Theorem mpocurryd 8085

Description: The currying of an operation given in maps-to notation, splitting the operation (function of two arguments) into a function of the first argument, producing a function over the second argument. (Contributed by AV, 27-Oct-2019.)

Hypotheses

Ref	Expression
mpocurryd.f	⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)
mpocurryd.c	⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ 𝑉)
mpocurryd.n	⊢ (𝜑 → 𝑌 ≠ ∅)

Assertion

Ref	Expression
mpocurryd	⊢ (𝜑 → curry 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐶)))

Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝑉,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝐶(𝑥,𝑦)

Proof of Theorem mpocurryd

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

Step	Hyp	Ref	Expression
1		df-cur 8083	. 2 ⊢ curry 𝐹 = (𝑥 ∈ dom dom 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧})
2		mpocurryd.c	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ 𝑉)
3		mpocurryd.f	. . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)
4	3	dmmpoga 7913	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ 𝑉 → dom 𝐹 = (𝑋 × 𝑌))
5	2, 4	syl 17	. . . . . 6 ⊢ (𝜑 → dom 𝐹 = (𝑋 × 𝑌))
6	5	dmeqd 5814	. . . . 5 ⊢ (𝜑 → dom dom 𝐹 = dom (𝑋 × 𝑌))
7		mpocurryd.n	. . . . . 6 ⊢ (𝜑 → 𝑌 ≠ ∅)
8		dmxp 5838	. . . . . 6 ⊢ (𝑌 ≠ ∅ → dom (𝑋 × 𝑌) = 𝑋)
9	7, 8	syl 17	. . . . 5 ⊢ (𝜑 → dom (𝑋 × 𝑌) = 𝑋)
10	6, 9	eqtrd 2778	. . . 4 ⊢ (𝜑 → dom dom 𝐹 = 𝑋)
11	10	mpteq1d 5169	. . 3 ⊢ (𝜑 → (𝑥 ∈ dom dom 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧}) = (𝑥 ∈ 𝑋 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧}))
12		df-mpt 5158	. . . . 5 ⊢ (𝑦 ∈ 𝑌 ↦ 𝐶) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝑌 ∧ 𝑧 = 𝐶)}
13	3	mpofun 7398	. . . . . . . 8 ⊢ Fun 𝐹
14		funbrfv2b 6827	. . . . . . . 8 ⊢ (Fun 𝐹 → (⟨𝑥, 𝑦⟩𝐹𝑧 ↔ (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
15	13, 14	mp1i 13	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (⟨𝑥, 𝑦⟩𝐹𝑧 ↔ (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
16	5	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → dom 𝐹 = (𝑋 × 𝑌))
17	16	eleq2d 2824	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌)))
18		opelxp 5625	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌) ↔ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌))
19	17, 18	bitrdi 287	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)))
20	19	anbi1d 630	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
21		an21 641	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ (𝑦 ∈ 𝑌 ∧ (𝑥 ∈ 𝑋 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
22		ibar 529	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 → ((𝐹‘⟨𝑥, 𝑦⟩) = 𝑧 ↔ (𝑥 ∈ 𝑋 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
23	22	bicomd 222	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑋 → ((𝑥 ∈ 𝑋 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧))
24	23	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧))
25	24	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → ((𝑥 ∈ 𝑋 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧))
26		df-ov 7278	. . . . . . . . . . . . 13 ⊢ (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
27		nfcv 2907	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑎𝐶
28		nfcv 2907	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑏𝐶
29		nfcv 2907	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥𝑏
30		nfcsb1v 3857	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐶
31	29, 30	nfcsbw 3859	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶
32		nfcsb1v 3857	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶
33		csbeq1a 3846	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → 𝐶 = ⦋𝑎 / 𝑥⦌𝐶)
34		csbeq1a 3846	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑏 → ⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
35	33, 34	sylan9eq 2798	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → 𝐶 = ⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
36	27, 28, 31, 32, 35	cbvmpo 7369	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) = (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑌 ↦ ⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
37	3, 36	eqtri 2766	. . . . . . . . . . . . . . 15 ⊢ 𝐹 = (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑌 ↦ ⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
38	37	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝐹 = (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑌 ↦ ⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶))
39	33	eqcomd 2744	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → ⦋𝑎 / 𝑥⦌𝐶 = 𝐶)
40	39	equcoms 2023	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑥 → ⦋𝑎 / 𝑥⦌𝐶 = 𝐶)
41	40	csbeq2dv 3839	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑥 → ⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑏 / 𝑦⦌𝐶)
42		csbeq1a 3846	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑏 → 𝐶 = ⦋𝑏 / 𝑦⦌𝐶)
43	42	eqcomd 2744	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑏 → ⦋𝑏 / 𝑦⦌𝐶 = 𝐶)
44	43	equcoms 2023	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑦 → ⦋𝑏 / 𝑦⦌𝐶 = 𝐶)
45	41, 44	sylan9eq 2798	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → ⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 = 𝐶)
46	45	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) ∧ (𝑎 = 𝑥 ∧ 𝑏 = 𝑦)) → ⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 = 𝐶)
47		simpr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
48	47	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝑥 ∈ 𝑋)
49		simpr 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑌)
50		rsp2 3138	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ 𝑉 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐶 ∈ 𝑉))
51	2, 50	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐶 ∈ 𝑉))
52	51	impl 456	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝐶 ∈ 𝑉)
53	38, 46, 48, 49, 52	ovmpod 7425	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (𝑥𝐹𝑦) = 𝐶)
54	26, 53	eqtr3id 2792	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (𝐹‘⟨𝑥, 𝑦⟩) = 𝐶)
55	54	eqeq1d 2740	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → ((𝐹‘⟨𝑥, 𝑦⟩) = 𝑧 ↔ 𝐶 = 𝑧))
56		eqcom 2745	. . . . . . . . . . 11 ⊢ (𝐶 = 𝑧 ↔ 𝑧 = 𝐶)
57	55, 56	bitrdi 287	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → ((𝐹‘⟨𝑥, 𝑦⟩) = 𝑧 ↔ 𝑧 = 𝐶))
58	25, 57	bitrd 278	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → ((𝑥 ∈ 𝑋 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ 𝑧 = 𝐶))
59	58	pm5.32da 579	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑌 ∧ (𝑥 ∈ 𝑋 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)) ↔ (𝑦 ∈ 𝑌 ∧ 𝑧 = 𝐶)))
60	21, 59	bitrid 282	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ (𝑦 ∈ 𝑌 ∧ 𝑧 = 𝐶)))
61	15, 20, 60	3bitrrd 306	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑌 ∧ 𝑧 = 𝐶) ↔ ⟨𝑥, 𝑦⟩𝐹𝑧))
62	61	opabbidv 5140	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝑌 ∧ 𝑧 = 𝐶)} = {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧})
63	12, 62	eqtr2id 2791	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧} = (𝑦 ∈ 𝑌 ↦ 𝐶))
64	63	mpteq2dva 5174	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧}) = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐶)))
65	11, 64	eqtrd 2778	. 2 ⊢ (𝜑 → (𝑥 ∈ dom dom 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧}) = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐶)))
66	1, 65	eqtrid 2790	1 ⊢ (𝜑 → curry 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐶)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ⦋csb 3832  ∅c0 4256  ⟨cop 4567   class class class wbr 5074  {copab 5136   ↦ cmpt 5157   × cxp 5587  dom cdm 5589  Fun wfun 6427  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  curry ccur 8081

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-cur 8083

This theorem is referenced by:  mpocurryvald  8086  curfv  35757