Theorem ov3 7587

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 3468	. 2 ⊢ ∃𝑧 𝑧 = 𝑆
3		nfv 1946	. . 3 ⊢ Ⅎ𝑧((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) ∧ (𝐶 ∈ 𝐻 ∧ 𝐷 ∈ 𝐻))
4		nfcv 2922	. . . . 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 7487	. . . . . 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 2920	. . . . 5 ⊢ Ⅎ𝑧𝐹
8		nfcv 2922	. . . . 5 ⊢ Ⅎ𝑧⟨𝐶, 𝐷⟩
9	4, 7, 8	nfov 7454	. . . 4 ⊢ Ⅎ𝑧(⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩)
10	9	nfeq1 2937	. . 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 2771	. . . . . 6 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → (𝑧 = 𝑅 ↔ 𝑧 = 𝑆))
13	12	copsex4g 5467	. . . . 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 5688	. . . . . 6 ⊢ ((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) → ⟨𝐴, 𝐵⟩ ∈ (𝐻 × 𝐻))
15		opelxpi 5688	. . . . . 6 ⊢ ((𝐶 ∈ 𝐻 ∧ 𝐷 ∈ 𝐻) → ⟨𝐶, 𝐷⟩ ∈ (𝐻 × 𝐻))
16		nfcv 2922	. . . . . . 7 ⊢ Ⅎ𝑥⟨𝐴, 𝐵⟩
17		nfcv 2922	. . . . . . 7 ⊢ Ⅎ𝑦⟨𝐴, 𝐵⟩
18		nfcv 2922	. . . . . . 7 ⊢ Ⅎ𝑦⟨𝐶, 𝐷⟩
19		nfv 1946	. . . . . . . 8 ⊢ Ⅎ𝑥∃𝑤∃𝑣∃𝑢∃𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅)
20		nfoprab1 7485	. . . . . . . . . . 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 2920	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐹
22		nfcv 2922	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑦
23	16, 21, 22	nfov 7454	. . . . . . . . 9 ⊢ Ⅎ𝑥(⟨𝐴, 𝐵⟩𝐹𝑦)
24	23	nfeq1 2937	. . . . . . . 8 ⊢ Ⅎ𝑥(⟨𝐴, 𝐵⟩𝐹𝑦) = 𝑧
25	19, 24	nfim 1928	. . . . . . 7 ⊢ Ⅎ𝑥(∃𝑤∃𝑣∃𝑢∃𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) → (⟨𝐴, 𝐵⟩𝐹𝑦) = 𝑧)
26		nfv 1946	. . . . . . . 8 ⊢ Ⅎ𝑦∃𝑤∃𝑣∃𝑢∃𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅)
27		nfoprab2 7486	. . . . . . . . . . 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 2920	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝐹
29	17, 28, 18	nfov 7454	. . . . . . . . 9 ⊢ Ⅎ𝑦(⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩)
30	29	nfeq1 2937	. . . . . . . 8 ⊢ Ⅎ𝑦(⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩) = 𝑧
31	26, 30	nfim 1928	. . . . . . 7 ⊢ Ⅎ𝑦(∃𝑤∃𝑣∃𝑢∃𝑓((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) → (⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩) = 𝑧)
32		eqeq1 2764	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑥 = ⟨𝑤, 𝑣⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩))
33	32	anbi1d 643	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)))
34	33	anbi1d 643	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ ((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅)))
35	34	4exbidv 1958	. . . . . . . 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 7431	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑥𝐹𝑦) = (⟨𝐴, 𝐵⟩𝐹𝑦))
37	36	eqeq1d 2762	. . . . . . . 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 2764	. . . . . . . . . . 11 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (𝑦 = ⟨𝑢, 𝑓⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩))
40	39	anbi2d 642	. . . . . . . . . 10 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → ((⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑤, 𝑣⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑢, 𝑓⟩)))
41	40	anbi1d 643	. . . . . . . . 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 1958	. . . . . . . 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 7432	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (⟨𝐴, 𝐵⟩𝐹𝑦) = (⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩))
44	43	eqeq1d 2762	. . . . . . . 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 3664	. . . . . . . . . . . 12 ⊢ ∃*𝑧 𝑧 = 𝑅
47	46	mosubop 5483	. . . . . . . . . . 11 ⊢ ∃*𝑧∃𝑢∃𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)
48	47	mosubop 5483	. . . . . . . . . 10 ⊢ ∃*𝑧∃𝑤∃𝑣(𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢∃𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅))
49		anass 474	. . . . . . . . . . . . . 14 ⊢ (((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
50	49	2exbii 1881	. . . . . . . . . . . . 13 ⊢ (∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ ∃𝑢∃𝑓(𝑥 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
51		19.42vv 1989	. . . . . . . . . . . . 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 278	. . . . . . . . . . . 12 ⊢ (∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅) ↔ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢∃𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = 𝑅)))
53	52	2exbii 1881	. . . . . . . . . . 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 2573	. . . . . . . . . 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 234	. . . . . . . . 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 7567	. . . . . . 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 3538	. . . . . 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 608	. . . . 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 263	. . . 4 ⊢ (((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) ∧ (𝐶 ∈ 𝐻 ∧ 𝐷 ∈ 𝐻)) → (𝑧 = 𝑆 → (⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩) = 𝑧))
61		eqeq2 2772	. . . 4 ⊢ (𝑧 = 𝑆 → ((⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩) = 𝑧 ↔ (⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩) = 𝑆))
62	60, 61	mpbidi 244	. . 3 ⊢ (((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) ∧ (𝐶 ∈ 𝐻 ∧ 𝐷 ∈ 𝐻)) → (𝑧 = 𝑆 → (⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩) = 𝑆))
63	3, 10, 62	exlimd 2254	. 2 ⊢ (((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) ∧ (𝐶 ∈ 𝐻 ∧ 𝐷 ∈ 𝐻)) → (∃𝑧 𝑧 = 𝑆 → (⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩) = 𝑆))
64	2, 63	mpi 21	1 ⊢ (((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) ∧ (𝐶 ∈ 𝐻 ∧ 𝐷 ∈ 𝐻)) → (⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩) = 𝑆)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ∃*wmo 2562  Vcvv 3450  ⟨cop 4834   × cxp 5649  (class class class)co 7424  {coprab 7425

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-ext 2732  ax-sep 5248  ax-pr 5391

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-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6497  df-fun 6545  df-fv 6551  df-ov 7427  df-oprab 7428

This theorem is used by:  addcnsr  11227  mulcnsr  11228