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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfid7 | Structured version Visualization version GIF version |
Description: Definition of identity relation as the trivial closure. (Contributed by RP, 26-Jul-2020.) |
Ref | Expression |
---|---|
dfid7 | ⊢ I = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfid4 5530 | . 2 ⊢ I = (𝑥 ∈ V ↦ 𝑥) | |
2 | ancom 461 | . . . . . . 7 ⊢ ((𝑥 ⊆ 𝑦 ∧ ⊤) ↔ (⊤ ∧ 𝑥 ⊆ 𝑦)) | |
3 | truan 1552 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ⊆ 𝑦) ↔ 𝑥 ⊆ 𝑦) | |
4 | 2, 3 | bitri 274 | . . . . . 6 ⊢ ((𝑥 ⊆ 𝑦 ∧ ⊤) ↔ 𝑥 ⊆ 𝑦) |
5 | 4 | abbii 2806 | . . . . 5 ⊢ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)} = {𝑦 ∣ 𝑥 ⊆ 𝑦} |
6 | 5 | inteqi 4909 | . . . 4 ⊢ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)} = ∩ {𝑦 ∣ 𝑥 ⊆ 𝑦} |
7 | vex 3447 | . . . . 5 ⊢ 𝑥 ∈ V | |
8 | 7 | intmin2 4934 | . . . 4 ⊢ ∩ {𝑦 ∣ 𝑥 ⊆ 𝑦} = 𝑥 |
9 | 6, 8 | eqtri 2764 | . . 3 ⊢ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)} = 𝑥 |
10 | 9 | mpteq2i 5208 | . 2 ⊢ (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)}) = (𝑥 ∈ V ↦ 𝑥) |
11 | 1, 10 | eqtr4i 2767 | 1 ⊢ I = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)}) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1541 ⊤wtru 1542 {cab 2713 Vcvv 3443 ⊆ wss 3908 ∩ cint 4905 ↦ cmpt 5186 I cid 5528 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3063 df-rab 3406 df-v 3445 df-in 3915 df-ss 3925 df-int 4906 df-opab 5166 df-mpt 5187 df-id 5529 |
This theorem is referenced by: (None) |
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