![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > dfid4 | Structured version Visualization version GIF version |
Description: The identity function expressed using maps-to notation. (Contributed by Scott Fenton, 15-Dec-2017.) |
Ref | Expression |
---|---|
dfid4 | ⊢ I = (𝑥 ∈ V ↦ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equcom 2015 | . . . 4 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) | |
2 | vex 3482 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | 2 | biantrur 530 | . . . 4 ⊢ (𝑦 = 𝑥 ↔ (𝑥 ∈ V ∧ 𝑦 = 𝑥)) |
4 | 1, 3 | bitri 275 | . . 3 ⊢ (𝑥 = 𝑦 ↔ (𝑥 ∈ V ∧ 𝑦 = 𝑥)) |
5 | 4 | opabbii 5215 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 = 𝑥)} |
6 | df-id 5583 | . 2 ⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} | |
7 | df-mpt 5232 | . 2 ⊢ (𝑥 ∈ V ↦ 𝑥) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 = 𝑥)} | |
8 | 5, 6, 7 | 3eqtr4i 2773 | 1 ⊢ I = (𝑥 ∈ V ↦ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 {copab 5210 ↦ cmpt 5231 I cid 5582 |
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 df-id 5583 |
This theorem is referenced by: dfid5 15063 dfid6 15064 dfid7 43602 |
Copyright terms: Public domain | W3C validator |