Theorem bj-idreseqb 37465

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 5959	. . 3 ⊢ Rel ( I ↾ 𝐶)
2	1	brrelex12i 5675	. 2 ⊢ (𝐴( I ↾ 𝐶)𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3		simpl 482	. . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝐶)
4	3	elexd 3451	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐴 ∈ V)
5		eleq1 2823	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
6	5	biimpac 478	. . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝐶)
7	6	elexd 3451	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐵 ∈ V)
8	4, 7	jca 511	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
9		brres 5940	. . . 4 ⊢ (𝐵 ∈ V → (𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 I 𝐵)))
10	9	adantl 481	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 I 𝐵)))
11		eqeq12 2752	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = 𝑦 ↔ 𝐴 = 𝐵))
12		df-id 5515	. . . . 5 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
13	11, 12	brabga 5478	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
14	13	anbi2d 631	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴 ∈ 𝐶 ∧ 𝐴 I 𝐵) ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵)))
15	10, 14	bitrd 279	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵)))
16	2, 8, 15	pm5.21nii 378	1 ⊢ (𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3427   class class class wbr 5074   I cid 5514   ↾ cres 5622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707  ax-sep 5220  ax-pr 5364

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ral 3050  df-rex 3060  df-rab 3388  df-v 3429  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-br 5075  df-opab 5137  df-id 5515  df-xp 5626  df-rel 5627  df-res 5632

This theorem is referenced by:  bj-elid7  37473