Theorem bj-idreseqb 37660

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 5993	. . 3 ⊢ Rel ( I ↾ 𝐶)
2	1	brrelex12i 5704	. 2 ⊢ (𝐴( I ↾ 𝐶)𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3		simpl 486	. . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝐶)
4	3	elexd 3479	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐴 ∈ V)
5		eleq1 2852	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
6	5	biimpac 482	. . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝐶)
7	6	elexd 3479	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐵 ∈ V)
8	4, 7	jca 519	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
9		brres 5974	. . . 4 ⊢ (𝐵 ∈ V → (𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 I 𝐵)))
10	9	adantl 485	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 I 𝐵)))
11		eqeq12 2781	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = 𝑦 ↔ 𝐴 = 𝐵))
12		df-id 5544	. . . . 5 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
13	11, 12	brabga 5506	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
14	13	anbi2d 639	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴 ∈ 𝐶 ∧ 𝐴 I 𝐵) ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵)))
15	10, 14	bitrd 281	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵)))
16	2, 8, 15	pm5.21nii 380	1 ⊢ (𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵))

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1562   ∈ wcel 2144  Vcvv 3456   class class class wbr 5102   I cid 5543   ↾ cres 5651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-opab 5165  df-id 5544  df-xp 5655  df-rel 5656  df-res 5661

This theorem is referenced by:  bj-elid7  37668