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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 5487 | . 2 ⊢ I = (𝑥 ∈ V ↦ 𝑥) | |
2 | ancom 461 | . . . . . . 7 ⊢ ((𝑥 ⊆ 𝑦 ∧ ⊤) ↔ (⊤ ∧ 𝑥 ⊆ 𝑦)) | |
3 | truan 1550 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ⊆ 𝑦) ↔ 𝑥 ⊆ 𝑦) | |
4 | 2, 3 | bitri 274 | . . . . . 6 ⊢ ((𝑥 ⊆ 𝑦 ∧ ⊤) ↔ 𝑥 ⊆ 𝑦) |
5 | 4 | abbii 2808 | . . . . 5 ⊢ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)} = {𝑦 ∣ 𝑥 ⊆ 𝑦} |
6 | 5 | inteqi 4885 | . . . 4 ⊢ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)} = ∩ {𝑦 ∣ 𝑥 ⊆ 𝑦} |
7 | vex 3435 | . . . . 5 ⊢ 𝑥 ∈ V | |
8 | 7 | intmin2 4908 | . . . 4 ⊢ ∩ {𝑦 ∣ 𝑥 ⊆ 𝑦} = 𝑥 |
9 | 6, 8 | eqtri 2766 | . . 3 ⊢ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)} = 𝑥 |
10 | 9 | mpteq2i 5180 | . 2 ⊢ (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)}) = (𝑥 ∈ V ↦ 𝑥) |
11 | 1, 10 | eqtr4i 2769 | 1 ⊢ I = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)}) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1539 ⊤wtru 1540 {cab 2715 Vcvv 3431 ⊆ wss 3888 ∩ cint 4881 ↦ cmpt 5158 I cid 5485 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rab 3073 df-v 3433 df-in 3895 df-ss 3905 df-int 4882 df-opab 5138 df-mpt 5159 df-id 5486 |
This theorem is referenced by: (None) |
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