Theorem bj-idreseq 37163

Description: Sufficient condition for the restricted identity relation to agree with equality. Note that the instance of bj-ideqg 37158 with V substituted for 𝑉 is a direct consequence of bj-idreseq 37163. This is a strengthening of resieq 6008 which should be proved from it (note that currently, resieq 6008 relies on ideq 5863). Note that the intersection in the antecedent is not very meaningful, but is a device to prove versions with either class assumed to be a set. It could be enough to prove the version with a disjunctive antecedent: ⊢ ((𝐴 ∈ 𝐶 ∨ 𝐵 ∈ 𝐶) → ...). (Contributed by BJ, 25-Dec-2023.)

Assertion

Ref	Expression
bj-idreseq	⊢ ((𝐴 ∩ 𝐵) ∈ 𝐶 → (𝐴( I ↾ 𝐶)𝐵 ↔ 𝐴 = 𝐵))

Proof of Theorem bj-idreseq

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

Step	Hyp	Ref	Expression
1		bj-brresdm 37147	. . . 4 ⊢ (𝐴( I ↾ 𝐶)𝐵 → 𝐴 ∈ 𝐶)
2		relres 6023	. . . . 5 ⊢ Rel ( I ↾ 𝐶)
3	2	brrelex2i 5742	. . . 4 ⊢ (𝐴( I ↾ 𝐶)𝐵 → 𝐵 ∈ V)
4	1, 3	jca 511	. . 3 ⊢ (𝐴( I ↾ 𝐶)𝐵 → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V))
5	4	adantl 481	. 2 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝐶 ∧ 𝐴( I ↾ 𝐶)𝐵) → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V))
6		eqimss 4042	. . . . . 6 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
7		dfss2 3969	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
8	6, 7	sylib 218	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = 𝐴)
9	8	adantl 481	. . . 4 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝐶 ∧ 𝐴 = 𝐵) → (𝐴 ∩ 𝐵) = 𝐴)
10		simpl 482	. . . 4 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝐶 ∧ 𝐴 = 𝐵) → (𝐴 ∩ 𝐵) ∈ 𝐶)
11	9, 10	eqeltrrd 2842	. . 3 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝐶)
12		eqimss2 4043	. . . . . . 7 ⊢ (𝐴 = 𝐵 → 𝐵 ⊆ 𝐴)
13		sseqin2 4223	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐵) = 𝐵)
14	12, 13	sylib 218	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = 𝐵)
15	14	adantl 481	. . . . 5 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝐶 ∧ 𝐴 = 𝐵) → (𝐴 ∩ 𝐵) = 𝐵)
16	15, 10	eqeltrrd 2842	. . . 4 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝐶)
17	16	elexd 3504	. . 3 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐵 ∈ V)
18	11, 17	jca 511	. 2 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝐶 ∧ 𝐴 = 𝐵) → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V))
19		brres 6004	. . . 4 ⊢ (𝐵 ∈ V → (𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 I 𝐵)))
20	19	adantl 481	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V) → (𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 I 𝐵)))
21		eqeq12 2754	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = 𝑦 ↔ 𝐴 = 𝐵))
22		df-id 5578	. . . . 5 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
23	21, 22	brabga 5539	. . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
24	23	anbi2d 630	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V) → ((𝐴 ∈ 𝐶 ∧ 𝐴 I 𝐵) ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵)))
25		simp3 1139	. . . . 5 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V) ∧ 𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
26	25	3expib 1123	. . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V) → ((𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵))
27		3simpb 1150	. . . . 5 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V ∧ 𝐴 = 𝐵) → (𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵))
28	27	3expia 1122	. . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V) → (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵)))
29	26, 28	impbid 212	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V) → ((𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵) ↔ 𝐴 = 𝐵))
30	20, 24, 29	3bitrd 305	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V) → (𝐴( I ↾ 𝐶)𝐵 ↔ 𝐴 = 𝐵))
31	5, 18, 30	pm5.21nd 802	1 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝐶 → (𝐴( I ↾ 𝐶)𝐵 ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3480   ∩ cin 3950   ⊆ wss 3951   class class class wbr 5143   I cid 5577   ↾ cres 5687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-res 5697

This theorem is referenced by: (None)