Theorem fvmpocurryd 8266

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 4391	. . . . 5 ⊢ ⦋𝐵 / 𝑏⦌⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑎⦌⦋𝐵 / 𝑏⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶
3		csbcow 3876	. . . . . 6 ⊢ ⦋𝐵 / 𝑏⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶
4	3	csbeq2i 3869	. . . . 5 ⊢ ⦋𝐴 / 𝑎⦌⦋𝐵 / 𝑏⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑎⦌⦋𝐵 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶
5		csbcom 4391	. . . . . 6 ⊢ ⦋𝐴 / 𝑎⦌⦋𝐵 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑎⦌⦋𝑎 / 𝑥⦌𝐶
6		csbcow 3876	. . . . . . 7 ⊢ ⦋𝐴 / 𝑎⦌⦋𝑎 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶
7	6	csbeq2i 3869	. . . . . 6 ⊢ ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑎⦌⦋𝑎 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝐶
8	5, 7	eqtri 2792	. . . . 5 ⊢ ⦋𝐴 / 𝑎⦌⦋𝐵 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝐶
9	2, 4, 8	3eqtri 2796	. . . 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 3885	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐶
13	12	nfel1 2947	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐶 ∈ 𝑉
14		nfcsb1v 3885	. . . . . . . 8 ⊢ Ⅎ𝑦⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝐶
15	14	nfel1 2947	. . . . . . 7 ⊢ Ⅎ𝑦⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝐶 ∈ 𝑉
16		csbeq1a 3875	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → 𝐶 = ⦋𝐴 / 𝑥⦌𝐶)
17	16	eleq1d 2854	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐶 ∈ 𝑉 ↔ ⦋𝐴 / 𝑥⦌𝐶 ∈ 𝑉))
18		csbeq1a 3875	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝐶)
19	18	eleq1d 2854	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (⦋𝐴 / 𝑥⦌𝐶 ∈ 𝑉 ↔ ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝐶 ∈ 𝑉))
20	13, 15, 17, 19	rspc2 3599	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ 𝑉 → ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝐶 ∈ 𝑉))
21	20	imp 411	. . . . 5 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ 𝑉) → ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝐶 ∈ 𝑉)
22	10, 1, 11, 21	syl21anc 850	. . . 4 ⊢ (𝜑 → ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝐶 ∈ 𝑉)
23	9, 22	eqeltrid 2873	. . 3 ⊢ (𝜑 → ⦋𝐵 / 𝑏⦌⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 ∈ 𝑉)
24		eqid 2769	. . . 4 ⊢ (𝑏 ∈ 𝑌 ↦ ⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶) = (𝑏 ∈ 𝑌 ↦ ⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
25	24	fvmpts 6994	. . 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 595	. 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 2931	. . . . . 6 ⊢ Ⅎ𝑎𝐶
29		nfcv 2931	. . . . . 6 ⊢ Ⅎ𝑏𝐶
30		nfcv 2931	. . . . . . 7 ⊢ Ⅎ𝑥𝑏
31		nfcsb1v 3885	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐶
32	30, 31	nfcsbw 3887	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶
33		nfcsb1v 3885	. . . . . 6 ⊢ Ⅎ𝑦⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶
34		csbeq1a 3875	. . . . . . 7 ⊢ (𝑥 = 𝑎 → 𝐶 = ⦋𝑎 / 𝑥⦌𝐶)
35		csbeq1a 3875	. . . . . . 7 ⊢ (𝑦 = 𝑏 → ⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
36	34, 35	sylan9eq 2824	. . . . . 6 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → 𝐶 = ⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
37	28, 29, 32, 33, 36	cbvmpo 7505	. . . . 5 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) = (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑌 ↦ ⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
38	27, 37	eqtri 2792	. . . 4 ⊢ 𝐹 = (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑌 ↦ ⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
39	31	nfel1 2947	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐶 ∈ 𝑉
40	33	nfel1 2947	. . . . . . 7 ⊢ Ⅎ𝑦⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 ∈ 𝑉
41	34	eleq1d 2854	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝐶 ∈ 𝑉 ↔ ⦋𝑎 / 𝑥⦌𝐶 ∈ 𝑉))
42	35	eleq1d 2854	. . . . . . 7 ⊢ (𝑦 = 𝑏 → (⦋𝑎 / 𝑥⦌𝐶 ∈ 𝑉 ↔ ⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 ∈ 𝑉))
43	39, 40, 41, 42	rspc2 3599	. . . . . 6 ⊢ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑌) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ 𝑉 → ⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 ∈ 𝑉))
44	11, 43	mpan9 515	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑌)) → ⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 ∈ 𝑉)
45	44	ralrimivva 3214	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑌 ⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 ∈ 𝑉)
46	1	ne0d 4303	. . . 4 ⊢ (𝜑 → 𝑌 ≠ ∅)
47		fvmpocurryd.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑊)
48	38, 45, 46, 47, 10	mpocurryvald 8265	. . 3 ⊢ (𝜑 → (curry 𝐹‘𝐴) = (𝑏 ∈ 𝑌 ↦ ⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶))
49	48	fveq1d 6884	. 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 3876	. . . . . . . 8 ⊢ ⦋𝑦 / 𝑏⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑦 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶
52		csbid 3874	. . . . . . . 8 ⊢ ⦋𝑦 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑎 / 𝑥⦌𝐶
53	51, 52	eqtr2i 2793	. . . . . . 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 3868	. . . . 5 ⊢ (𝜑 → ⦋𝑥 / 𝑎⦌⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑥 / 𝑎⦌⦋𝑦 / 𝑏⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
56		csbcow 3876	. . . . . 6 ⊢ ⦋𝑥 / 𝑎⦌⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑥 / 𝑥⦌𝐶
57		csbid 3874	. . . . . 6 ⊢ ⦋𝑥 / 𝑥⦌𝐶 = 𝐶
58	56, 57	eqtri 2792	. . . . 5 ⊢ ⦋𝑥 / 𝑎⦌⦋𝑎 / 𝑥⦌𝐶 = 𝐶
59		csbcom 4391	. . . . 5 ⊢ ⦋𝑥 / 𝑎⦌⦋𝑦 / 𝑏⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑦 / 𝑏⦌⦋𝑥 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶
60	55, 58, 59	3eqtr3g 2827	. . . 4 ⊢ (𝜑 → 𝐶 = ⦋𝑦 / 𝑏⦌⦋𝑥 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
61		csbeq1 3864	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ⦋𝑥 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
62	61	adantr 485	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ⦋𝑥 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
63	62	csbeq2dv 3868	. . . . 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 3864	. . . . . 6 ⊢ (𝑦 = 𝐵 → ⦋𝑦 / 𝑏⦌⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑏⦌⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
65	64	adantl 486	. . . . 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 2804	. . . 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 2824	. . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝐶 = ⦋𝐵 / 𝑏⦌⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
68		eqidd 2770	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑌 = 𝑌)
69		nfv 1941	. . 3 ⊢ Ⅎ𝑥𝜑
70		nfv 1941	. . 3 ⊢ Ⅎ𝑦𝜑
71		nfcv 2931	. . 3 ⊢ Ⅎ𝑦𝐴
72		nfcv 2931	. . 3 ⊢ Ⅎ𝑥𝐵
73		nfcv 2931	. . . . 5 ⊢ Ⅎ𝑥𝐴
74	73, 32	nfcsbw 3887	. . . 4 ⊢ Ⅎ𝑥⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶
75	72, 74	nfcsbw 3887	. . 3 ⊢ Ⅎ𝑥⦋𝐵 / 𝑏⦌⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶
76	9, 14	nfcxfr 2929	. . 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 7561	. 2 ⊢ (𝜑 → (𝐴𝐹𝐵) = ⦋𝐵 / 𝑏⦌⦋𝐴 / 𝑎⦌⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
78	26, 49, 77	3eqtr4d 2814	1 ⊢ (𝜑 → ((curry 𝐹‘𝐴)‘𝐵) = (𝐴𝐹𝐵))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ⦋csb 3861   ↦ cmpt 5196  ‘cfv 6537  (class class class)co 7411   ∈ cmpo 7413  curry ccur 8260

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7985  df-2nd 7986  df-cur 8262

This theorem is referenced by:  pmatcollpw3lem  22908  logbfval  26920