Theorem bj-idres 35258

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 5944 (see idinxpres 5943). See also elrid 5942 and elidinxp 5940. (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-idres	⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × 𝐴))

Proof of Theorem bj-idres

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-res 5592	. 2 ⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × V))
2		inss1 4159	. . . 4 ⊢ ( I ∩ (𝐴 × V)) ⊆ I
3		relinxp 5713	. . . . 5 ⊢ Rel ( I ∩ (𝐴 × V))
4		elin 3899	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ∩ (𝐴 × V)) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × V)))
5		bj-opelidb1 35251	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ I ↔ (𝑥 ∈ V ∧ 𝑥 = 𝑦))
6	5	simprbi 496	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ I → 𝑥 = 𝑦)
7		opelxp1 5621	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × V) → 𝑥 ∈ 𝐴)
8		simpr 484	. . . . . . . . 9 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
9		eleq1w 2821	. . . . . . . . . 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 5617	. . . . . 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 5686	. . . 4 ⊢ ( I ∩ (𝐴 × V)) ⊆ (𝐴 × 𝐴)
17	2, 16	ssini 4162	. . 3 ⊢ ( I ∩ (𝐴 × V)) ⊆ ( I ∩ (𝐴 × 𝐴))
18		ssv 3941	. . . 4 ⊢ 𝐴 ⊆ V
19		xpss2 5600	. . . 4 ⊢ (𝐴 ⊆ V → (𝐴 × 𝐴) ⊆ (𝐴 × V))
20		sslin 4165	. . . 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 3933	. 2 ⊢ ( I ∩ (𝐴 × V)) = ( I ∩ (𝐴 × 𝐴))
23	1, 22	eqtri 2766	1 ⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × 𝐴))

Syntax hints:   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ∩ cin 3882   ⊆ wss 3883  ⟨cop 4564   I cid 5479   × cxp 5578   ↾ cres 5582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-res 5592

This theorem is referenced by:  bj-opelidres  35259