Theorem bj-idreseq 37128

Description: Sufficient condition for the restricted identity relation to agree with equality. Note that the instance of bj-ideqg 37123 with V substituted for 𝑉 is a direct consequence of bj-idreseq 37128. This is a strengthening of resieq 6020 which should be proved from it (note that currently, resieq 6020 relies on ideq 5877). 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 37112	. . . 4 ⊢ (𝐴( I ↾ 𝐶)𝐵 → 𝐴 ∈ 𝐶)
2		relres 6035	. . . . 5 ⊢ Rel ( I ↾ 𝐶)
3	2	brrelex2i 5757	. . . 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 4067	. . . . . 6 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
7		dfss2 3994	. . . . . 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 2845	. . 3 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝐶)
12		eqimss2 4068	. . . . . . 7 ⊢ (𝐴 = 𝐵 → 𝐵 ⊆ 𝐴)
13		sseqin2 4244	. . . . . . 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 2845	. . . 4 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝐶)
17	16	elexd 3512	. . 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 6016	. . . 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 2757	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = 𝑦 ↔ 𝐴 = 𝐵))
22		df-id 5593	. . . . 5 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
23	21, 22	brabga 5553	. . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
24	23	anbi2d 629	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V) → ((𝐴 ∈ 𝐶 ∧ 𝐴 I 𝐵) ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵)))
25		simp3 1138	. . . . 5 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V) ∧ 𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
26	25	3expib 1122	. . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V) → ((𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵))
27		3simpb 1149	. . . . 5 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V ∧ 𝐴 = 𝐵) → (𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵))
28	27	3expia 1121	. . . 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 801	1 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝐶 → (𝐴( I ↾ 𝐶)𝐵 ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ∩ cin 3975   ⊆ wss 3976   class class class wbr 5166   I cid 5592   ↾ cres 5702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-res 5712

This theorem is referenced by: (None)