Theorem elidinxp 6038

Description: Characterization of the elements of the intersection of the identity relation with a Cartesian product. (Contributed by Peter Mazsa, 9-Sep-2022.)

Assertion

Ref	Expression
elidinxp	⊢ (𝐶 ∈ ( I ∩ (𝐴 × 𝐵)) ↔ ∃𝑥 ∈ (𝐴 ∩ 𝐵)𝐶 = ⟨𝑥, 𝑥⟩)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem elidinxp

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		risset 3238	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐵 𝑦 = 𝑥)
2	1	anbi2ci 637	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝐶 = ⟨𝑥, 𝑥⟩) ↔ (𝐶 = ⟨𝑥, 𝑥⟩ ∧ ∃𝑦 ∈ 𝐵 𝑦 = 𝑥))
3		r19.42v 3195	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 (𝐶 = ⟨𝑥, 𝑥⟩ ∧ 𝑦 = 𝑥) ↔ (𝐶 = ⟨𝑥, 𝑥⟩ ∧ ∃𝑦 ∈ 𝐵 𝑦 = 𝑥))
4		opeq2 4834	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ = ⟨𝑥, 𝑦⟩)
5	4	equcoms 2053	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → ⟨𝑥, 𝑥⟩ = ⟨𝑥, 𝑦⟩)
6	5	eqeq2d 2772	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝐶 = ⟨𝑥, 𝑥⟩ ↔ 𝐶 = ⟨𝑥, 𝑦⟩))
7	6	pm5.32ri 586	. . . . . 6 ⊢ ((𝐶 = ⟨𝑥, 𝑥⟩ ∧ 𝑦 = 𝑥) ↔ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝑦 = 𝑥))
8		vex 3455	. . . . . . . . 9 ⊢ 𝑦 ∈ V
9	8	ideq 5830	. . . . . . . 8 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
10		df-br 5104	. . . . . . . 8 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
11		equcom 2051	. . . . . . . 8 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
12	9, 10, 11	3bitr3i 304	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑦 = 𝑥)
13	12	anbi2i 635	. . . . . 6 ⊢ ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝑦 = 𝑥))
14	7, 13	bitr4i 281	. . . . 5 ⊢ ((𝐶 = ⟨𝑥, 𝑥⟩ ∧ 𝑦 = 𝑥) ↔ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
15	14	rexbii 3110	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 (𝐶 = ⟨𝑥, 𝑥⟩ ∧ 𝑦 = 𝑥) ↔ ∃𝑦 ∈ 𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
16	2, 3, 15	3bitr2i 302	. . 3 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝐶 = ⟨𝑥, 𝑥⟩) ↔ ∃𝑦 ∈ 𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
17	16	rexbii 3110	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ 𝐶 = ⟨𝑥, 𝑥⟩) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
18		rexin 4196	. 2 ⊢ (∃𝑥 ∈ (𝐴 ∩ 𝐵)𝐶 = ⟨𝑥, 𝑥⟩ ↔ ∃𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ 𝐶 = ⟨𝑥, 𝑥⟩))
19		elinxp 6010	. 2 ⊢ (𝐶 ∈ ( I ∩ (𝐴 × 𝐵)) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
20	17, 18, 19	3bitr4ri 307	1 ⊢ (𝐶 ∈ ( I ∩ (𝐴 × 𝐵)) ↔ ∃𝑥 ∈ (𝐴 ∩ 𝐵)𝐶 = ⟨𝑥, 𝑥⟩)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3087   ∩ cin 3898  ⟨cop 4590   class class class wbr 5103   I cid 5545   × cxp 5649

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658

This theorem is used by:  elidinxpid  6039  elrid  6040  idinxpres  6041