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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 6015 | . 2 ⊢ Rel ( I ↾ {𝐴}) | |
2 | relxp 5700 | . 2 ⊢ Rel ({𝐴} × {𝐴}) | |
3 | velsn 4648 | . . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
4 | velsn 4648 | . . . . 5 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴) | |
5 | 3, 4 | anbi12i 626 | . . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴)) |
6 | vex 3477 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
7 | 6 | ideq 5859 | . . . . . 6 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
8 | 3, 7 | anbi12i 626 | . . . . 5 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑥 I 𝑦) ↔ (𝑥 = 𝐴 ∧ 𝑥 = 𝑦)) |
9 | eqeq1 2732 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑦 ↔ 𝐴 = 𝑦)) | |
10 | eqcom 2735 | . . . . . . 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 5153 | . . . . 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 5997 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ( I ↾ {𝐴}) ↔ (𝑥 ∈ {𝐴} ∧ 〈𝑥, 𝑦〉 ∈ I )) |
18 | opelxp 5718 | . . 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 5796 | 1 ⊢ ( I ↾ {𝐴}) = ({𝐴} × {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1533 ∈ wcel 2098 {csn 4632 〈cop 4638 class class class wbr 5152 I cid 5579 × cxp 5680 ↾ cres 5684 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-br 5153 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-res 5694 |
This theorem is referenced by: residpr 7158 grp1inv 19011 psgnsn 19482 m1detdiag 22519 |
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