Theorem bj-idreseq 37728

Description: Sufficient condition for the restricted identity relation to agree with equality. Note that the instance of bj-ideqg 37723 with V substituted for 𝑉 is a direct consequence of bj-idreseq 37728. This is a strengthening of resieq 5990 which should be proved from it (note that currently, resieq 5990 relies on ideq 5839). 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 37712	. . . 4 ⊢ (𝐴( I ↾ 𝐶)𝐵 → 𝐴 ∈ 𝐶)
2		relres 6005	. . . . 5 ⊢ Rel ( I ↾ 𝐶)
3	2	brrelex2i 5719	. . . 4 ⊢ (𝐴( I ↾ 𝐶)𝐵 → 𝐵 ∈ V)
4	1, 3	jca 520	. . 3 ⊢ (𝐴( I ↾ 𝐶)𝐵 → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V))
5	4	adantl 486	. 2 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝐶 ∧ 𝐴( I ↾ 𝐶)𝐵) → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V))
6		eqimss 4003	. . . . . 6 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
7		dfss2 3931	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
8	6, 7	sylib 221	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = 𝐴)
9	8	adantl 486	. . . 4 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝐶 ∧ 𝐴 = 𝐵) → (𝐴 ∩ 𝐵) = 𝐴)
10		simpl 487	. . . 4 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝐶 ∧ 𝐴 = 𝐵) → (𝐴 ∩ 𝐵) ∈ 𝐶)
11	9, 10	eqeltrrd 2870	. . 3 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝐶)
12		eqimss2 4004	. . . . . . 7 ⊢ (𝐴 = 𝐵 → 𝐵 ⊆ 𝐴)
13		sseqin2 4184	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐵) = 𝐵)
14	12, 13	sylib 221	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = 𝐵)
15	14	adantl 486	. . . . 5 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝐶 ∧ 𝐴 = 𝐵) → (𝐴 ∩ 𝐵) = 𝐵)
16	15, 10	eqeltrrd 2870	. . . 4 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝐶)
17	16	elexd 3486	. . 3 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐵 ∈ V)
18	11, 17	jca 520	. 2 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝐶 ∧ 𝐴 = 𝐵) → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V))
19		brres 5986	. . . 4 ⊢ (𝐵 ∈ V → (𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 I 𝐵)))
20	19	adantl 486	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V) → (𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 I 𝐵)))
21		eqeq12 2786	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = 𝑦 ↔ 𝐴 = 𝐵))
22		df-id 5557	. . . . 5 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
23	21, 22	brabga 5519	. . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
24	23	anbi2d 641	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V) → ((𝐴 ∈ 𝐶 ∧ 𝐴 I 𝐵) ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵)))
25		simp3 1154	. . . . 5 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V) ∧ 𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
26	25	3expib 1138	. . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V) → ((𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵))
27		3simpb 1165	. . . . 5 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V ∧ 𝐴 = 𝐵) → (𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵))
28	27	3expia 1137	. . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V) → (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵)))
29	26, 28	impbid 215	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V) → ((𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵) ↔ 𝐴 = 𝐵))
30	20, 24, 29	3bitrd 308	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V) → (𝐴( I ↾ 𝐶)𝐵 ↔ 𝐴 = 𝐵))
31	5, 18, 30	pm5.21nd 813	1 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝐶 → (𝐴( I ↾ 𝐶)𝐵 ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463   ∩ cin 3912   ⊆ wss 3913   class class class wbr 5113   I cid 5556   ↾ cres 5664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-res 5674

This theorem is referenced by: (None)