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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 5970 | . 2 ⊢ Rel ( I ↾ {𝐴}) | |
2 | relxp 5655 | . 2 ⊢ Rel ({𝐴} × {𝐴}) | |
3 | velsn 4606 | . . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
4 | velsn 4606 | . . . . 5 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴) | |
5 | 3, 4 | anbi12i 628 | . . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴)) |
6 | vex 3451 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
7 | 6 | ideq 5812 | . . . . . 6 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
8 | 3, 7 | anbi12i 628 | . . . . 5 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑥 I 𝑦) ↔ (𝑥 = 𝐴 ∧ 𝑥 = 𝑦)) |
9 | eqeq1 2737 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑦 ↔ 𝐴 = 𝑦)) | |
10 | eqcom 2740 | . . . . . . 7 ⊢ (𝐴 = 𝑦 ↔ 𝑦 = 𝐴) | |
11 | 9, 10 | bitrdi 287 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑦 = 𝐴)) |
12 | 11 | pm5.32i 576 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝑦) ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴)) |
13 | 8, 12 | bitri 275 | . . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑥 I 𝑦) ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴)) |
14 | df-br 5110 | . . . . 5 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I ) | |
15 | 14 | anbi2i 624 | . . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑥 I 𝑦) ↔ (𝑥 ∈ {𝐴} ∧ ⟨𝑥, 𝑦⟩ ∈ I )) |
16 | 5, 13, 15 | 3bitr2ri 300 | . . 3 ⊢ ((𝑥 ∈ {𝐴} ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) |
17 | 6 | opelresi 5949 | . . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ {𝐴}) ↔ (𝑥 ∈ {𝐴} ∧ ⟨𝑥, 𝑦⟩ ∈ I )) |
18 | opelxp 5673 | . . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({𝐴} × {𝐴}) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) | |
19 | 16, 17, 18 | 3bitr4i 303 | . 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ {𝐴}) ↔ ⟨𝑥, 𝑦⟩ ∈ ({𝐴} × {𝐴})) |
20 | 1, 2, 19 | eqrelriiv 5750 | 1 ⊢ ( I ↾ {𝐴}) = ({𝐴} × {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 ∈ wcel 2107 {csn 4590 ⟨cop 4596 class class class wbr 5109 I cid 5534 × cxp 5635 ↾ cres 5639 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-id 5535 df-xp 5643 df-rel 5644 df-res 5649 |
This theorem is referenced by: residpr 7093 grp1inv 18863 psgnsn 19310 m1detdiag 21969 |
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