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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 4955 | . 2 ⊢ I = (𝑥 ∈ V ↦ 𝑥) | |
2 | ancom 465 | . . . . . . 7 ⊢ ((𝑥 ⊆ 𝑦 ∧ ⊤) ↔ (⊤ ∧ 𝑥 ⊆ 𝑦)) | |
3 | truan 1492 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ⊆ 𝑦) ↔ 𝑥 ⊆ 𝑦) | |
4 | 2, 3 | bitri 263 | . . . . . 6 ⊢ ((𝑥 ⊆ 𝑦 ∧ ⊤) ↔ 𝑥 ⊆ 𝑦) |
5 | 4 | abbii 2726 | . . . . 5 ⊢ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)} = {𝑦 ∣ 𝑥 ⊆ 𝑦} |
6 | 5 | inteqi 4414 | . . . 4 ⊢ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)} = ∩ {𝑦 ∣ 𝑥 ⊆ 𝑦} |
7 | vex 3176 | . . . . 5 ⊢ 𝑥 ∈ V | |
8 | 7 | intmin2 4439 | . . . 4 ⊢ ∩ {𝑦 ∣ 𝑥 ⊆ 𝑦} = 𝑥 |
9 | 6, 8 | eqtri 2632 | . . 3 ⊢ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)} = 𝑥 |
10 | 9 | mpteq2i 4669 | . 2 ⊢ (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)}) = (𝑥 ∈ V ↦ 𝑥) |
11 | 1, 10 | eqtr4i 2635 | 1 ⊢ I = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)}) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ⊤wtru 1476 {cab 2596 Vcvv 3173 ⊆ wss 3540 ∩ cint 4410 ↦ cmpt 4643 I cid 4948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rab 2905 df-v 3175 df-in 3547 df-ss 3554 df-int 4411 df-opab 4644 df-mpt 4645 df-id 4953 |
This theorem is referenced by: (None) |
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