Theorem bj-opelidres 37216

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 37217 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 37215	. . 3 ⊢ ( I ↾ 𝑉) = ( I ∩ (𝑉 × 𝑉))
2	1	eleq2i 2825	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ ( I ↾ 𝑉) ↔ ⟨𝐴, 𝐵⟩ ∈ ( I ∩ (𝑉 × 𝑉)))
3		elin 3915	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ ( I ∩ (𝑉 × 𝑉)) ↔ (⟨𝐴, 𝐵⟩ ∈ I ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑉)))
4		inex1g 5261	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V)
5		bj-opelid 37211	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ∈ V → (⟨𝐴, 𝐵⟩ ∈ I ↔ 𝐴 = 𝐵))
6	4, 5	syl 17	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (⟨𝐴, 𝐵⟩ ∈ I ↔ 𝐴 = 𝐵))
7		opelxp 5657	. . . . . 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 632	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((⟨𝐴, 𝐵⟩ ∈ I ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑉)) ↔ (𝐴 = 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))))
10		simpl 482	. . . . 5 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → 𝐴 = 𝐵)
11		eleq1 2821	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝑉 ↔ 𝐵 ∈ 𝑉))
12	11	biimpcd 249	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 → 𝐵 ∈ 𝑉))
13	12	anc2li 555	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)))
14	13	ancld 550	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 → (𝐴 = 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))))
15	10, 14	impbid2 226	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 = 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) ↔ 𝐴 = 𝐵))
16	9, 15	bitrd 279	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((⟨𝐴, 𝐵⟩ ∈ I ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑉 × 𝑉)) ↔ 𝐴 = 𝐵))
17	3, 16	bitrid 283	. 2 ⊢ (𝐴 ∈ 𝑉 → (⟨𝐴, 𝐵⟩ ∈ ( I ∩ (𝑉 × 𝑉)) ↔ 𝐴 = 𝐵))
18	2, 17	bitrid 283	1 ⊢ (𝐴 ∈ 𝑉 → (⟨𝐴, 𝐵⟩ ∈ ( I ↾ 𝑉) ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3438   ∩ cin 3898  ⟨cop 4583   I cid 5515   × cxp 5619   ↾ cres 5623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-opab 5158  df-id 5516  df-xp 5627  df-rel 5628  df-res 5633

This theorem is referenced by: (None)