Theorem elidinxp 6035

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 3237	. . . . 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 3194	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 (𝐶 = ⟨𝑥, 𝑥⟩ ∧ 𝑦 = 𝑥) ↔ (𝐶 = ⟨𝑥, 𝑥⟩ ∧ ∃𝑦 ∈ 𝐵 𝑦 = 𝑥))
4		opeq2 4840	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ = ⟨𝑥, 𝑦⟩)
5	4	equcoms 2052	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → ⟨𝑥, 𝑥⟩ = ⟨𝑥, 𝑦⟩)
6	5	eqeq2d 2771	. . . . . . 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 3454	. . . . . . . . 9 ⊢ 𝑦 ∈ V
9	8	ideq 5829	. . . . . . . 8 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
10		df-br 5103	. . . . . . . 8 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
11		equcom 2050	. . . . . . . 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 3109	. . . 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 3109	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ 𝐶 = ⟨𝑥, 𝑥⟩) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
18		rexin 4147	. 2 ⊢ (∃𝑥 ∈ (𝐴 ∩ 𝐵)𝐶 = ⟨𝑥, 𝑥⟩ ↔ ∃𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ 𝐶 = ⟨𝑥, 𝑥⟩))
19		elinxp 6007	. 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 1569   ∈ wcel 2144  ∃wrex 3086   ∩ cin 4089  ⟨cop 4834   class class class wbr 5102   I cid 5544   × cxp 5648

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-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-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 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657

This theorem is used by:  elidinxpid  6036  elrid  6037  idinxpres  6038