Theorem opiota 7739

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 3599	. . . . . 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 5321	. . . . . . . 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 2802	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝐶, 𝐷⟩ → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨𝑥, 𝑦⟩))
9		eqcom 2805	. . . . . . . . . . . 12 ⊢ (⟨𝐶, 𝐷⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐶, 𝐷⟩)
10		vex 3444	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
11		vex 3444	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
12	10, 11	opth 5333	. . . . . . . . . . . 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	syl6bb 290	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝐶, 𝐷⟩ → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)))
15	14	anbi1d 632	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝐶, 𝐷⟩ → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ((𝑥 = 𝐶 ∧ 𝑦 = 𝐷) ∧ 𝜑)))
16	15	2rexbidv 3259	. . . . . . . 8 ⊢ (𝑧 = ⟨𝐶, 𝐷⟩ → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ((𝑥 = 𝐶 ∧ 𝑦 = 𝐷) ∧ 𝜑)))
17	16	adantl 485	. . . . . . 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 2649	. . . . . . 7 ⊢ Ⅎ𝑧∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
19		nfvd 1916	. . . . . . 7 ⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → Ⅎ𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ((𝑥 = 𝐶 ∧ 𝑦 = 𝐷) ∧ 𝜑))
20		nfcvd 2956	. . . . . . 7 ⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → Ⅎ𝑧⟨𝐶, 𝐷⟩)
21	6, 7, 17, 18, 19, 20	iota2df 6311	. . . . . 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 2805	. . . . . . 7 ⊢ (⟨𝐶, 𝐷⟩ = 𝐼 ↔ 𝐼 = ⟨𝐶, 𝐷⟩)
23		opiota.1	. . . . . . . 8 ⊢ 𝐼 = (℩𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
24	23	eqeq1i 2803	. . . . . . 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	syl6bbr 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 514	. . . 4 ⊢ ((∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → (𝜒 ↔ ⟨𝐶, 𝐷⟩ = 𝐼))
28	27	pm5.32da 582	. . 3 ⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 𝜒) ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ ⟨𝐶, 𝐷⟩ = 𝐼)))
29		opelxpi 5556	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
30		simpl 486	. . . . . . . . . . 11 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 = ⟨𝑥, 𝑦⟩)
31	30	eleq1d 2874	. . . . . . . . . 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 3251	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 ∈ (𝐴 × 𝐵))
34	33	abssi 3997	. . . . . . 7 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ⊆ (𝐴 × 𝐵)
35		iotacl 6310	. . . . . . 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	sseldi 3913	. . . . . 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 2894	. . . . 5 ⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝐼 ∈ (𝐴 × 𝐵))
38		opelxp 5555	. . . . . 6 ⊢ (⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵))
39		eleq1 2877	. . . . . 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 566	. . 3 ⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (⟨𝐶, 𝐷⟩ = 𝐼 ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ ⟨𝐶, 𝐷⟩ = 𝐼)))
43		1st2nd2 7710	. . . . 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 2809	. . 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 1086	. 2 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ 𝜒) ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 𝜒))
48		opiota.2	. . . . 5 ⊢ 𝑋 = (1st ‘𝐼)
49	48	eqeq2i 2811	. . . 4 ⊢ (𝐶 = 𝑋 ↔ 𝐶 = (1st ‘𝐼))
50		opiota.3	. . . . 5 ⊢ 𝑌 = (2nd ‘𝐼)
51	50	eqeq2i 2811	. . . 4 ⊢ (𝐷 = 𝑌 ↔ 𝐷 = (2nd ‘𝐼))
52	49, 51	anbi12i 629	. . 3 ⊢ ((𝐶 = 𝑋 ∧ 𝐷 = 𝑌) ↔ (𝐶 = (1st ‘𝐼) ∧ 𝐷 = (2nd ‘𝐼)))
53		fvex 6658	. . . 4 ⊢ (1st ‘𝐼) ∈ V
54		fvex 6658	. . . 4 ⊢ (2nd ‘𝐼) ∈ V
55	53, 54	opth2 5337	. . 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)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∃!weu 2628  {cab 2776  ∃wrex 3107  Vcvv 3441  ⟨cop 4531   × cxp 5517  ℩cio 6281  ‘cfv 6324  1^st c1st 7669  2^nd c2nd 7670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fv 6332  df-1st 7671  df-2nd 7672

This theorem is referenced by:  oeeui  8211