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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 5461 | . 2 ⊢ I = (𝑥 ∈ V ↦ 𝑥) | |
2 | ancom 463 | . . . . . . 7 ⊢ ((𝑥 ⊆ 𝑦 ∧ ⊤) ↔ (⊤ ∧ 𝑥 ⊆ 𝑦)) | |
3 | truan 1548 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ⊆ 𝑦) ↔ 𝑥 ⊆ 𝑦) | |
4 | 2, 3 | bitri 277 | . . . . . 6 ⊢ ((𝑥 ⊆ 𝑦 ∧ ⊤) ↔ 𝑥 ⊆ 𝑦) |
5 | 4 | abbii 2886 | . . . . 5 ⊢ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)} = {𝑦 ∣ 𝑥 ⊆ 𝑦} |
6 | 5 | inteqi 4880 | . . . 4 ⊢ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)} = ∩ {𝑦 ∣ 𝑥 ⊆ 𝑦} |
7 | vex 3497 | . . . . 5 ⊢ 𝑥 ∈ V | |
8 | 7 | intmin2 4903 | . . . 4 ⊢ ∩ {𝑦 ∣ 𝑥 ⊆ 𝑦} = 𝑥 |
9 | 6, 8 | eqtri 2844 | . . 3 ⊢ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)} = 𝑥 |
10 | 9 | mpteq2i 5158 | . 2 ⊢ (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)}) = (𝑥 ∈ V ↦ 𝑥) |
11 | 1, 10 | eqtr4i 2847 | 1 ⊢ I = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)}) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ⊤wtru 1538 {cab 2799 Vcvv 3494 ⊆ wss 3936 ∩ cint 4876 ↦ 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-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-in 3943 df-ss 3952 df-int 4877 df-opab 5129 df-mpt 5147 df-id 5460 |
This theorem is referenced by: (None) |
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