Theorem opiota 8092

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 3615	. . . . . 6 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ((𝑥 = 𝐶 ∧ 𝑦 = 𝐷) ∧ 𝜑) ↔ 𝜒))
4	3	bicomd 226	. . . . 5 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝜒 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ((𝑥 = 𝐶 ∧ 𝑦 = 𝐷) ∧ 𝜑)))
5		opex 5436	. . . . . . . 8 ⊢ ⟨𝐶, 𝐷⟩ ∈ V
6	5	a1i 11	. . . . . . 7 ⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ⟨𝐶, 𝐷⟩ ∈ V)
7		id 23	. . . . . . 7 ⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
8		eqeq1 2767	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝐶, 𝐷⟩ → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨𝑥, 𝑦⟩))
9		eqcom 2770	. . . . . . . . . . . 12 ⊢ (⟨𝐶, 𝐷⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐶, 𝐷⟩)
10		vex 3458	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
11		vex 3458	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
12	10, 11	opth 5449	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝑥 = 𝐶 ∧ 𝑦 = 𝐷))
13	9, 12	bitri 278	. . . . . . . . . . 11 ⊢ (⟨𝐶, 𝐷⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑥 = 𝐶 ∧ 𝑦 = 𝐷))
14	8, 13	bitrdi 290	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝐶, 𝐷⟩ → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)))
15	14	anbi1d 643	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝐶, 𝐷⟩ → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ((𝑥 = 𝐶 ∧ 𝑦 = 𝐷) ∧ 𝜑)))
16	15	2rexbidv 3231	. . . . . . . 8 ⊢ (𝑧 = ⟨𝐶, 𝐷⟩ → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ((𝑥 = 𝐶 ∧ 𝑦 = 𝐷) ∧ 𝜑)))
17	16	adantl 487	. . . . . . 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 2617	. . . . . . 7 ⊢ Ⅎ𝑧∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
19		nfvd 1950	. . . . . . 7 ⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → Ⅎ𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ((𝑥 = 𝐶 ∧ 𝑦 = 𝐷) ∧ 𝜑))
20		nfcvd 2927	. . . . . . 7 ⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → Ⅎ𝑧⟨𝐶, 𝐷⟩)
21	6, 7, 17, 18, 19, 20	iota2df 6541	. . . . . 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 2770	. . . . . . 7 ⊢ (⟨𝐶, 𝐷⟩ = 𝐼 ↔ 𝐼 = ⟨𝐶, 𝐷⟩)
23		opiota.1	. . . . . . . 8 ⊢ 𝐼 = (℩𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
24	23	eqeq1i 2768	. . . . . . 7 ⊢ (𝐼 = ⟨𝐶, 𝐷⟩ ↔ (℩𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) = ⟨𝐶, 𝐷⟩)
25	22, 24	bitri 278	. . . . . 6 ⊢ (⟨𝐶, 𝐷⟩ = 𝐼 ↔ (℩𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) = ⟨𝐶, 𝐷⟩)
26	21, 25	bitr4di 292	. . . . 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 520	. . . 4 ⊢ ((∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → (𝜒 ↔ ⟨𝐶, 𝐷⟩ = 𝐼))
28	27	pm5.32da 590	. . 3 ⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 𝜒) ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ ⟨𝐶, 𝐷⟩ = 𝐼)))
29		opelxpi 5692	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
30		simpl 488	. . . . . . . . . . 11 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 = ⟨𝑥, 𝑦⟩)
31	30	eleq1d 2849	. . . . . . . . . 10 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (𝑧 ∈ (𝐴 × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
32	29, 31	syl5ibrcom 250	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 ∈ (𝐴 × 𝐵)))
33	32	rexlimivv 3208	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 ∈ (𝐴 × 𝐵))
34	33	abssi 3996	. . . . . . 7 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ⊆ (𝐴 × 𝐵)
35		iotacl 6540	. . . . . . 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 3911	. . . . . 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 2868	. . . . 5 ⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝐼 ∈ (𝐴 × 𝐵))
38		opelxp 5691	. . . . . 6 ⊢ (⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵))
39		eleq1 2852	. . . . . 6 ⊢ (⟨𝐶, 𝐷⟩ = 𝐼 → (⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵) ↔ 𝐼 ∈ (𝐴 × 𝐵)))
40	38, 39	bitr3id 288	. . . . 5 ⊢ (⟨𝐶, 𝐷⟩ = 𝐼 → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ↔ 𝐼 ∈ (𝐴 × 𝐵)))
41	37, 40	syl5ibrcom 250	. . . 4 ⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (⟨𝐶, 𝐷⟩ = 𝐼 → (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)))
42	41	pm4.71rd 572	. . 3 ⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (⟨𝐶, 𝐷⟩ = 𝐼 ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ ⟨𝐶, 𝐷⟩ = 𝐼)))
43		1st2nd2 8061	. . . . 5 ⊢ (𝐼 ∈ (𝐴 × 𝐵) → 𝐼 = ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩)
44	37, 43	syl 18	. . . 4 ⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝐼 = ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩)
45	44	eqeq2d 2774	. . 3 ⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (⟨𝐶, 𝐷⟩ = 𝐼 ↔ ⟨𝐶, 𝐷⟩ = ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩))
46	28, 42, 45	3bitr2d 310	. 2 ⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 𝜒) ↔ ⟨𝐶, 𝐷⟩ = ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩))
47		df-3an 1105	. 2 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ 𝜒) ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 𝜒))
48		opiota.2	. . . . 5 ⊢ 𝑋 = (1st ‘𝐼)
49	48	eqeq2i 2776	. . . 4 ⊢ (𝐶 = 𝑋 ↔ 𝐶 = (1st ‘𝐼))
50		opiota.3	. . . . 5 ⊢ 𝑌 = (2nd ‘𝐼)
51	50	eqeq2i 2776	. . . 4 ⊢ (𝐷 = 𝑌 ↔ 𝐷 = (2nd ‘𝐼))
52	49, 51	anbi12i 640	. . 3 ⊢ ((𝐶 = 𝑋 ∧ 𝐷 = 𝑌) ↔ (𝐶 = (1st ‘𝐼) ∧ 𝐷 = (2nd ‘𝐼)))
53		fvex 6918	. . . 4 ⊢ (1st ‘𝐼) ∈ V
54		fvex 6918	. . . 4 ⊢ (2nd ‘𝐼) ∈ V
55	53, 54	opth2 5453	. . 3 ⊢ (⟨𝐶, 𝐷⟩ = ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩ ↔ (𝐶 = (1st ‘𝐼) ∧ 𝐷 = (2nd ‘𝐼)))
56	52, 55	bitr4i 281	. 2 ⊢ ((𝐶 = 𝑋 ∧ 𝐷 = 𝑌) ↔ ⟨𝐶, 𝐷⟩ = ⟨(1st ‘𝐼), (2nd ‘𝐼)⟩)
57	46, 47, 56	3bitr4g 317	1 ⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ 𝜒) ↔ (𝐶 = 𝑋 ∧ 𝐷 = 𝑌)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∃!weu 2596  {cab 2741  ∃wrex 3090  Vcvv 3454  ⟨cop 4838   × cxp 5653  ℩cio 6507  ‘cfv 6555  1^st c1st 8020  2^nd c2nd 8021

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7772

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-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6509  df-fun 6557  df-fv 6563  df-1st 8022  df-2nd 8023

This theorem is used by:  oeeui  8636