Theorem bj-opelidres 38083

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 38084 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 38082	. . 3 ⊢ ( I ↾ 𝑉) = ( I ∩ (𝑉 × 𝑉))
2	1	eleq2i 2852	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ ( I ↾ 𝑉) ↔ ⟨𝐴, 𝐵⟩ ∈ ( I ∩ (𝑉 × 𝑉)))
3		elin 4093	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ ( I ∩ (𝑉 × 𝑉)) ↔ (⟨𝐴, 𝐵⟩ ∈ I ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑉)))
4		inex1g 5278	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V)
5		bj-opelid 38078	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ∈ V → (⟨𝐴, 𝐵⟩ ∈ I ↔ 𝐴 = 𝐵))
6	4, 5	syl 18	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (⟨𝐴, 𝐵⟩ ∈ I ↔ 𝐴 = 𝐵))
7		opelxp 5686	. . . . . 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 644	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((⟨𝐴, 𝐵⟩ ∈ I ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑉)) ↔ (𝐴 = 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))))
10		simpl 488	. . . . 5 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → 𝐴 = 𝐵)
11		eleq1 2848	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝑉 ↔ 𝐵 ∈ 𝑉))
12	11	biimpcd 252	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 → 𝐵 ∈ 𝑉))
13	12	anc2li 565	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)))
14	13	ancld 560	. . . . 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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ∩ cin 4089  ⟨cop 4589   I cid 5544   × cxp 5648   ↾ cres 5652

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-res 5662

This theorem is used by: (None)