Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > mptv | 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 3820 | . 2 ⊢ (𝑥 ∈ V ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 = 𝐵)} | |
2 | vex 2560 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | biantrur 287 | . . 3 ⊢ (𝑦 = 𝐵 ↔ (𝑥 ∈ V ∧ 𝑦 = 𝐵)) |
4 | 3 | opabbii 3824 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝐵} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 = 𝐵)} |
5 | 1, 4 | eqtr4i 2063 | 1 ⊢ (𝑥 ∈ V ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ 𝑦 = 𝐵} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 = wceq 1243 ∈ wcel 1393 Vcvv 2557 {copab 3817 ↦ cmpt 3818 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-v 2559 df-opab 3819 df-mpt 3820 |
This theorem is referenced by: df1st2 5840 df2nd2 5841 |
Copyright terms: Public domain | W3C validator |