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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 6041 (see idinxpres 6040). See also elrid 6039 and elidinxp 6037. (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-idres | ⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-res 5681 | . 2 ⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × V)) | |
2 | inss1 4223 | . . . 4 ⊢ ( I ∩ (𝐴 × V)) ⊆ I | |
3 | relinxp 5807 | . . . . 5 ⊢ Rel ( I ∩ (𝐴 × V)) | |
4 | elin 3959 | . . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ∩ (𝐴 × V)) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × V))) | |
5 | bj-opelidb1 36541 | . . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ I ↔ (𝑥 ∈ V ∧ 𝑥 = 𝑦)) | |
6 | 5 | simprbi 496 | . . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ I → 𝑥 = 𝑦) |
7 | opelxp1 5711 | . . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × V) → 𝑥 ∈ 𝐴) | |
8 | simpr 484 | . . . . . . . . 9 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
9 | eleq1w 2810 | . . . . . . . . . 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 595 | . . . . . . 7 ⊢ ((⟨𝑥, 𝑦⟩ ∈ I ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × V)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) |
13 | 4, 12 | sylbi 216 | . . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ∩ (𝐴 × V)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) |
14 | opelxpi 5706 | . . . . . 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 5780 | . . . 4 ⊢ ( I ∩ (𝐴 × V)) ⊆ (𝐴 × 𝐴) |
17 | 2, 16 | ssini 4226 | . . 3 ⊢ ( I ∩ (𝐴 × V)) ⊆ ( I ∩ (𝐴 × 𝐴)) |
18 | ssv 4001 | . . . 4 ⊢ 𝐴 ⊆ V | |
19 | xpss2 5689 | . . . 4 ⊢ (𝐴 ⊆ V → (𝐴 × 𝐴) ⊆ (𝐴 × V)) | |
20 | sslin 4229 | . . . 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 3993 | . 2 ⊢ ( I ∩ (𝐴 × V)) = ( I ∩ (𝐴 × 𝐴)) |
23 | 1, 22 | eqtri 2754 | 1 ⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ∩ cin 3942 ⊆ wss 3943 ⟨cop 4629 I cid 5566 × cxp 5667 ↾ cres 5671 |
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-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-res 5681 |
This theorem is referenced by: bj-opelidres 36549 |
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