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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 6056 (see idinxpres 6055). See also elrid 6054 and elidinxp 6052. (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-idres | ⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-res 5694 | . 2 ⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × V)) | |
2 | inss1 4231 | . . . 4 ⊢ ( I ∩ (𝐴 × V)) ⊆ I | |
3 | relinxp 5820 | . . . . 5 ⊢ Rel ( I ∩ (𝐴 × V)) | |
4 | elin 3965 | . . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ∩ (𝐴 × V)) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × V))) | |
5 | bj-opelidb1 36665 | . . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ I ↔ (𝑥 ∈ V ∧ 𝑥 = 𝑦)) | |
6 | 5 | simprbi 495 | . . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ I → 𝑥 = 𝑦) |
7 | opelxp1 5724 | . . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × V) → 𝑥 ∈ 𝐴) | |
8 | simpr 483 | . . . . . . . . 9 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
9 | eleq1w 2812 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
10 | 9 | biimpa 475 | . . . . . . . . 9 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴) |
11 | 8, 10 | jca 510 | . . . . . . . 8 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) |
12 | 6, 7, 11 | syl2an 594 | . . . . . . 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 5719 | . . . . . 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 5793 | . . . 4 ⊢ ( I ∩ (𝐴 × V)) ⊆ (𝐴 × 𝐴) |
17 | 2, 16 | ssini 4234 | . . 3 ⊢ ( I ∩ (𝐴 × V)) ⊆ ( I ∩ (𝐴 × 𝐴)) |
18 | ssv 4006 | . . . 4 ⊢ 𝐴 ⊆ V | |
19 | xpss2 5702 | . . . 4 ⊢ (𝐴 ⊆ V → (𝐴 × 𝐴) ⊆ (𝐴 × V)) | |
20 | sslin 4237 | . . . 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 3998 | . 2 ⊢ ( I ∩ (𝐴 × V)) = ( I ∩ (𝐴 × 𝐴)) |
23 | 1, 22 | eqtri 2756 | 1 ⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3473 ∩ cin 3948 ⊆ wss 3949 ⟨cop 4638 I cid 5579 × cxp 5680 ↾ cres 5684 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-res 5694 |
This theorem is referenced by: bj-opelidres 36673 |
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