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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 2014 | . . . 4 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) | |
2 | vex 3475 | . . . . 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 5576 | . 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 2766 | 1 ⊢ I = (𝑥 ∈ V ↦ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3471 {copab 5210 ↦ cmpt 5231 I cid 5575 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3473 df-opab 5211 df-mpt 5232 df-id 5576 |
This theorem is referenced by: dfid5 15006 dfid6 15007 dfid7 43042 |
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