Theorem opiota 8100

Description: The property of a uniquely specified ordered pair. The proof uses properties of the ℩ description binder. (Contributed by Mario Carneiro, 21-May-2015.)

Hypotheses

Ref	Expression
opiota.1	⊢ 𝐼 = (℩𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
opiota.2	⊢ 𝑋 = (1st ‘𝐼)
opiota.3	⊢ 𝑌 = (2nd ‘𝐼)
opiota.4	⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜓))
opiota.5	⊢ (𝑦 = 𝐷 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
opiota	⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ 𝜒) ↔ (𝐶 = 𝑋 ∧ 𝐷 = 𝑌)))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝜒,𝑦   𝜑,𝑧   𝑥,𝐷,𝑦,𝑧   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦,𝑧)   𝜒(𝑥,𝑧)   𝐼(𝑥,𝑦,𝑧)   𝑋(𝑥,𝑦,𝑧)   𝑌(𝑥,𝑦,𝑧)

Proof of Theorem opiota

Step	Hyp	Ref	Expression
1		opiota.4	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜓))
2		opiota.5	. . . . . . 7 ⊢ (𝑦 = 𝐷 → (𝜓 ↔ 𝜒))
3	1, 2	ceqsrex2v 3671	. . . . . 6 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ((𝑥 = 𝐶 ∧ 𝑦 = 𝐷) ∧ 𝜑) ↔ 𝜒))
4	3	bicomd 223	. . . . 5 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝜒 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ((𝑥 = 𝐶 ∧ 𝑦 = 𝐷) ∧ 𝜑)))
5		opex 5484	. . . . . . . 8 ⊢ ⟨𝐶, 𝐷⟩ ∈ V
6	5	a1i 11	. . . . . . 7 ⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ⟨𝐶, 𝐷⟩ ∈ V)
7		id 22	. . . . . . 7 ⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
8		eqeq1 2744	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝐶, 𝐷⟩ → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨𝑥, 𝑦⟩))
9		eqcom 2747	. . . . . . . . . . . 12 ⊢ (⟨𝐶, 𝐷⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐶, 𝐷⟩)
10		vex 3492	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
11		vex 3492	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
12	10, 11	opth 5496	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝑥 = 𝐶 ∧ 𝑦 = 𝐷))
13	9, 12	bitri 275	. . . . . . . . . . 11 ⊢ (⟨𝐶, 𝐷⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑥 = 𝐶 ∧ 𝑦 = 𝐷))
14	8, 13	bitrdi 287	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝐶, 𝐷⟩ → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)))
15	14	anbi1d 630	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝐶, 𝐷⟩ → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ((𝑥 = 𝐶 ∧ 𝑦 = 𝐷) ∧ 𝜑)))
16	15	2rexbidv 3228	. . . . . . . 8 ⊢ (𝑧 = ⟨𝐶, 𝐷⟩ → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ((𝑥 = 𝐶 ∧ 𝑦 = 𝐷) ∧ 𝜑)))
17	16	adantl 481	. . . . . . 7 ⊢ ((∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∧ 𝑧 = ⟨𝐶, 𝐷⟩) → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ((𝑥 = 𝐶 ∧ 𝑦 = 𝐷) ∧ 𝜑)))
18		nfeu1 2591	. . . . . . 7 ⊢ Ⅎ𝑧∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
19		nfvd 1914	. . . . . . 7 ⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → Ⅎ𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ((𝑥 = 𝐶 ∧ 𝑦 = 𝐷) ∧ 𝜑))
20		nfcvd 2909	. . . . . . 7 ⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → Ⅎ𝑧⟨𝐶, 𝐷⟩)
21	6, 7, 17, 18, 19, 20	iota2df 6560	. . . . . 6 ⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ((𝑥 = 𝐶 ∧ 𝑦 = 𝐷) ∧ 𝜑) ↔ (℩𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) = ⟨𝐶, 𝐷⟩))
22		eqcom 2747	. . . . . . 7 ⊢ (⟨𝐶, 𝐷⟩ = 𝐼 ↔ 𝐼 = ⟨𝐶, 𝐷⟩)
23		opiota.1	. . . . . . . 8 ⊢ 𝐼 = (℩𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
24	23	eqeq1i 2745	. . . . . . 7 ⊢ (𝐼 = ⟨𝐶, 𝐷⟩ ↔ (℩𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) = ⟨𝐶, 𝐷⟩)
25	22, 24	bitri 275	. . . . . 6 ⊢ (⟨𝐶, 𝐷⟩ = 𝐼 ↔ (℩𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) = ⟨𝐶, 𝐷⟩)
26	21, 25	bitr4di 289	. . . . 5 ⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ((𝑥 = 𝐶 ∧ 𝑦 = 𝐷) ∧ 𝜑) ↔ ⟨𝐶, 𝐷⟩ = 𝐼))
27	4, 26	sylan9bbr 510	. . . 4 ⊢ ((∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → (𝜒 ↔ ⟨𝐶, 𝐷⟩ = 𝐼))
28	27	pm5.32da 578	. . 3 ⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 𝜒) ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ ⟨𝐶, 𝐷⟩ = 𝐼)))
29		opelxpi 5737	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
30		simpl 482	. . . . . . . . . . 11 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 = ⟨𝑥, 𝑦⟩)
31	30	eleq1d 2829	. . . . . . . . . 10 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (𝑧 ∈ (𝐴 × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
32	29, 31	syl5ibrcom 247	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 ∈ (𝐴 × 𝐵)))
33	32	rexlimivv 3207	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 ∈ (𝐴 × 𝐵))
34	33	abssi 4093	. . . . . . 7 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ⊆ (𝐴 × 𝐵)
35		iotacl 6559	. . . . . . 7 ⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (℩𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)})
36	34, 35	sselid 4006	. . . . . 6 ⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (℩𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ∈ (𝐴 × 𝐵))
37	23, 36	eqeltrid 2848	. . . . 5 ⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝐼 ∈ (𝐴 × 𝐵))
38		opelxp 5736	. . . . . 6 ⊢ (⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵))
39		eleq1 2832	. . . . . 6 ⊢ (⟨𝐶, 𝐷⟩ = 𝐼 → (⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵) ↔ 𝐼 ∈ (𝐴 × 𝐵)))
40	38, 39	bitr3id 285	. . . . 5 ⊢ (⟨𝐶, 𝐷⟩ = 𝐼 → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ↔ 𝐼 ∈ (𝐴 × 𝐵)))
41	37, 40	syl5ibrcom 247	. . . 4 ⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (⟨𝐶, 𝐷⟩ = 𝐼 → (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)))
42	41	pm4.71rd 562	. . 3 ⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (⟨𝐶, 𝐷⟩ = 𝐼 ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ ⟨𝐶, 𝐷⟩ = 𝐼)))
43		1st2nd2 8069	. . . . 5 ⊢ (𝐼 ∈ (𝐴 × 𝐵) → 𝐼 = ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩)
44	37, 43	syl 17	. . . 4 ⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝐼 = ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩)
45	44	eqeq2d 2751	. . 3 ⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (⟨𝐶, 𝐷⟩ = 𝐼 ↔ ⟨𝐶, 𝐷⟩ = ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩))
46	28, 42, 45	3bitr2d 307	. 2 ⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 𝜒) ↔ ⟨𝐶, 𝐷⟩ = ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩))
47		df-3an 1089	. 2 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ 𝜒) ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 𝜒))
48		opiota.2	. . . . 5 ⊢ 𝑋 = (1st ‘𝐼)
49	48	eqeq2i 2753	. . . 4 ⊢ (𝐶 = 𝑋 ↔ 𝐶 = (1st ‘𝐼))
50		opiota.3	. . . . 5 ⊢ 𝑌 = (2nd ‘𝐼)
51	50	eqeq2i 2753	. . . 4 ⊢ (𝐷 = 𝑌 ↔ 𝐷 = (2nd ‘𝐼))
52	49, 51	anbi12i 627	. . 3 ⊢ ((𝐶 = 𝑋 ∧ 𝐷 = 𝑌) ↔ (𝐶 = (1st ‘𝐼) ∧ 𝐷 = (2nd ‘𝐼)))
53		fvex 6933	. . . 4 ⊢ (1st ‘𝐼) ∈ V
54		fvex 6933	. . . 4 ⊢ (2nd ‘𝐼) ∈ V
55	53, 54	opth2 5500	. . 3 ⊢ (⟨𝐶, 𝐷⟩ = ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩ ↔ (𝐶 = (1st ‘𝐼) ∧ 𝐷 = (2nd ‘𝐼)))
56	52, 55	bitr4i 278	. 2 ⊢ ((𝐶 = 𝑋 ∧ 𝐷 = 𝑌) ↔ ⟨𝐶, 𝐷⟩ = ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩)
57	46, 47, 56	3bitr4g 314	1 ⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ 𝜒) ↔ (𝐶 = 𝑋 ∧ 𝐷 = 𝑌)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∃!weu 2571  {cab 2717  ∃wrex 3076  Vcvv 3488  ⟨cop 4654   × cxp 5698  ℩cio 6523  ‘cfv 6573  1^st c1st 8028  2^nd c2nd 8029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fv 6581  df-1st 8030  df-2nd 8031

This theorem is referenced by:  oeeui  8658