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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 6010 | . 2 ⊢ Rel ( I ↾ {𝐴}) | |
2 | relxp 5694 | . 2 ⊢ Rel ({𝐴} × {𝐴}) | |
3 | velsn 4644 | . . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
4 | velsn 4644 | . . . . 5 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴) | |
5 | 3, 4 | anbi12i 627 | . . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴)) |
6 | vex 3478 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
7 | 6 | ideq 5852 | . . . . . 6 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
8 | 3, 7 | anbi12i 627 | . . . . 5 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑥 I 𝑦) ↔ (𝑥 = 𝐴 ∧ 𝑥 = 𝑦)) |
9 | eqeq1 2736 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑦 ↔ 𝐴 = 𝑦)) | |
10 | eqcom 2739 | . . . . . . 7 ⊢ (𝐴 = 𝑦 ↔ 𝑦 = 𝐴) | |
11 | 9, 10 | bitrdi 286 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑦 = 𝐴)) |
12 | 11 | pm5.32i 575 | . . . . 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 5149 | . . . . 5 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I ) | |
15 | 14 | anbi2i 623 | . . . 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 5989 | . . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ {𝐴}) ↔ (𝑥 ∈ {𝐴} ∧ ⟨𝑥, 𝑦⟩ ∈ I )) |
18 | opelxp 5712 | . . 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 5790 | 1 ⊢ ( I ↾ {𝐴}) = ({𝐴} × {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1541 ∈ wcel 2106 {csn 4628 ⟨cop 4634 class class class wbr 5148 I cid 5573 × cxp 5674 ↾ cres 5678 |
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-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-res 5688 |
This theorem is referenced by: residpr 7140 grp1inv 18930 psgnsn 19387 m1detdiag 22098 |
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