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Mirrors > Home > ILE Home > Th. List > restidsing | 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 4936 | . 2 ⊢ Rel ( I ↾ {𝐴}) | |
2 | relxp 4736 | . 2 ⊢ Rel ({𝐴} × {𝐴}) | |
3 | velsn 3610 | . . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
4 | velsn 3610 | . . . . 5 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴) | |
5 | 3, 4 | anbi12i 460 | . . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴)) |
6 | vex 2741 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
7 | 6 | ideq 4780 | . . . . . 6 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
8 | 3, 7 | anbi12i 460 | . . . . 5 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑥 I 𝑦) ↔ (𝑥 = 𝐴 ∧ 𝑥 = 𝑦)) |
9 | eqeq1 2184 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑦 ↔ 𝐴 = 𝑦)) | |
10 | eqcom 2179 | . . . . . . 7 ⊢ (𝐴 = 𝑦 ↔ 𝑦 = 𝐴) | |
11 | 9, 10 | bitrdi 196 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑦 = 𝐴)) |
12 | 11 | pm5.32i 454 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝑦) ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴)) |
13 | 8, 12 | bitri 184 | . . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑥 I 𝑦) ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐴)) |
14 | df-br 4005 | . . . . 5 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I ) | |
15 | 14 | anbi2i 457 | . . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑥 I 𝑦) ↔ (𝑥 ∈ {𝐴} ∧ ⟨𝑥, 𝑦⟩ ∈ I )) |
16 | 5, 13, 15 | 3bitr2ri 209 | . . 3 ⊢ ((𝑥 ∈ {𝐴} ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) |
17 | 6 | opelres 4913 | . . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ {𝐴}) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ 𝑥 ∈ {𝐴})) |
18 | 17 | biancomi 270 | . . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ {𝐴}) ↔ (𝑥 ∈ {𝐴} ∧ ⟨𝑥, 𝑦⟩ ∈ I )) |
19 | opelxp 4657 | . . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ({𝐴} × {𝐴}) ↔ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) | |
20 | 16, 18, 19 | 3bitr4i 212 | . 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ↾ {𝐴}) ↔ ⟨𝑥, 𝑦⟩ ∈ ({𝐴} × {𝐴})) |
21 | 1, 2, 20 | eqrelriiv 4721 | 1 ⊢ ( I ↾ {𝐴}) = ({𝐴} × {𝐴}) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 = wceq 1353 ∈ wcel 2148 {csn 3593 ⟨cop 3596 class class class wbr 4004 I cid 4289 × cxp 4625 ↾ cres 4629 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4122 ax-pow 4175 ax-pr 4210 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2740 df-un 3134 df-in 3136 df-ss 3143 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-br 4005 df-opab 4066 df-id 4294 df-xp 4633 df-rel 4634 df-res 4639 |
This theorem is referenced by: grp1inv 12977 |
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