐵)} | |
| 2 | vex 3484 | . . . 4 ⊢ 𝑥 ∈ V | |
| 3 | 2 | biantrur 530 | . . 3 ⊢ (𝑦 = 𝐵 ↔ (𝑥 ∈ V ∧ 𝑦 = 𝐵)) |
| 4 | 3 | opabbii 5210 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝐵} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 = 𝐵)} |
| 5 | 1, 4 | eqtr4i 2768 | 1 ⊢ (𝑥 ∈ V ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝐵} |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 {copab 5205 ↦ cmpt 5225 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-opab 5206 df-mpt 5226 |
| This theorem is referenced by: df1st2 8123 df2nd2 8124 fsplit 8142 rankf 9834 |
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