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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 2025 | . . . 4 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) | |
2 | vex 3497 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | 2 | biantrur 533 | . . . 4 ⊢ (𝑦 = 𝑥 ↔ (𝑥 ∈ V ∧ 𝑦 = 𝑥)) |
4 | 1, 3 | bitri 277 | . . 3 ⊢ (𝑥 = 𝑦 ↔ (𝑥 ∈ V ∧ 𝑦 = 𝑥)) |
5 | 4 | opabbii 5133 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 = 𝑥)} |
6 | df-id 5460 | . 2 ⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} | |
7 | df-mpt 5147 | . 2 ⊢ (𝑥 ∈ V ↦ 𝑥) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 = 𝑥)} | |
8 | 5, 6, 7 | 3eqtr4i 2854 | 1 ⊢ I = (𝑥 ∈ V ↦ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 {copab 5128 ↦ cmpt 5146 I cid 5459 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-v 3496 df-opab 5129 df-mpt 5147 df-id 5460 |
This theorem is referenced by: dfid5 14386 dfid6 14387 dfid7 39992 |
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