Theorem bj-idreseqb 37902

Description: Characterization for two classes to be related under the restricted identity relation. (Contributed by BJ, 24-Dec-2023.)

Assertion

Ref	Expression
bj-idreseqb	⊢ (𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵))

Proof of Theorem bj-idreseqb

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

Step	Hyp	Ref	Expression
1		relres 6002	. . 3 ⊢ Rel ( I ↾ 𝐶)
2	1	brrelex12i 5714	. 2 ⊢ (𝐴( I ↾ 𝐶)𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3		simpl 488	. . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝐶)
4	3	elexd 3476	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐴 ∈ V)
5		eleq1 2850	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
6	5	biimpac 484	. . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝐶)
7	6	elexd 3476	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐵 ∈ V)
8	4, 7	jca 521	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
9		brres 5983	. . . 4 ⊢ (𝐵 ∈ V → (𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 I 𝐵)))
10	9	adantl 487	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 I 𝐵)))
11		eqeq12 2779	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = 𝑦 ↔ 𝐴 = 𝐵))
12		df-id 5554	. . . . 5 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
13	11, 12	brabga 5516	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
14	13	anbi2d 642	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴 ∈ 𝐶 ∧ 𝐴 I 𝐵) ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵)))
15	10, 14	bitrd 282	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵)))
16	2, 8, 15	pm5.21nii 381	1 ⊢ (𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3453   class class class wbr 5107   I cid 5553   ↾ cres 5661

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-id 5554  df-xp 5665  df-rel 5666  df-res 5671

This theorem is used by:  bj-elid7  37910