Theorem bj-opelidres 37749

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 37750 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 37748	. . 3 ⊢ ( I ↾ 𝑉) = ( I ∩ (𝑉 × 𝑉))
2	1	eleq2i 2853	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ ( I ↾ 𝑉) ↔ ⟨𝐴, 𝐵⟩ ∈ ( I ∩ (𝑉 × 𝑉)))
3		elin 3920	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ ( I ∩ (𝑉 × 𝑉)) ↔ (⟨𝐴, 𝐵⟩ ∈ I ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑉)))
4		inex1g 5287	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V)
5		bj-opelid 37744	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ∈ V → (⟨𝐴, 𝐵⟩ ∈ I ↔ 𝐴 = 𝐵))
6	4, 5	syl 18	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (⟨𝐴, 𝐵⟩ ∈ I ↔ 𝐴 = 𝐵))
7		opelxp 5697	. . . . . 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 643	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((⟨𝐴, 𝐵⟩ ∈ I ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑉)) ↔ (𝐴 = 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))))
10		simpl 487	. . . . 5 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → 𝐴 = 𝐵)
11		eleq1 2849	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝑉 ↔ 𝐵 ∈ 𝑉))
12	11	biimpcd 252	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 → 𝐵 ∈ 𝑉))
13	12	anc2li 564	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)))
14	13	ancld 559	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 → (𝐴 = 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))))
15	10, 14	impbid2 229	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 = 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) ↔ 𝐴 = 𝐵))
16	9, 15	bitrd 282	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((⟨𝐴, 𝐵⟩ ∈ I ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑉)) ↔ 𝐴 = 𝐵))
17	3, 16	bitrid 286	. 2 ⊢ (𝐴 ∈ 𝑉 → (⟨𝐴, 𝐵⟩ ∈ ( I ∩ (𝑉 × 𝑉)) ↔ 𝐴 = 𝐵))
18	2, 17	bitrid 286	1 ⊢ (𝐴 ∈ 𝑉 → (⟨𝐴, 𝐵⟩ ∈ ( I ↾ 𝑉) ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1568   ∈ wcel 2141  Vcvv 3453   ∩ cin 3903  ⟨cop 4594   I cid 5555   × cxp 5659   ↾ cres 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-opab 5173  df-id 5556  df-xp 5667  df-rel 5668  df-res 5673

This theorem is referenced by: (None)