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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 2017 | . . . 4 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) | |
| 2 | vex 3484 | . . . . 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 5210 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 = 𝑥)} |
| 6 | df-id 5578 | . 2 ⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} | |
| 7 | df-mpt 5226 | . 2 ⊢ (𝑥 ∈ V ↦ 𝑥) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 = 𝑥)} | |
| 8 | 5, 6, 7 | 3eqtr4i 2775 | 1 ⊢ I = (𝑥 ∈ V ↦ 𝑥) |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 {copab 5205 ↦ cmpt 5225 I cid 5577 |
| 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 df-id 5578 |
| This theorem is referenced by: dfid5 15066 dfid6 15067 dfid7 43625 |
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