Theorem ov3 7313

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
ov3.1	⊢ 𝑆 ∈ V
ov3.2	⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → 𝑅 = 𝑆)
ov3.3	⊢ 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
ov3	⊢ (((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 ov3

Step	Hyp	Ref	Expression
1		ov3.1	. . 3 ⊢ 𝑆 ∈ V
2	1	isseti 3510	. 2 ⊢ ∃𝑧 𝑧 = 𝑆
3		nfv 1915	. . 3 ⊢ Ⅎ𝑧((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) ∧ (𝐶 ∈ 𝐻 ∧ 𝐷 ∈ 𝐻))
4		nfcv 2979	. . . . 5 ⊢ Ⅎ𝑧⟨𝐴, 𝐵⟩
5		ov3.3	. . . . . 6 ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅))}
6		nfoprab3 7219	. . . . . 6 ⊢ Ⅎ𝑧{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅))}
7	5, 6	nfcxfr 2977	. . . . 5 ⊢ Ⅎ𝑧𝐹
8		nfcv 2979	. . . . 5 ⊢ Ⅎ𝑧⟨𝐶, 𝐷⟩
9	4, 7, 8	nfov 7188	. . . 4 ⊢ Ⅎ𝑧(⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩)
10	9	nfeq1 2995	. . 3 ⊢ Ⅎ𝑧(⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩) = 𝑆
11		ov3.2	. . . . . . 7 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → 𝑅 = 𝑆)
12	11	eqeq2d 2834	. . . . . 6 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → (𝑧 = 𝑅 ↔ 𝑧 = 𝑆))
13	12	copsex4g 5387	. . . . 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))
14		opelxpi 5594	. . . . . 6 ⊢ ((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) → ⟨𝐴, 𝐵⟩ ∈ (𝐻 × 𝐻))
15		opelxpi 5594	. . . . . 6 ⊢ ((𝐶 ∈ 𝐻 ∧ 𝐷 ∈ 𝐻) → ⟨𝐶, 𝐷⟩ ∈ (𝐻 × 𝐻))
16		nfcv 2979	. . . . . . 7 ⊢ Ⅎ𝑥⟨𝐴, 𝐵⟩
17		nfcv 2979	. . . . . . 7 ⊢ Ⅎ𝑦⟨𝐴, 𝐵⟩
18		nfcv 2979	. . . . . . 7 ⊢ Ⅎ𝑦⟨𝐶, 𝐷⟩
19		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑥∃𝑤∃𝑣∃𝑢∃𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅)
20		nfoprab1 7217	. . . . . . . . . . 11 ⊢ Ⅎ𝑥{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅))}
21	5, 20	nfcxfr 2977	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐹
22		nfcv 2979	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑦
23	16, 21, 22	nfov 7188	. . . . . . . . 9 ⊢ Ⅎ𝑥(⟨𝐴, 𝐵⟩𝐹𝑦)
24	23	nfeq1 2995	. . . . . . . 8 ⊢ Ⅎ𝑥(⟨𝐴, 𝐵⟩𝐹𝑦) = 𝑧
25	19, 24	nfim 1897	. . . . . . 7 ⊢ Ⅎ𝑥(∃𝑤∃𝑣∃𝑢∃𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) → (⟨𝐴, 𝐵⟩𝐹𝑦) = 𝑧)
26		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑦∃𝑤∃𝑣∃𝑢∃𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅)
27		nfoprab2 7218	. . . . . . . . . . 11 ⊢ Ⅎ𝑦{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅))}
28	5, 27	nfcxfr 2977	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝐹
29	17, 28, 18	nfov 7188	. . . . . . . . 9 ⊢ Ⅎ𝑦(⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩)
30	29	nfeq1 2995	. . . . . . . 8 ⊢ Ⅎ𝑦(⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩) = 𝑧
31	26, 30	nfim 1897	. . . . . . 7 ⊢ Ⅎ𝑦(∃𝑤∃𝑣∃𝑢∃𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) → (⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩) = 𝑧)
32		eqeq1 2827	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑥 = ⟨𝑤, 𝑣⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩))
33	32	anbi1d 631	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)))
34	33	anbi1d 631	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ ((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅)))
35	34	4exbidv 1927	. . . . . . . 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)))
36		oveq1 7165	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑥𝐹𝑦) = (⟨𝐴, 𝐵⟩𝐹𝑦))
37	36	eqeq1d 2825	. . . . . . . 8 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ((𝑥𝐹𝑦) = 𝑧 ↔ (⟨𝐴, 𝐵⟩𝐹𝑦) = 𝑧))
38	35, 37	imbi12d 347	. . . . . . 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)))
39		eqeq1 2827	. . . . . . . . . . 11 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (𝑦 = ⟨𝑢, 𝑓⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩))
40	39	anbi2d 630	. . . . . . . . . 10 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → ((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩)))
41	40	anbi1d 631	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ ((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅)))
42	41	4exbidv 1927	. . . . . . . 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)))
43		oveq2 7166	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (⟨𝐴, 𝐵⟩𝐹𝑦) = (⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩))
44	43	eqeq1d 2825	. . . . . . . 8 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → ((⟨𝐴, 𝐵⟩𝐹𝑦) = 𝑧 ↔ (⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩) = 𝑧))
45	42, 44	imbi12d 347	. . . . . . 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)))
46		moeq 3700	. . . . . . . . . . . 12 ⊢ ∃*𝑧 𝑧 = 𝑅
47	46	mosubop 5403	. . . . . . . . . . 11 ⊢ ∃*𝑧∃𝑢∃𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)
48	47	mosubop 5403	. . . . . . . . . 10 ⊢ ∃*𝑧∃𝑤∃𝑣(𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢∃𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅))
49		anass 471	. . . . . . . . . . . . . 14 ⊢ (((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
50	49	2exbii 1849	. . . . . . . . . . . . 13 ⊢ (∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ ∃𝑢∃𝑓(𝑥 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
51		19.42vv 1958	. . . . . . . . . . . . 13 ⊢ (∃𝑢∃𝑓(𝑥 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)) ↔ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢∃𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
52	50, 51	bitri 277	. . . . . . . . . . . 12 ⊢ (∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢∃𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
53	52	2exbii 1849	. . . . . . . . . . 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)))
54	53	mobii 2631	. . . . . . . . . 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)))
55	48, 54	mpbir 233	. . . . . . . . 9 ⊢ ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅)
56	55	a1i 11	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) → ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅))
57	56, 5	ovidi 7295	. . . . . . 7 ⊢ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) → (∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) → (𝑥𝐹𝑦) = 𝑧))
58	16, 17, 18, 25, 31, 38, 45, 57	vtocl2gaf 3578	. . . . . 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))
59	14, 15, 58	syl2an 597	. . . . 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))
60	13, 59	sylbird 262	. . . 4 ⊢ (((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) ∧ (𝐶 ∈ 𝐻 ∧ 𝐷 ∈ 𝐻)) → (𝑧 = 𝑆 → (⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩) = 𝑧))
61		eqeq2 2835	. . . 4 ⊢ (𝑧 = 𝑆 → ((⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩) = 𝑧 ↔ (⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩) = 𝑆))
62	60, 61	mpbidi 243	. . 3 ⊢ (((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) ∧ (𝐶 ∈ 𝐻 ∧ 𝐷 ∈ 𝐻)) → (𝑧 = 𝑆 → (⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩) = 𝑆))
63	3, 10, 62	exlimd 2218	. 2 ⊢ (((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) ∧ (𝐶 ∈ 𝐻 ∧ 𝐷 ∈ 𝐻)) → (∃𝑧 𝑧 = 𝑆 → (⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩) = 𝑆))
64	2, 63	mpi 20	1 ⊢ (((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) ∧ (𝐶 ∈ 𝐻 ∧ 𝐷 ∈ 𝐻)) → (⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩) = 𝑆)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114  ∃*wmo 2620  Vcvv 3496  ⟨cop 4575   × cxp 5555  (class class class)co 7158  {coprab 7159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-iota 6316  df-fun 6359  df-fv 6365  df-ov 7161  df-oprab 7162

This theorem is referenced by:  addcnsr  10559  mulcnsr  10560