Theorem ovi3 5915

Description: The value of an operation class abstraction. Special case. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 29-Dec-2014.)

Hypotheses

Ref	Expression
ovi3.1	⊢ (((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) ∧ (𝐶 ∈ 𝐻 ∧ 𝐷 ∈ 𝐻)) → 𝑆 ∈ (𝐻 × 𝐻))
ovi3.2	⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → 𝑅 = 𝑆)
ovi3.3	⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅))}

Assertion

Ref	Expression
ovi3	⊢ (((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) ∧ (𝐶 ∈ 𝐻 ∧ 𝐷 ∈ 𝐻)) → (⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩) = 𝑆)

Distinct variable groups:   𝑢,𝑓,𝑣,𝑤,𝑥,𝑦,𝑧,𝐴   𝐵,𝑓,𝑢,𝑣,𝑤,𝑥,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝐶,𝑓,𝑢,𝑣,𝑤,𝑦,𝑧   𝐷,𝑓,𝑢,𝑣,𝑤,𝑦,𝑧   𝑓,𝐻,𝑢,𝑣,𝑤,𝑥,𝑦,𝑧   𝑆,𝑓,𝑢,𝑣,𝑤,𝑧

Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑥)   𝑅(𝑤,𝑣,𝑢,𝑓)   𝑆(𝑥,𝑦)   𝐹(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑓)

Proof of Theorem ovi3

Step	Hyp	Ref	Expression
1		ovi3.1	. . . 4 ⊢ (((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) ∧ (𝐶 ∈ 𝐻 ∧ 𝐷 ∈ 𝐻)) → 𝑆 ∈ (𝐻 × 𝐻))
2		elex 2700	. . . 4 ⊢ (𝑆 ∈ (𝐻 × 𝐻) → 𝑆 ∈ V)
3	1, 2	syl 14	. . 3 ⊢ (((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) ∧ (𝐶 ∈ 𝐻 ∧ 𝐷 ∈ 𝐻)) → 𝑆 ∈ V)
4		isset 2695	. . 3 ⊢ (𝑆 ∈ V ↔ ∃𝑧 𝑧 = 𝑆)
5	3, 4	sylib 121	. 2 ⊢ (((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) ∧ (𝐶 ∈ 𝐻 ∧ 𝐷 ∈ 𝐻)) → ∃𝑧 𝑧 = 𝑆)
6		nfv 1509	. . 3 ⊢ Ⅎ𝑧((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) ∧ (𝐶 ∈ 𝐻 ∧ 𝐷 ∈ 𝐻))
7		nfcv 2282	. . . . 5 ⊢ Ⅎ𝑧⟨𝐴, 𝐵⟩
8		ovi3.3	. . . . . 6 ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅))}
9		nfoprab3 5830	. . . . . 6 ⊢ Ⅎ𝑧{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅))}
10	8, 9	nfcxfr 2279	. . . . 5 ⊢ Ⅎ𝑧𝐹
11		nfcv 2282	. . . . 5 ⊢ Ⅎ𝑧⟨𝐶, 𝐷⟩
12	7, 10, 11	nfov 5809	. . . 4 ⊢ Ⅎ𝑧(⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩)
13	12	nfeq1 2292	. . 3 ⊢ Ⅎ𝑧(⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩) = 𝑆
14		ovi3.2	. . . . . . 7 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → 𝑅 = 𝑆)
15	14	eqeq2d 2152	. . . . . 6 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → (𝑧 = 𝑅 ↔ 𝑧 = 𝑆))
16	15	copsex4g 4177	. . . . 5 ⊢ (((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) ∧ (𝐶 ∈ 𝐻 ∧ 𝐷 ∈ 𝐻)) → (∃𝑤∃𝑣∃𝑢∃𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ 𝑧 = 𝑆))
17		opelxpi 4579	. . . . . 6 ⊢ ((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) → ⟨𝐴, 𝐵⟩ ∈ (𝐻 × 𝐻))
18		opelxpi 4579	. . . . . 6 ⊢ ((𝐶 ∈ 𝐻 ∧ 𝐷 ∈ 𝐻) → ⟨𝐶, 𝐷⟩ ∈ (𝐻 × 𝐻))
19		nfcv 2282	. . . . . . 7 ⊢ Ⅎ𝑥⟨𝐴, 𝐵⟩
20		nfcv 2282	. . . . . . 7 ⊢ Ⅎ𝑦⟨𝐴, 𝐵⟩
21		nfcv 2282	. . . . . . 7 ⊢ Ⅎ𝑦⟨𝐶, 𝐷⟩
22		nfv 1509	. . . . . . . 8 ⊢ Ⅎ𝑥∃𝑤∃𝑣∃𝑢∃𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅)
23		nfoprab1 5828	. . . . . . . . . . 11 ⊢ Ⅎ𝑥{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅))}
24	8, 23	nfcxfr 2279	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐹
25		nfcv 2282	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑦
26	19, 24, 25	nfov 5809	. . . . . . . . 9 ⊢ Ⅎ𝑥(⟨𝐴, 𝐵⟩𝐹𝑦)
27	26	nfeq1 2292	. . . . . . . 8 ⊢ Ⅎ𝑥(⟨𝐴, 𝐵⟩𝐹𝑦) = 𝑧
28	22, 27	nfim 1552	. . . . . . 7 ⊢ Ⅎ𝑥(∃𝑤∃𝑣∃𝑢∃𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) → (⟨𝐴, 𝐵⟩𝐹𝑦) = 𝑧)
29		nfv 1509	. . . . . . . 8 ⊢ Ⅎ𝑦∃𝑤∃𝑣∃𝑢∃𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅)
30		nfoprab2 5829	. . . . . . . . . . 11 ⊢ Ⅎ𝑦{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅))}
31	8, 30	nfcxfr 2279	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝐹
32	20, 31, 21	nfov 5809	. . . . . . . . 9 ⊢ Ⅎ𝑦(⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩)
33	32	nfeq1 2292	. . . . . . . 8 ⊢ Ⅎ𝑦(⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩) = 𝑧
34	29, 33	nfim 1552	. . . . . . 7 ⊢ Ⅎ𝑦(∃𝑤∃𝑣∃𝑢∃𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) → (⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩) = 𝑧)
35		eqeq1 2147	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑥 = ⟨𝑤, 𝑣⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩))
36	35	anbi1d 461	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)))
37	36	anbi1d 461	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ ((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅)))
38	37	4exbidv 1843	. . . . . . . 8 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ ∃𝑤∃𝑣∃𝑢∃𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅)))
39		oveq1 5789	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑥𝐹𝑦) = (⟨𝐴, 𝐵⟩𝐹𝑦))
40	39	eqeq1d 2149	. . . . . . . 8 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ((𝑥𝐹𝑦) = 𝑧 ↔ (⟨𝐴, 𝐵⟩𝐹𝑦) = 𝑧))
41	38, 40	imbi12d 233	. . . . . . 7 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ((∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) → (𝑥𝐹𝑦) = 𝑧) ↔ (∃𝑤∃𝑣∃𝑢∃𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) → (⟨𝐴, 𝐵⟩𝐹𝑦) = 𝑧)))
42		eqeq1 2147	. . . . . . . . . . 11 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (𝑦 = ⟨𝑢, 𝑓⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩))
43	42	anbi2d 460	. . . . . . . . . 10 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → ((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩)))
44	43	anbi1d 461	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ ((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅)))
45	44	4exbidv 1843	. . . . . . . 8 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (∃𝑤∃𝑣∃𝑢∃𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ ∃𝑤∃𝑣∃𝑢∃𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅)))
46		oveq2 5790	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (⟨𝐴, 𝐵⟩𝐹𝑦) = (⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩))
47	46	eqeq1d 2149	. . . . . . . 8 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → ((⟨𝐴, 𝐵⟩𝐹𝑦) = 𝑧 ↔ (⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩) = 𝑧))
48	45, 47	imbi12d 233	. . . . . . 7 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → ((∃𝑤∃𝑣∃𝑢∃𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) → (⟨𝐴, 𝐵⟩𝐹𝑦) = 𝑧) ↔ (∃𝑤∃𝑣∃𝑢∃𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) → (⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩) = 𝑧)))
49		moeq 2863	. . . . . . . . . . . 12 ⊢ ∃*𝑧 𝑧 = 𝑅
50	49	mosubop 4613	. . . . . . . . . . 11 ⊢ ∃*𝑧∃𝑢∃𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)
51	50	mosubop 4613	. . . . . . . . . 10 ⊢ ∃*𝑧∃𝑤∃𝑣(𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢∃𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅))
52		anass 399	. . . . . . . . . . . . . 14 ⊢ (((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
53	52	2exbii 1586	. . . . . . . . . . . . 13 ⊢ (∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ ∃𝑢∃𝑓(𝑥 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
54		19.42vv 1884	. . . . . . . . . . . . 13 ⊢ (∃𝑢∃𝑓(𝑥 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)) ↔ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢∃𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
55	53, 54	bitri 183	. . . . . . . . . . . 12 ⊢ (∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢∃𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
56	55	2exbii 1586	. . . . . . . . . . 11 ⊢ (∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ ∃𝑤∃𝑣(𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢∃𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
57	56	mobii 2037	. . . . . . . . . 10 ⊢ (∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ ∃*𝑧∃𝑤∃𝑣(𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢∃𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
58	51, 57	mpbir 145	. . . . . . . . 9 ⊢ ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅)
59	58	a1i 9	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) → ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅))
60	59, 8	ovidi 5897	. . . . . . 7 ⊢ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) → (∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) → (𝑥𝐹𝑦) = 𝑧))
61	19, 20, 21, 28, 34, 41, 48, 60	vtocl2gaf 2756	. . . . . 6 ⊢ ((⟨𝐴, 𝐵⟩ ∈ (𝐻 × 𝐻) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝐻 × 𝐻)) → (∃𝑤∃𝑣∃𝑢∃𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) → (⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩) = 𝑧))
62	17, 18, 61	syl2an 287	. . . . 5 ⊢ (((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) ∧ (𝐶 ∈ 𝐻 ∧ 𝐷 ∈ 𝐻)) → (∃𝑤∃𝑣∃𝑢∃𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) → (⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩) = 𝑧))
63	16, 62	sylbird 169	. . . 4 ⊢ (((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) ∧ (𝐶 ∈ 𝐻 ∧ 𝐷 ∈ 𝐻)) → (𝑧 = 𝑆 → (⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩) = 𝑧))
64		eqeq2 2150	. . . 4 ⊢ (𝑧 = 𝑆 → ((⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩) = 𝑧 ↔ (⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩) = 𝑆))
65	63, 64	mpbidi 150	. . 3 ⊢ (((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) ∧ (𝐶 ∈ 𝐻 ∧ 𝐷 ∈ 𝐻)) → (𝑧 = 𝑆 → (⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩) = 𝑆))
66	6, 13, 65	exlimd 1577	. 2 ⊢ (((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) ∧ (𝐶 ∈ 𝐻 ∧ 𝐷 ∈ 𝐻)) → (∃𝑧 𝑧 = 𝑆 → (⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩) = 𝑆))
67	5, 66	mpd 13	1 ⊢ (((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) ∧ (𝐶 ∈ 𝐻 ∧ 𝐷 ∈ 𝐻)) → (⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩) = 𝑆)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332  ∃wex 1469   ∈ wcel 1481  ∃*wmo 2001  Vcvv 2689  ⟨cop 3535   × cxp 4545  (class class class)co 5782  {coprab 5783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-setind 4460

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-iota 5096  df-fun 5133  df-fv 5139  df-ov 5785  df-oprab 5786

This theorem is referenced by:  oviec  6543  addcnsr  7666  mulcnsr  7667