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Mirrors > Home > MPE Home > Th. List > restidsing | Structured version Visualization version GIF version |
Description: Restriction of the identity to a singleton. (Contributed by FL, 2-Aug-2009.) (Proof shortened by JJ, 25-Aug-2021.) (Proof shortened by Peter Mazsa, 6-Oct-2022.) |
Ref | Expression |
---|---|
restidsing | ⊢ ( I ↾ {𝐴}) = ({𝐴} × {𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relres 6009 | . 2 ⊢ Rel ( I ↾ {𝐴}) | |
2 | relxp 5693 | . 2 ⊢ Rel ({𝐴} × {𝐴}) | |
3 | velsn 4643 | . . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
4 | velsn 4643 | . . . . 5 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴) | |
5 | 3, 4 | anbi12i 625 | . . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴)) |
6 | vex 3476 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
7 | 6 | ideq 5851 | . . . . . 6 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
8 | 3, 7 | anbi12i 625 | . . . . 5 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑥 I 𝑦) ↔ (𝑥 = 𝐴 ∧ 𝑥 = 𝑦)) |
9 | eqeq1 2734 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑦 ↔ 𝐴 = 𝑦)) | |
10 | eqcom 2737 | . . . . . . 7 ⊢ (𝐴 = 𝑦 ↔ 𝑦 = 𝐴) | |
11 | 9, 10 | bitrdi 286 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑦 = 𝐴)) |
12 | 11 | pm5.32i 573 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝑦) ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴)) |
13 | 8, 12 | bitri 274 | . . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑥 I 𝑦) ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴)) |
14 | df-br 5148 | . . . . 5 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I ) | |
15 | 14 | anbi2i 621 | . . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑥 I 𝑦) ↔ (𝑥 ∈ {𝐴} ∧ ⟨𝑥, 𝑦⟩ ∈ I )) |
16 | 5, 13, 15 | 3bitr2ri 299 | . . 3 ⊢ ((𝑥 ∈ {𝐴} ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) |
17 | 6 | opelresi 5988 | . . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ {𝐴}) ↔ (𝑥 ∈ {𝐴} ∧ ⟨𝑥, 𝑦⟩ ∈ I )) |
18 | opelxp 5711 | . . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({𝐴} × {𝐴}) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) | |
19 | 16, 17, 18 | 3bitr4i 302 | . 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ {𝐴}) ↔ ⟨𝑥, 𝑦⟩ ∈ ({𝐴} × {𝐴})) |
20 | 1, 2, 19 | eqrelriiv 5789 | 1 ⊢ ( I ↾ {𝐴}) = ({𝐴} × {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1539 ∈ wcel 2104 {csn 4627 ⟨cop 4633 class class class wbr 5147 I cid 5572 × cxp 5673 ↾ cres 5677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-res 5687 |
This theorem is referenced by: residpr 7142 grp1inv 18967 psgnsn 19429 m1detdiag 22319 |
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