Theorem opreu2reuALT 33073

Description: Correspondence between uniqueness of ordered pairs and double restricted existential uniqueness quantification. Alternate proof of one direction only, use opreu2reurex 6297 instead. (Contributed by Thierry Arnoux, 4-Jul-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
opsbc2ie.a	⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑 ↔ 𝜒))

Assertion

Ref	Expression
opreu2reuALT	⊢ ((∃!𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ ∃!𝑏 ∈ 𝐵 ∃𝑎 ∈ 𝐴 𝜒) → ∃!𝑝 ∈ (𝐴 × 𝐵)𝜑)

Distinct variable groups:   𝑎,𝑏,𝑝   𝜑,𝑎,𝑏   𝐴,𝑎,𝑏,𝑝   𝐵,𝑎,𝑏,𝑝   𝜒,𝑎,𝑏,𝑝

Allowed substitution hint:   𝜑(𝑝)

Proof of Theorem opreu2reuALT

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2reu4 4479	. 2 ⊢ ((∃!𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ ∃!𝑏 ∈ 𝐵 ∃𝑎 ∈ 𝐴 𝜒) ↔ (∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))))
2		simpllr 788	. . . . . 6 ⊢ ((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) → 𝑥 ∈ 𝐴)
3		simplr 781	. . . . . 6 ⊢ ((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) → 𝑦 ∈ 𝐵)
4		opelxpi 5687	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
5	2, 3, 4	syl2anc 596	. . . . 5 ⊢ ((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
6		nfre1 3287	. . . . . . . . 9 ⊢ Ⅎ𝑎∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒
7		nfv 1946	. . . . . . . . 9 ⊢ Ⅎ𝑎 𝑥 ∈ 𝐴
8	6, 7	nfan 1931	. . . . . . . 8 ⊢ Ⅎ𝑎(∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴)
9		nfv 1946	. . . . . . . 8 ⊢ Ⅎ𝑎 𝑦 ∈ 𝐵
10	8, 9	nfan 1931	. . . . . . 7 ⊢ Ⅎ𝑎((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵)
11		nfra1 3286	. . . . . . 7 ⊢ Ⅎ𝑎∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))
12	10, 11	nfan 1931	. . . . . 6 ⊢ Ⅎ𝑎(((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦)))
13		nfcv 2922	. . . . . . 7 ⊢ Ⅎ𝑎𝑦
14		nfsbc1v 3758	. . . . . . 7 ⊢ Ⅎ𝑎[𝑥 / 𝑎]𝜒
15	13, 14	nfsbc 3763	. . . . . 6 ⊢ Ⅎ𝑎[𝑦 / 𝑏][𝑥 / 𝑎]𝜒
16		nfcv 2922	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑏𝐴
17		nfre1 3287	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑏∃𝑏 ∈ 𝐵 𝜒
18	16, 17	nfrexw 3310	. . . . . . . . . . . 12 ⊢ Ⅎ𝑏∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒
19		nfv 1946	. . . . . . . . . . . 12 ⊢ Ⅎ𝑏 𝑥 ∈ 𝐴
20	18, 19	nfan 1931	. . . . . . . . . . 11 ⊢ Ⅎ𝑏(∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴)
21		nfv 1946	. . . . . . . . . . 11 ⊢ Ⅎ𝑏 𝑦 ∈ 𝐵
22	20, 21	nfan 1931	. . . . . . . . . 10 ⊢ Ⅎ𝑏((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵)
23		nfra1 3286	. . . . . . . . . . 11 ⊢ Ⅎ𝑏∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))
24	16, 23	nfral 3359	. . . . . . . . . 10 ⊢ Ⅎ𝑏∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))
25	22, 24	nfan 1931	. . . . . . . . 9 ⊢ Ⅎ𝑏(((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦)))
26		nfv 1946	. . . . . . . . 9 ⊢ Ⅎ𝑏 𝑎 ∈ 𝐴
27	25, 26	nfan 1931	. . . . . . . 8 ⊢ Ⅎ𝑏((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) ∧ 𝑎 ∈ 𝐴)
28	27, 17	nfan 1931	. . . . . . 7 ⊢ Ⅎ𝑏(((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) ∧ 𝑎 ∈ 𝐴) ∧ ∃𝑏 ∈ 𝐵 𝜒)
29		nfsbc1v 3758	. . . . . . 7 ⊢ Ⅎ𝑏[𝑦 / 𝑏][𝑥 / 𝑎]𝜒
30		rspa 3251	. . . . . . . . . . . 12 ⊢ ((∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦)) ∧ 𝑎 ∈ 𝐴) → ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦)))
31	30	ad5ant23 772	. . . . . . . . . . 11 ⊢ (((((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵) ∧ 𝜒) → ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦)))
32		simplr 781	. . . . . . . . . . 11 ⊢ (((((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵) ∧ 𝜒) → 𝑏 ∈ 𝐵)
33		simpr 490	. . . . . . . . . . 11 ⊢ (((((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵) ∧ 𝜒) → 𝜒)
34		rspa 3251	. . . . . . . . . . . 12 ⊢ ((∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦)) ∧ 𝑏 ∈ 𝐵) → (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦)))
35	34	imp 412	. . . . . . . . . . 11 ⊢ (((∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦)) ∧ 𝑏 ∈ 𝐵) ∧ 𝜒) → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))
36	31, 32, 33, 35	syl21anc 851	. . . . . . . . . 10 ⊢ (((((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵) ∧ 𝜒) → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))
37	36	simprd 501	. . . . . . . . 9 ⊢ (((((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵) ∧ 𝜒) → 𝑏 = 𝑦)
38	36	simpld 500	. . . . . . . . . 10 ⊢ (((((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵) ∧ 𝜒) → 𝑎 = 𝑥)
39		sbceq1a 3749	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (𝜒 ↔ [𝑥 / 𝑎]𝜒))
40	39	biimpa 482	. . . . . . . . . 10 ⊢ ((𝑎 = 𝑥 ∧ 𝜒) → [𝑥 / 𝑎]𝜒)
41	38, 33, 40	syl2anc 596	. . . . . . . . 9 ⊢ (((((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵) ∧ 𝜒) → [𝑥 / 𝑎]𝜒)
42		sbceq1a 3749	. . . . . . . . . 10 ⊢ (𝑏 = 𝑦 → ([𝑥 / 𝑎]𝜒 ↔ [𝑦 / 𝑏][𝑥 / 𝑎]𝜒))
43	42	biimpa 482	. . . . . . . . 9 ⊢ ((𝑏 = 𝑦 ∧ [𝑥 / 𝑎]𝜒) → [𝑦 / 𝑏][𝑥 / 𝑎]𝜒)
44	37, 41, 43	syl2anc 596	. . . . . . . 8 ⊢ (((((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) ∧ 𝑎 ∈ 𝐴) ∧ 𝑏 ∈ 𝐵) ∧ 𝜒) → [𝑦 / 𝑏][𝑥 / 𝑎]𝜒)
45	44	adantllr 732	. . . . . . 7 ⊢ ((((((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) ∧ 𝑎 ∈ 𝐴) ∧ ∃𝑏 ∈ 𝐵 𝜒) ∧ 𝑏 ∈ 𝐵) ∧ 𝜒) → [𝑦 / 𝑏][𝑥 / 𝑎]𝜒)
46		simpr 490	. . . . . . 7 ⊢ ((((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) ∧ 𝑎 ∈ 𝐴) ∧ ∃𝑏 ∈ 𝐵 𝜒) → ∃𝑏 ∈ 𝐵 𝜒)
47	28, 29, 45, 46	r19.29af2 3270	. . . . . 6 ⊢ ((((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) ∧ 𝑎 ∈ 𝐴) ∧ ∃𝑏 ∈ 𝐵 𝜒) → [𝑦 / 𝑏][𝑥 / 𝑎]𝜒)
48		simplll 787	. . . . . 6 ⊢ ((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) → ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒)
49	12, 15, 47, 48	r19.29af2 3270	. . . . 5 ⊢ ((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) → [𝑦 / 𝑏][𝑥 / 𝑎]𝜒)
50		1st2nd2 8040	. . . . . . . . 9 ⊢ (𝑝 ∈ (𝐴 × 𝐵) → 𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
51	50	ad2antlr 740	. . . . . . . 8 ⊢ ((((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) ∧ 𝑝 ∈ (𝐴 × 𝐵)) ∧ 𝜑) → 𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
52		nfv 1946	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑎 𝑝 ∈ (𝐴 × 𝐵)
53	12, 52	nfan 1931	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑎((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) ∧ 𝑝 ∈ (𝐴 × 𝐵))
54		nfv 1946	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑎𝜑
55	53, 54	nfan 1931	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑎(((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) ∧ 𝑝 ∈ (𝐴 × 𝐵)) ∧ 𝜑)
56		nfv 1946	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑏 𝑝 ∈ (𝐴 × 𝐵)
57	25, 56	nfan 1931	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑏((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) ∧ 𝑝 ∈ (𝐴 × 𝐵))
58		nfv 1946	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑏𝜑
59	57, 58	nfan 1931	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑏(((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) ∧ 𝑝 ∈ (𝐴 × 𝐵)) ∧ 𝜑)
60		nfv 1946	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑎(𝜑 → ((1st ‘𝑝) = 𝑥 ∧ (2nd ‘𝑝) = 𝑦))
61		nfv 1946	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑏(𝜑 → ((1st ‘𝑝) = 𝑥 ∧ (2nd ‘𝑝) = 𝑦))
62		xp1st 8033	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ (𝐴 × 𝐵) → (1st ‘𝑝) ∈ 𝐴)
63	62	ad2antlr 740	. . . . . . . . . . . . 13 ⊢ ((((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) ∧ 𝑝 ∈ (𝐴 × 𝐵)) ∧ 𝜑) → (1st ‘𝑝) ∈ 𝐴)
64		xp2nd 8034	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ (𝐴 × 𝐵) → (2nd ‘𝑝) ∈ 𝐵)
65	64	ad2antlr 740	. . . . . . . . . . . . 13 ⊢ ((((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) ∧ 𝑝 ∈ (𝐴 × 𝐵)) ∧ 𝜑) → (2nd ‘𝑝) ∈ 𝐵)
66		eqcom 2767	. . . . . . . . . . . . . . . . 17 ⊢ ((1st ‘𝑝) = 𝑎 ↔ 𝑎 = (1st ‘𝑝))
67		eqcom 2767	. . . . . . . . . . . . . . . . 17 ⊢ ((2nd ‘𝑝) = 𝑏 ↔ 𝑏 = (2nd ‘𝑝))
68		eqopi 8037	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑝 ∈ (𝐴 × 𝐵) ∧ ((1st ‘𝑝) = 𝑎 ∧ (2nd ‘𝑝) = 𝑏)) → 𝑝 = ⟨𝑎, 𝑏⟩)
69		opsbc2ie.a	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑 ↔ 𝜒))
70	68, 69	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑝 ∈ (𝐴 × 𝐵) ∧ ((1st ‘𝑝) = 𝑎 ∧ (2nd ‘𝑝) = 𝑏)) → (𝜑 ↔ 𝜒))
71	70	bicomd 226	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑝 ∈ (𝐴 × 𝐵) ∧ ((1st ‘𝑝) = 𝑎 ∧ (2nd ‘𝑝) = 𝑏)) → (𝜒 ↔ 𝜑))
72	71	ancoms 464	. . . . . . . . . . . . . . . . . 18 ⊢ ((((1st ‘𝑝) = 𝑎 ∧ (2nd ‘𝑝) = 𝑏) ∧ 𝑝 ∈ (𝐴 × 𝐵)) → (𝜒 ↔ 𝜑))
73	72	ex 418	. . . . . . . . . . . . . . . . 17 ⊢ (((1st ‘𝑝) = 𝑎 ∧ (2nd ‘𝑝) = 𝑏) → (𝑝 ∈ (𝐴 × 𝐵) → (𝜒 ↔ 𝜑)))
74	66, 67, 73	syl2anbr 611	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 = (1st ‘𝑝) ∧ 𝑏 = (2nd ‘𝑝)) → (𝑝 ∈ (𝐴 × 𝐵) → (𝜒 ↔ 𝜑)))
75	74	impcom 413	. . . . . . . . . . . . . . 15 ⊢ ((𝑝 ∈ (𝐴 × 𝐵) ∧ (𝑎 = (1st ‘𝑝) ∧ 𝑏 = (2nd ‘𝑝))) → (𝜒 ↔ 𝜑))
76	75	ad4ant24 767	. . . . . . . . . . . . . 14 ⊢ (((((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) ∧ 𝑝 ∈ (𝐴 × 𝐵)) ∧ 𝜑) ∧ (𝑎 = (1st ‘𝑝) ∧ 𝑏 = (2nd ‘𝑝))) → (𝜒 ↔ 𝜑))
77		simpl 488	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 = (1st ‘𝑝) ∧ 𝑏 = (2nd ‘𝑝)) → 𝑎 = (1st ‘𝑝))
78	77	eqeq1d 2762	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 = (1st ‘𝑝) ∧ 𝑏 = (2nd ‘𝑝)) → (𝑎 = 𝑥 ↔ (1st ‘𝑝) = 𝑥))
79		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 = (1st ‘𝑝) ∧ 𝑏 = (2nd ‘𝑝)) → 𝑏 = (2nd ‘𝑝))
80	79	eqeq1d 2762	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 = (1st ‘𝑝) ∧ 𝑏 = (2nd ‘𝑝)) → (𝑏 = 𝑦 ↔ (2nd ‘𝑝) = 𝑦))
81	78, 80	anbi12d 644	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = (1st ‘𝑝) ∧ 𝑏 = (2nd ‘𝑝)) → ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) ↔ ((1st ‘𝑝) = 𝑥 ∧ (2nd ‘𝑝) = 𝑦)))
82	81	adantl 487	. . . . . . . . . . . . . 14 ⊢ (((((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) ∧ 𝑝 ∈ (𝐴 × 𝐵)) ∧ 𝜑) ∧ (𝑎 = (1st ‘𝑝) ∧ 𝑏 = (2nd ‘𝑝))) → ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) ↔ ((1st ‘𝑝) = 𝑥 ∧ (2nd ‘𝑝) = 𝑦)))
83	76, 82	imbi12d 347	. . . . . . . . . . . . 13 ⊢ (((((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) ∧ 𝑝 ∈ (𝐴 × 𝐵)) ∧ 𝜑) ∧ (𝑎 = (1st ‘𝑝) ∧ 𝑏 = (2nd ‘𝑝))) → ((𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦)) ↔ (𝜑 → ((1st ‘𝑝) = 𝑥 ∧ (2nd ‘𝑝) = 𝑦))))
84		simpllr 788	. . . . . . . . . . . . 13 ⊢ ((((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) ∧ 𝑝 ∈ (𝐴 × 𝐵)) ∧ 𝜑) → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦)))
85	55, 59, 60, 61, 63, 65, 83, 84	rspc2daf 33063	. . . . . . . . . . . 12 ⊢ ((((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) ∧ 𝑝 ∈ (𝐴 × 𝐵)) ∧ 𝜑) → (𝜑 → ((1st ‘𝑝) = 𝑥 ∧ (2nd ‘𝑝) = 𝑦)))
86	85	com12 33	. . . . . . . . . . 11 ⊢ (𝜑 → ((((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) ∧ 𝑝 ∈ (𝐴 × 𝐵)) ∧ 𝜑) → ((1st ‘𝑝) = 𝑥 ∧ (2nd ‘𝑝) = 𝑦)))
87	86	anabsi7 684	. . . . . . . . . 10 ⊢ ((((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) ∧ 𝑝 ∈ (𝐴 × 𝐵)) ∧ 𝜑) → ((1st ‘𝑝) = 𝑥 ∧ (2nd ‘𝑝) = 𝑦))
88	87	simpld 500	. . . . . . . . 9 ⊢ ((((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) ∧ 𝑝 ∈ (𝐴 × 𝐵)) ∧ 𝜑) → (1st ‘𝑝) = 𝑥)
89	87	simprd 501	. . . . . . . . 9 ⊢ ((((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) ∧ 𝑝 ∈ (𝐴 × 𝐵)) ∧ 𝜑) → (2nd ‘𝑝) = 𝑦)
90	88, 89	opeq12d 4840	. . . . . . . 8 ⊢ ((((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) ∧ 𝑝 ∈ (𝐴 × 𝐵)) ∧ 𝜑) → ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩ = ⟨𝑥, 𝑦⟩)
91	51, 90	eqtrd 2795	. . . . . . 7 ⊢ ((((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) ∧ 𝑝 ∈ (𝐴 × 𝐵)) ∧ 𝜑) → 𝑝 = ⟨𝑥, 𝑦⟩)
92	91	ex 418	. . . . . 6 ⊢ (((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) ∧ 𝑝 ∈ (𝐴 × 𝐵)) → (𝜑 → 𝑝 = ⟨𝑥, 𝑦⟩))
93	92	ralrimiva 3154	. . . . 5 ⊢ ((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) → ∀𝑝 ∈ (𝐴 × 𝐵)(𝜑 → 𝑝 = ⟨𝑥, 𝑦⟩))
94	5, 49, 93	3jca 1146	. . . 4 ⊢ ((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) → (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜒 ∧ ∀𝑝 ∈ (𝐴 × 𝐵)(𝜑 → 𝑝 = ⟨𝑥, 𝑦⟩)))
95	69	opsbc2ie 33072	. . . . 5 ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ [𝑦 / 𝑏][𝑥 / 𝑎]𝜒))
96	95	eqreu 3686	. . . 4 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜒 ∧ ∀𝑝 ∈ (𝐴 × 𝐵)(𝜑 → 𝑝 = ⟨𝑥, 𝑦⟩)) → ∃!𝑝 ∈ (𝐴 × 𝐵)𝜑)
97	94, 96	syl 18	. . 3 ⊢ ((((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) → ∃!𝑝 ∈ (𝐴 × 𝐵)𝜑)
98	97	r19.29ffa 33068	. 2 ⊢ ((∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 (𝜒 → (𝑎 = 𝑥 ∧ 𝑏 = 𝑦))) → ∃!𝑝 ∈ (𝐴 × 𝐵)𝜑)
99	1, 98	sylbi 220	1 ⊢ ((∃!𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ ∃!𝑏 ∈ 𝐵 ∃𝑎 ∈ 𝐴 𝜒) → ∃!𝑝 ∈ (𝐴 × 𝐵)𝜑)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  ∃!wreu 3363  [wsbc 3738  ⟨cop 4589   × cxp 5648  ‘cfv 6538  1^st c1st 7999  2^nd c2nd 8000

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-10 2177  ax-11 2193  ax-12 2212  ax-13 2401  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

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-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6540  df-fv 6546  df-1st 8001  df-2nd 8002

This theorem is used by: (None)