Theorem curry1 7503

Description: Composition with ^◡(2^nd ↾ ({𝐶} × V)) turns any binary operation 𝐹 with a constant first operand into a function 𝐺 of the second operand only. This transformation is called "currying." (Contributed by NM, 28-Mar-2008.) (Revised by Mario Carneiro, 26-Dec-2014.)

Hypothesis

Ref	Expression
curry1.1	⊢ 𝐺 = (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))

Assertion

Ref	Expression
curry1	⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐹   𝑥,𝐺

Proof of Theorem curry1

Step	Hyp	Ref	Expression
1		fnfun 6199	. . . . 5 ⊢ (𝐹 Fn (𝐴 × 𝐵) → Fun 𝐹)
2		2ndconst 7500	. . . . . 6 ⊢ (𝐶 ∈ 𝐴 → (2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V)
3		dff1o3 6359	. . . . . . 7 ⊢ ((2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V ↔ ((2nd ↾ ({𝐶} × V)):({𝐶} × V)–onto→V ∧ Fun ◡(2nd ↾ ({𝐶} × V))))
4	3	simprbi 486	. . . . . 6 ⊢ ((2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V → Fun ◡(2nd ↾ ({𝐶} × V)))
5	2, 4	syl 17	. . . . 5 ⊢ (𝐶 ∈ 𝐴 → Fun ◡(2nd ↾ ({𝐶} × V)))
6		funco 6141	. . . . 5 ⊢ ((Fun 𝐹 ∧ Fun ◡(2nd ↾ ({𝐶} × V))) → Fun (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))))
7	1, 5, 6	syl2an 585	. . . 4 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → Fun (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))))
8		dmco 5857	. . . . 5 ⊢ dom (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) = (◡◡(2nd ↾ ({𝐶} × V)) “ dom 𝐹)
9		fndm 6201	. . . . . . . 8 ⊢ (𝐹 Fn (𝐴 × 𝐵) → dom 𝐹 = (𝐴 × 𝐵))
10	9	adantr 468	. . . . . . 7 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → dom 𝐹 = (𝐴 × 𝐵))
11	10	imaeq2d 5676	. . . . . 6 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (◡◡(2nd ↾ ({𝐶} × V)) “ dom 𝐹) = (◡◡(2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)))
12		imacnvcnv 5810	. . . . . . . . 9 ⊢ (◡◡(2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = ((2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵))
13		df-ima 5324	. . . . . . . . 9 ⊢ ((2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = ran ((2nd ↾ ({𝐶} × V)) ↾ (𝐴 × 𝐵))
14		resres 5613	. . . . . . . . . 10 ⊢ ((2nd ↾ ({𝐶} × V)) ↾ (𝐴 × 𝐵)) = (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵)))
15	14	rneqi 5553	. . . . . . . . 9 ⊢ ran ((2nd ↾ ({𝐶} × V)) ↾ (𝐴 × 𝐵)) = ran (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵)))
16	12, 13, 15	3eqtri 2832	. . . . . . . 8 ⊢ (◡◡(2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = ran (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵)))
17		inxp 5456	. . . . . . . . . . . . 13 ⊢ (({𝐶} × V) ∩ (𝐴 × 𝐵)) = (({𝐶} ∩ 𝐴) × (V ∩ 𝐵))
18		incom 4004	. . . . . . . . . . . . . . 15 ⊢ (V ∩ 𝐵) = (𝐵 ∩ V)
19		inv1 4168	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∩ V) = 𝐵
20	18, 19	eqtri 2828	. . . . . . . . . . . . . 14 ⊢ (V ∩ 𝐵) = 𝐵
21	20	xpeq2i 5337	. . . . . . . . . . . . 13 ⊢ (({𝐶} ∩ 𝐴) × (V ∩ 𝐵)) = (({𝐶} ∩ 𝐴) × 𝐵)
22	17, 21	eqtri 2828	. . . . . . . . . . . 12 ⊢ (({𝐶} × V) ∩ (𝐴 × 𝐵)) = (({𝐶} ∩ 𝐴) × 𝐵)
23		snssi 4529	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ 𝐴 → {𝐶} ⊆ 𝐴)
24		df-ss 3783	. . . . . . . . . . . . . 14 ⊢ ({𝐶} ⊆ 𝐴 ↔ ({𝐶} ∩ 𝐴) = {𝐶})
25	23, 24	sylib 209	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ 𝐴 → ({𝐶} ∩ 𝐴) = {𝐶})
26	25	xpeq1d 5339	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ 𝐴 → (({𝐶} ∩ 𝐴) × 𝐵) = ({𝐶} × 𝐵))
27	22, 26	syl5eq 2852	. . . . . . . . . . 11 ⊢ (𝐶 ∈ 𝐴 → (({𝐶} × V) ∩ (𝐴 × 𝐵)) = ({𝐶} × 𝐵))
28	27	reseq2d 5597	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝐴 → (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵))) = (2nd ↾ ({𝐶} × 𝐵)))
29	28	rneqd 5554	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝐴 → ran (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵))) = ran (2nd ↾ ({𝐶} × 𝐵)))
30		2ndconst 7500	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝐴 → (2nd ↾ ({𝐶} × 𝐵)):({𝐶} × 𝐵)–1-1-onto→𝐵)
31		f1ofo 6360	. . . . . . . . . 10 ⊢ ((2nd ↾ ({𝐶} × 𝐵)):({𝐶} × 𝐵)–1-1-onto→𝐵 → (2nd ↾ ({𝐶} × 𝐵)):({𝐶} × 𝐵)–onto→𝐵)
32		forn 6334	. . . . . . . . . 10 ⊢ ((2nd ↾ ({𝐶} × 𝐵)):({𝐶} × 𝐵)–onto→𝐵 → ran (2nd ↾ ({𝐶} × 𝐵)) = 𝐵)
33	30, 31, 32	3syl 18	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝐴 → ran (2nd ↾ ({𝐶} × 𝐵)) = 𝐵)
34	29, 33	eqtrd 2840	. . . . . . . 8 ⊢ (𝐶 ∈ 𝐴 → ran (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵))) = 𝐵)
35	16, 34	syl5eq 2852	. . . . . . 7 ⊢ (𝐶 ∈ 𝐴 → (◡◡(2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = 𝐵)
36	35	adantl 469	. . . . . 6 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (◡◡(2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = 𝐵)
37	11, 36	eqtrd 2840	. . . . 5 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (◡◡(2nd ↾ ({𝐶} × V)) “ dom 𝐹) = 𝐵)
38	8, 37	syl5eq 2852	. . . 4 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → dom (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) = 𝐵)
39		curry1.1	. . . . . 6 ⊢ 𝐺 = (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))
40	39	fneq1i 6196	. . . . 5 ⊢ (𝐺 Fn 𝐵 ↔ (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) Fn 𝐵)
41		df-fn 6104	. . . . 5 ⊢ ((𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) Fn 𝐵 ↔ (Fun (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) ∧ dom (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) = 𝐵))
42	40, 41	bitri 266	. . . 4 ⊢ (𝐺 Fn 𝐵 ↔ (Fun (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) ∧ dom (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) = 𝐵))
43	7, 38, 42	sylanbrc 574	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → 𝐺 Fn 𝐵)
44		dffn5 6462	. . 3 ⊢ (𝐺 Fn 𝐵 ↔ 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐺‘𝑥)))
45	43, 44	sylib 209	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐺‘𝑥)))
46	39	fveq1i 6409	. . . . 5 ⊢ (𝐺‘𝑥) = ((𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))‘𝑥)
47		dff1o4 6361	. . . . . . . . 9 ⊢ ((2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V ↔ ((2nd ↾ ({𝐶} × V)) Fn ({𝐶} × V) ∧ ◡(2nd ↾ ({𝐶} × V)) Fn V))
48	2, 47	sylib 209	. . . . . . . 8 ⊢ (𝐶 ∈ 𝐴 → ((2nd ↾ ({𝐶} × V)) Fn ({𝐶} × V) ∧ ◡(2nd ↾ ({𝐶} × V)) Fn V))
49	48	simprd 485	. . . . . . 7 ⊢ (𝐶 ∈ 𝐴 → ◡(2nd ↾ ({𝐶} × V)) Fn V)
50		fvco2 6494	. . . . . . . 8 ⊢ ((◡(2nd ↾ ({𝐶} × V)) Fn V ∧ 𝑥 ∈ V) → ((𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))‘𝑥) = (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥)))
51	50	elvd 3396	. . . . . . 7 ⊢ (◡(2nd ↾ ({𝐶} × V)) Fn V → ((𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))‘𝑥) = (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥)))
52	49, 51	syl 17	. . . . . 6 ⊢ (𝐶 ∈ 𝐴 → ((𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))‘𝑥) = (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥)))
53	52	ad2antlr 709	. . . . 5 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → ((𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))‘𝑥) = (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥)))
54	46, 53	syl5eq 2852	. . . 4 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥)))
55	2	adantr 468	. . . . . . . . 9 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → (2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V)
56		snidg 4400	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ {𝐶})
57		vex 3394	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
58	56, 57	jctir 512	. . . . . . . . . . 11 ⊢ (𝐶 ∈ 𝐴 → (𝐶 ∈ {𝐶} ∧ 𝑥 ∈ V))
59		opelxp 5346	. . . . . . . . . . 11 ⊢ (⟨𝐶, 𝑥⟩ ∈ ({𝐶} × V) ↔ (𝐶 ∈ {𝐶} ∧ 𝑥 ∈ V))
60	58, 59	sylibr 225	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝐴 → ⟨𝐶, 𝑥⟩ ∈ ({𝐶} × V))
61	60	adantr 468	. . . . . . . . 9 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ⟨𝐶, 𝑥⟩ ∈ ({𝐶} × V))
62	55, 61	jca 503	. . . . . . . 8 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ((2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V ∧ ⟨𝐶, 𝑥⟩ ∈ ({𝐶} × V)))
63		fvres 6427	. . . . . . . . . . 11 ⊢ (⟨𝐶, 𝑥⟩ ∈ ({𝐶} × V) → ((2nd ↾ ({𝐶} × V))‘⟨𝐶, 𝑥⟩) = (2nd ‘⟨𝐶, 𝑥⟩))
64	60, 63	syl 17	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝐴 → ((2nd ↾ ({𝐶} × V))‘⟨𝐶, 𝑥⟩) = (2nd ‘⟨𝐶, 𝑥⟩))
65		op2ndg 7411	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑥 ∈ V) → (2nd ‘⟨𝐶, 𝑥⟩) = 𝑥)
66	65	elvd 3396	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝐴 → (2nd ‘⟨𝐶, 𝑥⟩) = 𝑥)
67	64, 66	eqtrd 2840	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝐴 → ((2nd ↾ ({𝐶} × V))‘⟨𝐶, 𝑥⟩) = 𝑥)
68	67	adantr 468	. . . . . . . 8 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ((2nd ↾ ({𝐶} × V))‘⟨𝐶, 𝑥⟩) = 𝑥)
69		f1ocnvfv 6758	. . . . . . . 8 ⊢ (((2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V ∧ ⟨𝐶, 𝑥⟩ ∈ ({𝐶} × V)) → (((2nd ↾ ({𝐶} × V))‘⟨𝐶, 𝑥⟩) = 𝑥 → (◡(2nd ↾ ({𝐶} × V))‘𝑥) = ⟨𝐶, 𝑥⟩))
70	62, 68, 69	sylc 65	. . . . . . 7 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → (◡(2nd ↾ ({𝐶} × V))‘𝑥) = ⟨𝐶, 𝑥⟩)
71	70	fveq2d 6412	. . . . . 6 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥)) = (𝐹‘⟨𝐶, 𝑥⟩))
72	71	adantll 696	. . . . 5 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥)) = (𝐹‘⟨𝐶, 𝑥⟩))
73		df-ov 6877	. . . . 5 ⊢ (𝐶𝐹𝑥) = (𝐹‘⟨𝐶, 𝑥⟩)
74	72, 73	syl6eqr 2858	. . . 4 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥)) = (𝐶𝐹𝑥))
75	54, 74	eqtrd 2840	. . 3 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐶𝐹𝑥))
76	75	mpteq2dva 4938	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ 𝐵 ↦ (𝐺‘𝑥)) = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥)))
77	45, 76	eqtrd 2840	1 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥)))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1637   ∈ wcel 2156  Vcvv 3391   ∩ cin 3768   ⊆ wss 3769  {csn 4370  ⟨cop 4376   ↦ cmpt 4923   × cxp 5309  ^◡ccnv 5310  dom cdm 5311  ran crn 5312   ↾ cres 5313   “ cima 5314   ∘ ccom 5315  Fun wfun 6095   Fn wfn 6096  –onto→wfo 6099  –1-1-onto→wf1o 6100  ‘cfv 6101  (class class class)co 6874  2^nd c2nd 7397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6877  df-1st 7398  df-2nd 7399

This theorem is referenced by:  curry1val  7504  curry1f  7505