Theorem mpt2curryd 7395

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
mpt2curryd.f	⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)
mpt2curryd.c	⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ 𝑉)
mpt2curryd.n	⊢ (𝜑 → 𝑌 ≠ ∅)

Assertion

Ref	Expression
mpt2curryd	⊢ (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 mpt2curryd

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

Step	Hyp	Ref	Expression
1		df-cur 7393	. 2 ⊢ curry 𝐹 = (𝑥 ∈ dom dom 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧})
2		mpt2curryd.c	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ 𝑉)
3		mpt2curryd.f	. . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶)
4	3	dmmpt2ga 7242	. . . . . . 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 5326	. . . . 5 ⊢ (𝜑 → dom dom 𝐹 = dom (𝑋 × 𝑌))
7		mpt2curryd.n	. . . . . 6 ⊢ (𝜑 → 𝑌 ≠ ∅)
8		dmxp 5344	. . . . . 6 ⊢ (𝑌 ≠ ∅ → dom (𝑋 × 𝑌) = 𝑋)
9	7, 8	syl 17	. . . . 5 ⊢ (𝜑 → dom (𝑋 × 𝑌) = 𝑋)
10	6, 9	eqtrd 2656	. . . 4 ⊢ (𝜑 → dom dom 𝐹 = 𝑋)
11	10	mpteq1d 4738	. . 3 ⊢ (𝜑 → (𝑥 ∈ dom dom 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧}) = (𝑥 ∈ 𝑋 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧}))
12		df-mpt 4730	. . . . 5 ⊢ (𝑦 ∈ 𝑌 ↦ 𝐶) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝑌 ∧ 𝑧 = 𝐶)}
13	3	mpt2fun 6762	. . . . . . . 8 ⊢ Fun 𝐹
14		funbrfv2b 6240	. . . . . . . 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 2687	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌)))
18		opelxp 5146	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌) ↔ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌))
19	17, 18	syl6bb 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)))
20	19	anbi1d 741	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
21		an32 839	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ ((𝑥 ∈ 𝑋 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ∧ 𝑦 ∈ 𝑌))
22		ancom 466	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝑋 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ∧ 𝑦 ∈ 𝑌) ↔ (𝑦 ∈ 𝑌 ∧ (𝑥 ∈ 𝑋 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
23	21, 22	bitri 264	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ (𝑦 ∈ 𝑌 ∧ (𝑥 ∈ 𝑋 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
24		ibar 525	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 → ((𝐹‘⟨𝑥, 𝑦⟩) = 𝑧 ↔ (𝑥 ∈ 𝑋 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
25	24	bicomd 213	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑋 → ((𝑥 ∈ 𝑋 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧))
26	25	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧))
27	26	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → ((𝑥 ∈ 𝑋 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧))
28		df-ov 6653	. . . . . . . . . . . . 13 ⊢ (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
29		nfcv 2764	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑎𝐶
30		nfcv 2764	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑏𝐶
31		nfcv 2764	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥𝑏
32		nfcsb1v 3549	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐶
33	31, 32	nfcsb 3551	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶
34		nfcsb1v 3549	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶
35		csbeq1a 3542	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → 𝐶 = ⦋𝑎 / 𝑥⦌𝐶)
36		csbeq1a 3542	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑏 → ⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
37	35, 36	sylan9eq 2676	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → 𝐶 = ⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
38	29, 30, 33, 34, 37	cbvmpt2 6734	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) = (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑌 ↦ ⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
39	3, 38	eqtri 2644	. . . . . . . . . . . . . . 15 ⊢ 𝐹 = (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑌 ↦ ⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶)
40	39	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝐹 = (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑌 ↦ ⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶))
41	35	eqcomd 2628	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → ⦋𝑎 / 𝑥⦌𝐶 = 𝐶)
42	41	equcoms 1947	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑥 → ⦋𝑎 / 𝑥⦌𝐶 = 𝐶)
43	42	csbeq2dv 3992	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑥 → ⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑏 / 𝑦⦌𝐶)
44		csbeq1a 3542	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑏 → 𝐶 = ⦋𝑏 / 𝑦⦌𝐶)
45	44	eqcomd 2628	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑏 → ⦋𝑏 / 𝑦⦌𝐶 = 𝐶)
46	45	equcoms 1947	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑦 → ⦋𝑏 / 𝑦⦌𝐶 = 𝐶)
47	43, 46	sylan9eq 2676	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → ⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 = 𝐶)
48	47	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) ∧ (𝑎 = 𝑥 ∧ 𝑏 = 𝑦)) → ⦋𝑏 / 𝑦⦌⦋𝑎 / 𝑥⦌𝐶 = 𝐶)
49		simpr 477	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
50	49	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝑥 ∈ 𝑋)
51		simpr 477	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑌)
52		rsp2 2936	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ 𝑉 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐶 ∈ 𝑉))
53	2, 52	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐶 ∈ 𝑉))
54	53	impl 650	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝐶 ∈ 𝑉)
55	40, 48, 50, 51, 54	ovmpt2d 6788	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (𝑥𝐹𝑦) = 𝐶)
56	28, 55	syl5eqr 2670	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (𝐹‘⟨𝑥, 𝑦⟩) = 𝐶)
57	56	eqeq1d 2624	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → ((𝐹‘⟨𝑥, 𝑦⟩) = 𝑧 ↔ 𝐶 = 𝑧))
58		eqcom 2629	. . . . . . . . . . 11 ⊢ (𝐶 = 𝑧 ↔ 𝑧 = 𝐶)
59	57, 58	syl6bb 276	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → ((𝐹‘⟨𝑥, 𝑦⟩) = 𝑧 ↔ 𝑧 = 𝐶))
60	27, 59	bitrd 268	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → ((𝑥 ∈ 𝑋 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ 𝑧 = 𝐶))
61	60	pm5.32da 673	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑌 ∧ (𝑥 ∈ 𝑋 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)) ↔ (𝑦 ∈ 𝑌 ∧ 𝑧 = 𝐶)))
62	23, 61	syl5bb 272	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ (𝑦 ∈ 𝑌 ∧ 𝑧 = 𝐶)))
63	15, 20, 62	3bitrrd 295	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑌 ∧ 𝑧 = 𝐶) ↔ ⟨𝑥, 𝑦⟩𝐹𝑧))
64	63	opabbidv 4716	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝑌 ∧ 𝑧 = 𝐶)} = {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧})
65	12, 64	syl5req 2669	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧} = (𝑦 ∈ 𝑌 ↦ 𝐶))
66	65	mpteq2dva 4744	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧}) = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐶)))
67	11, 66	eqtrd 2656	. 2 ⊢ (𝜑 → (𝑥 ∈ dom dom 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧}) = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐶)))
68	1, 67	syl5eq 2668	1 ⊢ (𝜑 → curry 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐶)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ⦋csb 3533  ∅c0 3915  ⟨cop 4183   class class class wbr 4653  {copab 4712   ↦ cmpt 4729   × cxp 5112  dom cdm 5114  Fun wfun 5882  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  curry ccur 7391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-cur 7393

This theorem is referenced by:  mpt2curryvald  7396  curfv  33389