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Mirrors > Home > MPE Home > Th. List > dfid4 | Structured version Visualization version GIF version |
Description: The identity function using maps-to notation. (Contributed by Scott Fenton, 15-Dec-2017.) |
Ref | Expression |
---|---|
dfid4 | ⊢ I = (𝑥 ∈ V ↦ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equcom 2003 | . . . 4 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) | |
2 | vex 3439 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | 2 | biantrur 531 | . . . 4 ⊢ (𝑦 = 𝑥 ↔ (𝑥 ∈ V ∧ 𝑦 = 𝑥)) |
4 | 1, 3 | bitri 276 | . . 3 ⊢ (𝑥 = 𝑦 ↔ (𝑥 ∈ V ∧ 𝑦 = 𝑥)) |
5 | 4 | opabbii 5031 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 = 𝑥)} |
6 | df-id 5351 | . 2 ⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} | |
7 | df-mpt 5044 | . 2 ⊢ (𝑥 ∈ V ↦ 𝑥) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 = 𝑥)} | |
8 | 5, 6, 7 | 3eqtr4i 2828 | 1 ⊢ I = (𝑥 ∈ V ↦ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1522 ∈ wcel 2080 Vcvv 3436 {copab 5026 ↦ cmpt 5043 I cid 5350 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-12 2140 ax-ext 2768 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1763 df-nf 1767 df-sb 2042 df-clab 2775 df-cleq 2787 df-clel 2862 df-v 3438 df-opab 5027 df-mpt 5044 df-id 5351 |
This theorem is referenced by: dfid5 14220 dfid6 14221 cnmptid 21953 dfid7 39470 |
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