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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 5915 (see idinxpres 5914). See also elrid 5913 and elidinxp 5911. (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-idres | ⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-res 5567 | . 2 ⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × V)) | |
2 | inss1 4205 | . . . 4 ⊢ ( I ∩ (𝐴 × V)) ⊆ I | |
3 | relinxp 5687 | . . . . 5 ⊢ Rel ( I ∩ (𝐴 × V)) | |
4 | elin 4169 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉 ∈ ( I ∩ (𝐴 × V)) ↔ (〈𝑥, 𝑦〉 ∈ I ∧ 〈𝑥, 𝑦〉 ∈ (𝐴 × V))) | |
5 | bj-opelidb1 34448 | . . . . . . . . 9 ⊢ (〈𝑥, 𝑦〉 ∈ I ↔ (𝑥 ∈ V ∧ 𝑥 = 𝑦)) | |
6 | 5 | simprbi 499 | . . . . . . . 8 ⊢ (〈𝑥, 𝑦〉 ∈ I → 𝑥 = 𝑦) |
7 | opelxp1 5596 | . . . . . . . 8 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 × V) → 𝑥 ∈ 𝐴) | |
8 | simpr 487 | . . . . . . . . 9 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
9 | eleq1w 2895 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
10 | 9 | biimpa 479 | . . . . . . . . 9 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴) |
11 | 8, 10 | jca 514 | . . . . . . . 8 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) |
12 | 6, 7, 11 | syl2an 597 | . . . . . . 7 ⊢ ((〈𝑥, 𝑦〉 ∈ I ∧ 〈𝑥, 𝑦〉 ∈ (𝐴 × V)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) |
13 | 4, 12 | sylbi 219 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ ( I ∩ (𝐴 × V)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) |
14 | opelxpi 5592 | . . . . . 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 5660 | . . . 4 ⊢ ( I ∩ (𝐴 × V)) ⊆ (𝐴 × 𝐴) |
17 | 2, 16 | ssini 4208 | . . 3 ⊢ ( I ∩ (𝐴 × V)) ⊆ ( I ∩ (𝐴 × 𝐴)) |
18 | ssv 3991 | . . . 4 ⊢ 𝐴 ⊆ V | |
19 | xpss2 5575 | . . . 4 ⊢ (𝐴 ⊆ V → (𝐴 × 𝐴) ⊆ (𝐴 × V)) | |
20 | sslin 4211 | . . . 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 3983 | . 2 ⊢ ( I ∩ (𝐴 × V)) = ( I ∩ (𝐴 × 𝐴)) |
23 | 1, 22 | eqtri 2844 | 1 ⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∩ cin 3935 ⊆ wss 3936 〈cop 4573 I cid 5459 × cxp 5553 ↾ cres 5557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-res 5567 |
This theorem is referenced by: bj-opelidres 34456 |
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