Theorem curry1 7915

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 6517	. . . . 5 ⊢ (𝐹 Fn (𝐴 × 𝐵) → Fun 𝐹)
2		2ndconst 7912	. . . . . 6 ⊢ (𝐶 ∈ 𝐴 → (2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V)
3		dff1o3 6706	. . . . . . 7 ⊢ ((2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V ↔ ((2nd ↾ ({𝐶} × V)):({𝐶} × V)–onto→V ∧ Fun ◡(2nd ↾ ({𝐶} × V))))
4	3	simprbi 496	. . . . . 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 6458	. . . . 5 ⊢ ((Fun 𝐹 ∧ Fun ◡(2nd ↾ ({𝐶} × V))) → Fun (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))))
7	1, 5, 6	syl2an 595	. . . 4 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → Fun (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))))
8		dmco 6147	. . . . 5 ⊢ dom (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) = (◡◡(2nd ↾ ({𝐶} × V)) “ dom 𝐹)
9		fndm 6520	. . . . . . . 8 ⊢ (𝐹 Fn (𝐴 × 𝐵) → dom 𝐹 = (𝐴 × 𝐵))
10	9	adantr 480	. . . . . . 7 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → dom 𝐹 = (𝐴 × 𝐵))
11	10	imaeq2d 5958	. . . . . 6 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (◡◡(2nd ↾ ({𝐶} × V)) “ dom 𝐹) = (◡◡(2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)))
12		imacnvcnv 6098	. . . . . . . . 9 ⊢ (◡◡(2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = ((2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵))
13		df-ima 5593	. . . . . . . . 9 ⊢ ((2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = ran ((2nd ↾ ({𝐶} × V)) ↾ (𝐴 × 𝐵))
14		resres 5893	. . . . . . . . . 10 ⊢ ((2nd ↾ ({𝐶} × V)) ↾ (𝐴 × 𝐵)) = (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵)))
15	14	rneqi 5835	. . . . . . . . 9 ⊢ ran ((2nd ↾ ({𝐶} × V)) ↾ (𝐴 × 𝐵)) = ran (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵)))
16	12, 13, 15	3eqtri 2770	. . . . . . . 8 ⊢ (◡◡(2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = ran (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵)))
17		inxp 5730	. . . . . . . . . . . . 13 ⊢ (({𝐶} × V) ∩ (𝐴 × 𝐵)) = (({𝐶} ∩ 𝐴) × (V ∩ 𝐵))
18		incom 4131	. . . . . . . . . . . . . . 15 ⊢ (V ∩ 𝐵) = (𝐵 ∩ V)
19		inv1 4325	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∩ V) = 𝐵
20	18, 19	eqtri 2766	. . . . . . . . . . . . . 14 ⊢ (V ∩ 𝐵) = 𝐵
21	20	xpeq2i 5607	. . . . . . . . . . . . 13 ⊢ (({𝐶} ∩ 𝐴) × (V ∩ 𝐵)) = (({𝐶} ∩ 𝐴) × 𝐵)
22	17, 21	eqtri 2766	. . . . . . . . . . . 12 ⊢ (({𝐶} × V) ∩ (𝐴 × 𝐵)) = (({𝐶} ∩ 𝐴) × 𝐵)
23		snssi 4738	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ 𝐴 → {𝐶} ⊆ 𝐴)
24		df-ss 3900	. . . . . . . . . . . . . 14 ⊢ ({𝐶} ⊆ 𝐴 ↔ ({𝐶} ∩ 𝐴) = {𝐶})
25	23, 24	sylib 217	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ 𝐴 → ({𝐶} ∩ 𝐴) = {𝐶})
26	25	xpeq1d 5609	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ 𝐴 → (({𝐶} ∩ 𝐴) × 𝐵) = ({𝐶} × 𝐵))
27	22, 26	eqtrid 2790	. . . . . . . . . . 11 ⊢ (𝐶 ∈ 𝐴 → (({𝐶} × V) ∩ (𝐴 × 𝐵)) = ({𝐶} × 𝐵))
28	27	reseq2d 5880	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝐴 → (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵))) = (2nd ↾ ({𝐶} × 𝐵)))
29	28	rneqd 5836	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝐴 → ran (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵))) = ran (2nd ↾ ({𝐶} × 𝐵)))
30		2ndconst 7912	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝐴 → (2nd ↾ ({𝐶} × 𝐵)):({𝐶} × 𝐵)–1-1-onto→𝐵)
31		f1ofo 6707	. . . . . . . . . 10 ⊢ ((2nd ↾ ({𝐶} × 𝐵)):({𝐶} × 𝐵)–1-1-onto→𝐵 → (2nd ↾ ({𝐶} × 𝐵)):({𝐶} × 𝐵)–onto→𝐵)
32		forn 6675	. . . . . . . . . 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 2778	. . . . . . . 8 ⊢ (𝐶 ∈ 𝐴 → ran (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵))) = 𝐵)
35	16, 34	eqtrid 2790	. . . . . . 7 ⊢ (𝐶 ∈ 𝐴 → (◡◡(2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = 𝐵)
36	35	adantl 481	. . . . . 6 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (◡◡(2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = 𝐵)
37	11, 36	eqtrd 2778	. . . . 5 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (◡◡(2nd ↾ ({𝐶} × V)) “ dom 𝐹) = 𝐵)
38	8, 37	eqtrid 2790	. . . 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 6514	. . . . 5 ⊢ (𝐺 Fn 𝐵 ↔ (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) Fn 𝐵)
41		df-fn 6421	. . . . 5 ⊢ ((𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) Fn 𝐵 ↔ (Fun (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) ∧ dom (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) = 𝐵))
42	40, 41	bitri 274	. . . 4 ⊢ (𝐺 Fn 𝐵 ↔ (Fun (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) ∧ dom (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) = 𝐵))
43	7, 38, 42	sylanbrc 582	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → 𝐺 Fn 𝐵)
44		dffn5 6810	. . 3 ⊢ (𝐺 Fn 𝐵 ↔ 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐺‘𝑥)))
45	43, 44	sylib 217	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐺‘𝑥)))
46	39	fveq1i 6757	. . . . 5 ⊢ (𝐺‘𝑥) = ((𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))‘𝑥)
47		dff1o4 6708	. . . . . . . 8 ⊢ ((2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V ↔ ((2nd ↾ ({𝐶} × V)) Fn ({𝐶} × V) ∧ ◡(2nd ↾ ({𝐶} × V)) Fn V))
48	2, 47	sylib 217	. . . . . . 7 ⊢ (𝐶 ∈ 𝐴 → ((2nd ↾ ({𝐶} × V)) Fn ({𝐶} × V) ∧ ◡(2nd ↾ ({𝐶} × V)) Fn V))
49		fvco2 6847	. . . . . . . 8 ⊢ ((◡(2nd ↾ ({𝐶} × V)) Fn V ∧ 𝑥 ∈ V) → ((𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))‘𝑥) = (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥)))
50	49	elvd 3429	. . . . . . 7 ⊢ (◡(2nd ↾ ({𝐶} × V)) Fn V → ((𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))‘𝑥) = (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥)))
51	48, 50	simpl2im 503	. . . . . 6 ⊢ (𝐶 ∈ 𝐴 → ((𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))‘𝑥) = (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥)))
52	51	ad2antlr 723	. . . . 5 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → ((𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))‘𝑥) = (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥)))
53	46, 52	eqtrid 2790	. . . 4 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥)))
54	2	adantr 480	. . . . . . . . 9 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → (2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V)
55		snidg 4592	. . . . . . . . . . 11 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ {𝐶})
56		vex 3426	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
57		opelxp 5616	. . . . . . . . . . 11 ⊢ (⟨𝐶, 𝑥⟩ ∈ ({𝐶} × V) ↔ (𝐶 ∈ {𝐶} ∧ 𝑥 ∈ V))
58	55, 56, 57	sylanblrc 589	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝐴 → ⟨𝐶, 𝑥⟩ ∈ ({𝐶} × V))
59	58	adantr 480	. . . . . . . . 9 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ⟨𝐶, 𝑥⟩ ∈ ({𝐶} × V))
60	54, 59	jca 511	. . . . . . . 8 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ((2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V ∧ ⟨𝐶, 𝑥⟩ ∈ ({𝐶} × V)))
61	58	fvresd 6776	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝐴 → ((2nd ↾ ({𝐶} × V))‘⟨𝐶, 𝑥⟩) = (2nd ‘⟨𝐶, 𝑥⟩))
62		op2ndg 7817	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑥 ∈ V) → (2nd ‘⟨𝐶, 𝑥⟩) = 𝑥)
63	62	elvd 3429	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝐴 → (2nd ‘⟨𝐶, 𝑥⟩) = 𝑥)
64	61, 63	eqtrd 2778	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝐴 → ((2nd ↾ ({𝐶} × V))‘⟨𝐶, 𝑥⟩) = 𝑥)
65	64	adantr 480	. . . . . . . 8 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ((2nd ↾ ({𝐶} × V))‘⟨𝐶, 𝑥⟩) = 𝑥)
66		f1ocnvfv 7131	. . . . . . . 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⟩))
67	60, 65, 66	sylc 65	. . . . . . 7 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → (◡(2nd ↾ ({𝐶} × V))‘𝑥) = ⟨𝐶, 𝑥⟩)
68	67	fveq2d 6760	. . . . . 6 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥)) = (𝐹‘⟨𝐶, 𝑥⟩))
69	68	adantll 710	. . . . 5 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥)) = (𝐹‘⟨𝐶, 𝑥⟩))
70		df-ov 7258	. . . . 5 ⊢ (𝐶𝐹𝑥) = (𝐹‘⟨𝐶, 𝑥⟩)
71	69, 70	eqtr4di 2797	. . . 4 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥)) = (𝐶𝐹𝑥))
72	53, 71	eqtrd 2778	. . 3 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐶𝐹𝑥))
73	72	mpteq2dva 5170	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ 𝐵 ↦ (𝐺‘𝑥)) = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥)))
74	45, 73	eqtrd 2778	1 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ∩ cin 3882   ⊆ wss 3883  {csn 4558  ⟨cop 4564   ↦ cmpt 5153   × cxp 5578  ^◡ccnv 5579  dom cdm 5580  ran crn 5581   ↾ cres 5582   “ cima 5583   ∘ ccom 5584  Fun wfun 6412   Fn wfn 6413  –onto→wfo 6416  –1-1-onto→wf1o 6417  ‘cfv 6418  (class class class)co 7255  2^nd c2nd 7803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-1st 7804  df-2nd 7805

This theorem is referenced by:  curry1val  7916  curry1f  7917