Theorem bj-opelidres 37658

Description: Characterization of the ordered pairs in the restricted identity relation when the intersection of their component belongs to the restricting class. TODO: prove bj-idreseq 37659 from it. (Contributed by BJ, 29-Mar-2020.)

Assertion

Ref	Expression
bj-opelidres	⊢ (𝐴 ∈ 𝑉 → (⟨𝐴, 𝐵⟩ ∈ ( I ↾ 𝑉) ↔ 𝐴 = 𝐵))

Proof of Theorem bj-opelidres

Step	Hyp	Ref	Expression
1		bj-idres 37657	. . 3 ⊢ ( I ↾ 𝑉) = ( I ∩ (𝑉 × 𝑉))
2	1	eleq2i 2856	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ ( I ↾ 𝑉) ↔ ⟨𝐴, 𝐵⟩ ∈ ( I ∩ (𝑉 × 𝑉)))
3		elin 3922	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ ( I ∩ (𝑉 × 𝑉)) ↔ (⟨𝐴, 𝐵⟩ ∈ I ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑉)))
4		inex1g 5277	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V)
5		bj-opelid 37653	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ∈ V → (⟨𝐴, 𝐵⟩ ∈ I ↔ 𝐴 = 𝐵))
6	4, 5	syl 17	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (⟨𝐴, 𝐵⟩ ∈ I ↔ 𝐴 = 𝐵))
7		opelxp 5685	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑉) ↔ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))
8	7	a1i 11	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑉) ↔ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)))
9	6, 8	anbi12d 641	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((⟨𝐴, 𝐵⟩ ∈ I ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑉)) ↔ (𝐴 = 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))))
10		simpl 486	. . . . 5 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → 𝐴 = 𝐵)
11		eleq1 2852	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝑉 ↔ 𝐵 ∈ 𝑉))
12	11	biimpcd 251	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 → 𝐵 ∈ 𝑉))
13	12	anc2li 563	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)))
14	13	ancld 558	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 → (𝐴 = 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))))
15	10, 14	impbid2 228	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 = 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) ↔ 𝐴 = 𝐵))
16	9, 15	bitrd 281	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((⟨𝐴, 𝐵⟩ ∈ I ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑉)) ↔ 𝐴 = 𝐵))
17	3, 16	bitrid 285	. 2 ⊢ (𝐴 ∈ 𝑉 → (⟨𝐴, 𝐵⟩ ∈ ( I ∩ (𝑉 × 𝑉)) ↔ 𝐴 = 𝐵))
18	2, 17	bitrid 285	1 ⊢ (𝐴 ∈ 𝑉 → (⟨𝐴, 𝐵⟩ ∈ ( I ↾ 𝑉) ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1562   ∈ wcel 2144  Vcvv 3456   ∩ cin 3905  ⟨cop 4590   I cid 5543   × cxp 5647   ↾ 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-10 2177  ax-11 2193  ax-12 2214  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-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  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-opab 5165  df-id 5544  df-xp 5655  df-rel 5656  df-res 5661

This theorem is referenced by: (None)