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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 6068 (see idinxpres 6067). See also elrid 6066 and elidinxp 6064. (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-idres | ⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-res 5701 | . 2 ⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × V)) | |
2 | inss1 4245 | . . . 4 ⊢ ( I ∩ (𝐴 × V)) ⊆ I | |
3 | relinxp 5827 | . . . . 5 ⊢ Rel ( I ∩ (𝐴 × V)) | |
4 | elin 3979 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉 ∈ ( I ∩ (𝐴 × V)) ↔ (〈𝑥, 𝑦〉 ∈ I ∧ 〈𝑥, 𝑦〉 ∈ (𝐴 × V))) | |
5 | bj-opelidb1 37136 | . . . . . . . . 9 ⊢ (〈𝑥, 𝑦〉 ∈ I ↔ (𝑥 ∈ V ∧ 𝑥 = 𝑦)) | |
6 | 5 | simprbi 496 | . . . . . . . 8 ⊢ (〈𝑥, 𝑦〉 ∈ I → 𝑥 = 𝑦) |
7 | opelxp1 5731 | . . . . . . . 8 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 × V) → 𝑥 ∈ 𝐴) | |
8 | simpr 484 | . . . . . . . . 9 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
9 | eleq1w 2822 | . . . . . . . . . 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 5726 | . . . . . 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 5800 | . . . 4 ⊢ ( I ∩ (𝐴 × V)) ⊆ (𝐴 × 𝐴) |
17 | 2, 16 | ssini 4248 | . . 3 ⊢ ( I ∩ (𝐴 × V)) ⊆ ( I ∩ (𝐴 × 𝐴)) |
18 | ssv 4020 | . . . 4 ⊢ 𝐴 ⊆ V | |
19 | xpss2 5709 | . . . 4 ⊢ (𝐴 ⊆ V → (𝐴 × 𝐴) ⊆ (𝐴 × V)) | |
20 | sslin 4251 | . . . 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 4012 | . 2 ⊢ ( I ∩ (𝐴 × V)) = ( I ∩ (𝐴 × 𝐴)) |
23 | 1, 22 | eqtri 2763 | 1 ⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∩ cin 3962 ⊆ wss 3963 〈cop 4637 I cid 5582 × cxp 5687 ↾ cres 5691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-res 5701 |
This theorem is referenced by: bj-opelidres 37144 |
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