Theorem fvmpocurryd 8256

Description: The value of the value of a curried operation given in maps-to notation is the operation value of the original operation. (Contributed by AV, 27-Oct-2019.)

Hypotheses

Ref	Expression
fvmpocurryd.f	⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)
fvmpocurryd.c	⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ 𝑉)
fvmpocurryd.y	⊢ (𝜑 → 𝑌 ∈ 𝑊)
fvmpocurryd.a	⊢ (𝜑 → 𝐴 ∈ 𝑋)
fvmpocurryd.b	⊢ (𝜑 → 𝐵 ∈ 𝑌)

Assertion

Ref	Expression
fvmpocurryd	⊢ (𝜑 → ((curry 𝐹‘𝐴)‘𝐵) = (𝐴𝐹𝐵))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑉,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem fvmpocurryd

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvmpocurryd.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑌)
2		csbcom 4418	. . . . 5 ⊢ ⦋𝐵 / 𝑏⦌⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑎⦌⦋𝐵 / 𝑏⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶
3		csbcow 3909	. . . . . 6 ⊢ ⦋𝐵 / 𝑏⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶
4	3	csbeq2i 3902	. . . . 5 ⊢ ⦋𝐴 / 𝑎⦌⦋𝐵 / 𝑏⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑎⦌⦋𝐵 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶
5		csbcom 4418	. . . . . 6 ⊢ ⦋𝐴 / 𝑎⦌⦋𝐵 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑎⦌⦋𝑎 / 𝑥⦌𝐶
6		csbcow 3909	. . . . . . 7 ⊢ ⦋𝐴 / 𝑎⦌⦋𝑎 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶
7	6	csbeq2i 3902	. . . . . 6 ⊢ ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑎⦌⦋𝑎 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝐶
8	5, 7	eqtri 2761	. . . . 5 ⊢ ⦋𝐴 / 𝑎⦌⦋𝐵 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝐶
9	2, 4, 8	3eqtri 2765	. . . 4 ⊢ ⦋𝐵 / 𝑏⦌⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝐶
10		fvmpocurryd.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
11		fvmpocurryd.c	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ 𝑉)
12		nfcsb1v 3919	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐶
13	12	nfel1 2920	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐶 ∈ 𝑉
14		nfcsb1v 3919	. . . . . . . 8 ⊢ Ⅎ𝑦⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝐶
15	14	nfel1 2920	. . . . . . 7 ⊢ Ⅎ𝑦⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝐶 ∈ 𝑉
16		csbeq1a 3908	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → 𝐶 = ⦋𝐴 / 𝑥⦌𝐶)
17	16	eleq1d 2819	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐶 ∈ 𝑉 ↔ ⦋𝐴 / 𝑥⦌𝐶 ∈ 𝑉))
18		csbeq1a 3908	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝐶)
19	18	eleq1d 2819	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (⦋𝐴 / 𝑥⦌𝐶 ∈ 𝑉 ↔ ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝐶 ∈ 𝑉))
20	13, 15, 17, 19	rspc2 3621	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ 𝑉 → ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝐶 ∈ 𝑉))
21	20	imp 408	. . . . 5 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ 𝑉) → ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝐶 ∈ 𝑉)
22	10, 1, 11, 21	syl21anc 837	. . . 4 ⊢ (𝜑 → ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝐶 ∈ 𝑉)
23	9, 22	eqeltrid 2838	. . 3 ⊢ (𝜑 → ⦋𝐵 / 𝑏⦌⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 ∈ 𝑉)
24		eqid 2733	. . . 4 ⊢ (𝑏 ∈ 𝑌 ↦ ⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶) = (𝑏 ∈ 𝑌 ↦ ⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
25	24	fvmpts 7002	. . 3 ⊢ ((𝐵 ∈ 𝑌 ∧ ⦋𝐵 / 𝑏⦌⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 ∈ 𝑉) → ((𝑏 ∈ 𝑌 ↦ ⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)‘𝐵) = ⦋𝐵 / 𝑏⦌⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
26	1, 23, 25	syl2anc 585	. 2 ⊢ (𝜑 → ((𝑏 ∈ 𝑌 ↦ ⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)‘𝐵) = ⦋𝐵 / 𝑏⦌⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
27		fvmpocurryd.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)
28		nfcv 2904	. . . . . 6 ⊢ Ⅎ𝑎𝐶
29		nfcv 2904	. . . . . 6 ⊢ Ⅎ𝑏𝐶
30		nfcv 2904	. . . . . . 7 ⊢ Ⅎ𝑥𝑏
31		nfcsb1v 3919	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐶
32	30, 31	nfcsbw 3921	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶
33		nfcsb1v 3919	. . . . . 6 ⊢ Ⅎ𝑦⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶
34		csbeq1a 3908	. . . . . . 7 ⊢ (𝑥 = 𝑎 → 𝐶 = ⦋𝑎 / 𝑥⦌𝐶)
35		csbeq1a 3908	. . . . . . 7 ⊢ (𝑦 = 𝑏 → ⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
36	34, 35	sylan9eq 2793	. . . . . 6 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → 𝐶 = ⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
37	28, 29, 32, 33, 36	cbvmpo 7503	. . . . 5 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) = (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑌 ↦ ⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
38	27, 37	eqtri 2761	. . . 4 ⊢ 𝐹 = (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑌 ↦ ⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
39	31	nfel1 2920	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐶 ∈ 𝑉
40	33	nfel1 2920	. . . . . . 7 ⊢ Ⅎ𝑦⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 ∈ 𝑉
41	34	eleq1d 2819	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝐶 ∈ 𝑉 ↔ ⦋𝑎 / 𝑥⦌𝐶 ∈ 𝑉))
42	35	eleq1d 2819	. . . . . . 7 ⊢ (𝑦 = 𝑏 → (⦋𝑎 / 𝑥⦌𝐶 ∈ 𝑉 ↔ ⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 ∈ 𝑉))
43	39, 40, 41, 42	rspc2 3621	. . . . . 6 ⊢ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑌) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ 𝑉 → ⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 ∈ 𝑉))
44	11, 43	mpan9 508	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑌)) → ⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 ∈ 𝑉)
45	44	ralrimivva 3201	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑌 ⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 ∈ 𝑉)
46	1	ne0d 4336	. . . 4 ⊢ (𝜑 → 𝑌 ≠ ∅)
47		fvmpocurryd.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑊)
48	38, 45, 46, 47, 10	mpocurryvald 8255	. . 3 ⊢ (𝜑 → (curry 𝐹‘𝐴) = (𝑏 ∈ 𝑌 ↦ ⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶))
49	48	fveq1d 6894	. 2 ⊢ (𝜑 → ((curry 𝐹‘𝐴)‘𝐵) = ((𝑏 ∈ 𝑌 ↦ ⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)‘𝐵))
50	27	a1i 11	. . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶))
51		csbcow 3909	. . . . . . . 8 ⊢ ⦋𝑦 / 𝑏⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑦 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶
52		csbid 3907	. . . . . . . 8 ⊢ ⦋𝑦 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑎 / 𝑥⦌𝐶
53	51, 52	eqtr2i 2762	. . . . . . 7 ⊢ ⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑦 / 𝑏⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶
54	53	a1i 11	. . . . . 6 ⊢ (𝜑 → ⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑦 / 𝑏⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
55	54	csbeq2dv 3901	. . . . 5 ⊢ (𝜑 → ⦋𝑥 / 𝑎⦌⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑥 / 𝑎⦌⦋𝑦 / 𝑏⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
56		csbcow 3909	. . . . . 6 ⊢ ⦋𝑥 / 𝑎⦌⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑥 / 𝑥⦌𝐶
57		csbid 3907	. . . . . 6 ⊢ ⦋𝑥 / 𝑥⦌𝐶 = 𝐶
58	56, 57	eqtri 2761	. . . . 5 ⊢ ⦋𝑥 / 𝑎⦌⦋𝑎 / 𝑥⦌𝐶 = 𝐶
59		csbcom 4418	. . . . 5 ⊢ ⦋𝑥 / 𝑎⦌⦋𝑦 / 𝑏⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑦 / 𝑏⦌⦋𝑥 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶
60	55, 58, 59	3eqtr3g 2796	. . . 4 ⊢ (𝜑 → 𝐶 = ⦋𝑦 / 𝑏⦌⦋𝑥 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
61		csbeq1 3897	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ⦋𝑥 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
62	61	adantr 482	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ⦋𝑥 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
63	62	csbeq2dv 3901	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ⦋𝑦 / 𝑏⦌⦋𝑥 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑦 / 𝑏⦌⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
64		csbeq1 3897	. . . . . 6 ⊢ (𝑦 = 𝐵 → ⦋𝑦 / 𝑏⦌⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑏⦌⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
65	64	adantl 483	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ⦋𝑦 / 𝑏⦌⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑏⦌⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
66	63, 65	eqtrd 2773	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ⦋𝑦 / 𝑏⦌⦋𝑥 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑏⦌⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
67	60, 66	sylan9eq 2793	. . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝐶 = ⦋𝐵 / 𝑏⦌⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
68		eqidd 2734	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑌 = 𝑌)
69		nfv 1918	. . 3 ⊢ Ⅎ𝑥𝜑
70		nfv 1918	. . 3 ⊢ Ⅎ𝑦𝜑
71		nfcv 2904	. . 3 ⊢ Ⅎ𝑦𝐴
72		nfcv 2904	. . 3 ⊢ Ⅎ𝑥𝐵
73		nfcv 2904	. . . . 5 ⊢ Ⅎ𝑥𝐴
74	73, 32	nfcsbw 3921	. . . 4 ⊢ Ⅎ𝑥⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶
75	72, 74	nfcsbw 3921	. . 3 ⊢ Ⅎ𝑥⦋𝐵 / 𝑏⦌⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶
76	9, 14	nfcxfr 2902	. . 3 ⊢ Ⅎ𝑦⦋𝐵 / 𝑏⦌⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶
77	50, 67, 68, 10, 1, 23, 69, 70, 71, 72, 75, 76	ovmpodxf 7558	. 2 ⊢ (𝜑 → (𝐴𝐹𝐵) = ⦋𝐵 / 𝑏⦌⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
78	26, 49, 77	3eqtr4d 2783	1 ⊢ (𝜑 → ((curry 𝐹‘𝐴)‘𝐵) = (𝐴𝐹𝐵))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  ⦋csb 3894   ↦ cmpt 5232  ‘cfv 6544  (class class class)co 7409   ∈ cmpo 7411  curry ccur 8250

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-cur 8252

This theorem is referenced by:  pmatcollpw3lem  22285  logbfval  26295