Theorem bj-idres 38082

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 6039 (see idinxpres 6038). See also elrid 6037 and elidinxp 6035. (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 5662	. 2 ⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × V))
2		inss1 4133	. . . 4 ⊢ ( I ∩ (𝐴 × V)) ⊆ I
3		relinxp 5791	. . . . 5 ⊢ Rel ( I ∩ (𝐴 × V))
4		elin 4093	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ∩ (𝐴 × V)) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × V)))
5		bj-opelidb1 38075	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ I ↔ (𝑥 ∈ V ∧ 𝑥 = 𝑦))
6	5	simprbi 503	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ I → 𝑥 = 𝑦)
7		opelxp1 5692	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × V) → 𝑥 ∈ 𝐴)
8		simpr 490	. . . . . . . . 9 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
9		eleq1w 2843	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
10	9	biimpa 482	. . . . . . . . 9 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴)
11	8, 10	jca 521	. . . . . . . 8 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴))
12	6, 7, 11	syl2an 608	. . . . . . 7 ⊢ ((⟨𝑥, 𝑦⟩ ∈ I ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × V)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴))
13	4, 12	sylbi 220	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ∩ (𝐴 × V)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴))
14		opelxpi 5687	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴))
15	13, 14	syl 18	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ( I ∩ (𝐴 × V)) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴))
16	3, 15	relssi 5762	. . . 4 ⊢ ( I ∩ (𝐴 × V)) ⊆ (𝐴 × 𝐴)
17	2, 16	ssini 4136	. . 3 ⊢ ( I ∩ (𝐴 × V)) ⊆ ( I ∩ (𝐴 × 𝐴))
18		ssv 3932	. . . 4 ⊢ 𝐴 ⊆ V
19		xpss2 5670	. . . 4 ⊢ (𝐴 ⊆ V → (𝐴 × 𝐴) ⊆ (𝐴 × V))
20		sslin 4139	. . . 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 3924	. 2 ⊢ ( I ∩ (𝐴 × V)) = ( I ∩ (𝐴 × 𝐴))
23	1, 22	eqtri 2783	1 ⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × 𝐴))

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3895   ∩ cin 4089  ⟨cop 4589   I cid 5544   × cxp 5648   ↾ cres 5652

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-res 5662

This theorem is used by:  bj-opelidres  38083