Theorem elidinxp 5997

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 3208	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐵 𝑦 = 𝑥)
2	1	anbi2ci 625	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝐶 = ⟨𝑥, 𝑥⟩) ↔ (𝐶 = ⟨𝑥, 𝑥⟩ ∧ ∃𝑦 ∈ 𝐵 𝑦 = 𝑥))
3		r19.42v 3165	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 (𝐶 = ⟨𝑥, 𝑥⟩ ∧ 𝑦 = 𝑥) ↔ (𝐶 = ⟨𝑥, 𝑥⟩ ∧ ∃𝑦 ∈ 𝐵 𝑦 = 𝑥))
4		opeq2 4825	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ = ⟨𝑥, 𝑦⟩)
5	4	equcoms 2021	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → ⟨𝑥, 𝑥⟩ = ⟨𝑥, 𝑦⟩)
6	5	eqeq2d 2744	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝐶 = ⟨𝑥, 𝑥⟩ ↔ 𝐶 = ⟨𝑥, 𝑦⟩))
7	6	pm5.32ri 575	. . . . . 6 ⊢ ((𝐶 = ⟨𝑥, 𝑥⟩ ∧ 𝑦 = 𝑥) ↔ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝑦 = 𝑥))
8		vex 3441	. . . . . . . . 9 ⊢ 𝑦 ∈ V
9	8	ideq 5796	. . . . . . . 8 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
10		df-br 5094	. . . . . . . 8 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
11		equcom 2019	. . . . . . . 8 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
12	9, 10, 11	3bitr3i 301	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑦 = 𝑥)
13	12	anbi2i 623	. . . . . 6 ⊢ ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ) ↔ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝑦 = 𝑥))
14	7, 13	bitr4i 278	. . . . 5 ⊢ ((𝐶 = ⟨𝑥, 𝑥⟩ ∧ 𝑦 = 𝑥) ↔ (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
15	14	rexbii 3080	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 (𝐶 = ⟨𝑥, 𝑥⟩ ∧ 𝑦 = 𝑥) ↔ ∃𝑦 ∈ 𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
16	2, 3, 15	3bitr2i 299	. . 3 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝐶 = ⟨𝑥, 𝑥⟩) ↔ ∃𝑦 ∈ 𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
17	16	rexbii 3080	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ 𝐶 = ⟨𝑥, 𝑥⟩) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
18		rexin 4199	. 2 ⊢ (∃𝑥 ∈ (𝐴 ∩ 𝐵)𝐶 = ⟨𝑥, 𝑥⟩ ↔ ∃𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ 𝐶 = ⟨𝑥, 𝑥⟩))
19		elinxp 5972	. 2 ⊢ (𝐶 ∈ ( I ∩ (𝐴 × 𝐵)) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ I ))
20	17, 18, 19	3bitr4ri 304	1 ⊢ (𝐶 ∈ ( I ∩ (𝐴 × 𝐵)) ↔ ∃𝑥 ∈ (𝐴 ∩ 𝐵)𝐶 = ⟨𝑥, 𝑥⟩)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3057   ∩ cin 3897  ⟨cop 4581   class class class wbr 5093   I cid 5513   × cxp 5617

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-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

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-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-opab 5156  df-id 5514  df-xp 5625  df-rel 5626

This theorem is referenced by:  elidinxpid  5998  elrid  5999  idinxpres  6000