Theorem bj-idres 35340

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 5954 (see idinxpres 5953). See also elrid 5952 and elidinxp 5950. (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 5602	. 2 ⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × V))
2		inss1 4168	. . . 4 ⊢ ( I ∩ (𝐴 × V)) ⊆ I
3		relinxp 5723	. . . . 5 ⊢ Rel ( I ∩ (𝐴 × V))
4		elin 3908	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ∩ (𝐴 × V)) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × V)))
5		bj-opelidb1 35333	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ I ↔ (𝑥 ∈ V ∧ 𝑥 = 𝑦))
6	5	simprbi 497	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ I → 𝑥 = 𝑦)
7		opelxp1 5631	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × V) → 𝑥 ∈ 𝐴)
8		simpr 485	. . . . . . . . 9 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
9		eleq1w 2823	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
10	9	biimpa 477	. . . . . . . . 9 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴)
11	8, 10	jca 512	. . . . . . . 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 216	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ∩ (𝐴 × V)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴))
14		opelxpi 5627	. . . . . 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 5696	. . . 4 ⊢ ( I ∩ (𝐴 × V)) ⊆ (𝐴 × 𝐴)
17	2, 16	ssini 4171	. . 3 ⊢ ( I ∩ (𝐴 × V)) ⊆ ( I ∩ (𝐴 × 𝐴))
18		ssv 3950	. . . 4 ⊢ 𝐴 ⊆ V
19		xpss2 5610	. . . 4 ⊢ (𝐴 ⊆ V → (𝐴 × 𝐴) ⊆ (𝐴 × V))
20		sslin 4174	. . . 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 3942	. 2 ⊢ ( I ∩ (𝐴 × V)) = ( I ∩ (𝐴 × 𝐴))
23	1, 22	eqtri 2768	1 ⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × 𝐴))

Syntax hints:   ∧ wa 396   = wceq 1542   ∈ wcel 2110  Vcvv 3431   ∩ cin 3891   ⊆ wss 3892  ⟨cop 4573   I cid 5489   × cxp 5588   ↾ cres 5592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-opab 5142  df-id 5490  df-xp 5596  df-rel 5597  df-res 5602

This theorem is referenced by:  bj-opelidres  35341