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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-idres | Structured version Visualization version GIF version | ||
| Description: Alternate expression for
the restricted identity relation.  The
       advantage of that expression is to expose it as a "bounded"
class, being
       included in the Cartesian square of the restricting class.  (Contributed
       by BJ, 27-Dec-2023.) This is an alternate of idinxpresid 6065 (see idinxpres 6064). See also elrid 6063 and elidinxp 6061. (Proof modification is discouraged.) | 
| Ref | Expression | 
|---|---|
| bj-idres | ⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × 𝐴)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-res 5696 | . 2 ⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × V)) | |
| 2 | inss1 4236 | . . . 4 ⊢ ( I ∩ (𝐴 × V)) ⊆ I | |
| 3 | relinxp 5823 | . . . . 5 ⊢ Rel ( I ∩ (𝐴 × V)) | |
| 4 | elin 3966 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉 ∈ ( I ∩ (𝐴 × V)) ↔ (〈𝑥, 𝑦〉 ∈ I ∧ 〈𝑥, 𝑦〉 ∈ (𝐴 × V))) | |
| 5 | bj-opelidb1 37155 | . . . . . . . . 9 ⊢ (〈𝑥, 𝑦〉 ∈ I ↔ (𝑥 ∈ V ∧ 𝑥 = 𝑦)) | |
| 6 | 5 | simprbi 496 | . . . . . . . 8 ⊢ (〈𝑥, 𝑦〉 ∈ I → 𝑥 = 𝑦) | 
| 7 | opelxp1 5726 | . . . . . . . 8 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 × V) → 𝑥 ∈ 𝐴) | |
| 8 | simpr 484 | . . . . . . . . 9 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
| 9 | eleq1w 2823 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
| 10 | 9 | biimpa 476 | . . . . . . . . 9 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴) | 
| 11 | 8, 10 | jca 511 | . . . . . . . 8 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) | 
| 12 | 6, 7, 11 | syl2an 596 | . . . . . . 7 ⊢ ((〈𝑥, 𝑦〉 ∈ I ∧ 〈𝑥, 𝑦〉 ∈ (𝐴 × V)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) | 
| 13 | 4, 12 | sylbi 217 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ ( I ∩ (𝐴 × V)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) | 
| 14 | opelxpi 5721 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐴)) | |
| 15 | 13, 14 | syl 17 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ ( I ∩ (𝐴 × V)) → 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐴)) | 
| 16 | 3, 15 | relssi 5796 | . . . 4 ⊢ ( I ∩ (𝐴 × V)) ⊆ (𝐴 × 𝐴) | 
| 17 | 2, 16 | ssini 4239 | . . 3 ⊢ ( I ∩ (𝐴 × V)) ⊆ ( I ∩ (𝐴 × 𝐴)) | 
| 18 | ssv 4007 | . . . 4 ⊢ 𝐴 ⊆ V | |
| 19 | xpss2 5704 | . . . 4 ⊢ (𝐴 ⊆ V → (𝐴 × 𝐴) ⊆ (𝐴 × V)) | |
| 20 | sslin 4242 | . . . 4 ⊢ ((𝐴 × 𝐴) ⊆ (𝐴 × V) → ( I ∩ (𝐴 × 𝐴)) ⊆ ( I ∩ (𝐴 × V))) | |
| 21 | 18, 19, 20 | mp2b 10 | . . 3 ⊢ ( I ∩ (𝐴 × 𝐴)) ⊆ ( I ∩ (𝐴 × V)) | 
| 22 | 17, 21 | eqssi 3999 | . 2 ⊢ ( I ∩ (𝐴 × V)) = ( I ∩ (𝐴 × 𝐴)) | 
| 23 | 1, 22 | eqtri 2764 | 1 ⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × 𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ∩ cin 3949 ⊆ wss 3950 〈cop 4631 I cid 5576 × cxp 5682 ↾ cres 5686 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-res 5696 | 
| This theorem is referenced by: bj-opelidres 37163 | 
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