Theorem bj-idreseq 36043

Description: Sufficient condition for the restricted identity relation to agree with equality. Note that the instance of bj-ideqg 36038 with V substituted for 𝑉 is a direct consequence of bj-idreseq 36043. This is a strengthening of resieq 5993 which should be proved from it (note that currently, resieq 5993 relies on ideq 5853). 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 36027	. . . 4 ⊢ (𝐴( I ↾ 𝐶)𝐵 → 𝐴 ∈ 𝐶)
2		relres 6011	. . . . 5 ⊢ Rel ( I ↾ 𝐶)
3	2	brrelex2i 5734	. . . 4 ⊢ (𝐴( I ↾ 𝐶)𝐵 → 𝐵 ∈ V)
4	1, 3	jca 513	. . 3 ⊢ (𝐴( I ↾ 𝐶)𝐵 → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V))
5	4	adantl 483	. 2 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝐶 ∧ 𝐴( I ↾ 𝐶)𝐵) → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V))
6		eqimss 4041	. . . . . 6 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
7		df-ss 3966	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
8	6, 7	sylib 217	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = 𝐴)
9	8	adantl 483	. . . 4 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝐶 ∧ 𝐴 = 𝐵) → (𝐴 ∩ 𝐵) = 𝐴)
10		simpl 484	. . . 4 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝐶 ∧ 𝐴 = 𝐵) → (𝐴 ∩ 𝐵) ∈ 𝐶)
11	9, 10	eqeltrrd 2835	. . 3 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝐶)
12		eqimss2 4042	. . . . . . 7 ⊢ (𝐴 = 𝐵 → 𝐵 ⊆ 𝐴)
13		sseqin2 4216	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐵) = 𝐵)
14	12, 13	sylib 217	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = 𝐵)
15	14	adantl 483	. . . . 5 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝐶 ∧ 𝐴 = 𝐵) → (𝐴 ∩ 𝐵) = 𝐵)
16	15, 10	eqeltrrd 2835	. . . 4 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝐶)
17	16	elexd 3495	. . 3 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐵 ∈ V)
18	11, 17	jca 513	. 2 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝐶 ∧ 𝐴 = 𝐵) → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V))
19		brres 5989	. . . 4 ⊢ (𝐵 ∈ V → (𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 I 𝐵)))
20	19	adantl 483	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V) → (𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 I 𝐵)))
21		eqeq12 2750	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = 𝑦 ↔ 𝐴 = 𝐵))
22		df-id 5575	. . . . 5 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
23	21, 22	brabga 5535	. . . 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 211	. . 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 801	1 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝐶 → (𝐴( I ↾ 𝐶)𝐵 ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3475   ∩ cin 3948   ⊆ wss 3949   class class class wbr 5149   I cid 5574   ↾ cres 5679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-res 5689

This theorem is referenced by: (None)