Theorem curry1 8087

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 6625	. . . . 5 ⊢ (𝐹 Fn (𝐴 × 𝐵) → Fun 𝐹)
2		2ndconst 8084	. . . . . 6 ⊢ (𝐶 ∈ 𝐴 → (2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V)
3		dff1o3 6817	. . . . . . 7 ⊢ ((2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V ↔ ((2nd ↾ ({𝐶} × V)):({𝐶} × V)–onto→V ∧ Fun ◡(2nd ↾ ({𝐶} × V))))
4	3	simprbi 502	. . . . . 6 ⊢ ((2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V → Fun ◡(2nd ↾ ({𝐶} × V)))
5	2, 4	syl 18	. . . . 5 ⊢ (𝐶 ∈ 𝐴 → Fun ◡(2nd ↾ ({𝐶} × V)))
6		funco 6565	. . . . 5 ⊢ ((Fun 𝐹 ∧ Fun ◡(2nd ↾ ({𝐶} × V))) → Fun (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))))
7	1, 5, 6	syl2an 607	. . . 4 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → Fun (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))))
8		dmco 6245	. . . . 5 ⊢ dom (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) = (◡◡(2nd ↾ ({𝐶} × V)) “ dom 𝐹)
9		fndm 6628	. . . . . . . 8 ⊢ (𝐹 Fn (𝐴 × 𝐵) → dom 𝐹 = (𝐴 × 𝐵))
10	9	adantr 485	. . . . . . 7 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → dom 𝐹 = (𝐴 × 𝐵))
11	10	imaeq2d 6052	. . . . . 6 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (◡◡(2nd ↾ ({𝐶} × V)) “ dom 𝐹) = (◡◡(2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)))
12		imacnvcnv 6196	. . . . . . . . 9 ⊢ (◡◡(2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = ((2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵))
13		df-ima 5664	. . . . . . . . 9 ⊢ ((2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = ran ((2nd ↾ ({𝐶} × V)) ↾ (𝐴 × 𝐵))
14		resres 5981	. . . . . . . . . 10 ⊢ ((2nd ↾ ({𝐶} × V)) ↾ (𝐴 × 𝐵)) = (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵)))
15	14	rneqi 5917	. . . . . . . . 9 ⊢ ran ((2nd ↾ ({𝐶} × V)) ↾ (𝐴 × 𝐵)) = ran (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵)))
16	12, 13, 15	3eqtri 2792	. . . . . . . 8 ⊢ (◡◡(2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = ran (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵)))
17		inxp 5808	. . . . . . . . . . . . 13 ⊢ (({𝐶} × V) ∩ (𝐴 × 𝐵)) = (({𝐶} ∩ 𝐴) × (V ∩ 𝐵))
18		incom 4164	. . . . . . . . . . . . . . 15 ⊢ (V ∩ 𝐵) = (𝐵 ∩ V)
19		inv1 4355	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∩ V) = 𝐵
20	18, 19	eqtri 2788	. . . . . . . . . . . . . 14 ⊢ (V ∩ 𝐵) = 𝐵
21	20	xpeq2i 5678	. . . . . . . . . . . . 13 ⊢ (({𝐶} ∩ 𝐴) × (V ∩ 𝐵)) = (({𝐶} ∩ 𝐴) × 𝐵)
22	17, 21	eqtri 2788	. . . . . . . . . . . 12 ⊢ (({𝐶} × V) ∩ (𝐴 × 𝐵)) = (({𝐶} ∩ 𝐴) × 𝐵)
23		snssi 4747	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ 𝐴 → {𝐶} ⊆ 𝐴)
24		dfss2 3925	. . . . . . . . . . . . . 14 ⊢ ({𝐶} ⊆ 𝐴 ↔ ({𝐶} ∩ 𝐴) = {𝐶})
25	23, 24	sylib 221	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ 𝐴 → ({𝐶} ∩ 𝐴) = {𝐶})
26	25	xpeq1d 5680	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ 𝐴 → (({𝐶} ∩ 𝐴) × 𝐵) = ({𝐶} × 𝐵))
27	22, 26	eqtrid 2812	. . . . . . . . . . 11 ⊢ (𝐶 ∈ 𝐴 → (({𝐶} × V) ∩ (𝐴 × 𝐵)) = ({𝐶} × 𝐵))
28	27	reseq2d 5968	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝐴 → (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵))) = (2nd ↾ ({𝐶} × 𝐵)))
29	28	rneqd 5918	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝐴 → ran (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵))) = ran (2nd ↾ ({𝐶} × 𝐵)))
30		2ndconst 8084	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝐴 → (2nd ↾ ({𝐶} × 𝐵)):({𝐶} × 𝐵)–1-1-onto→𝐵)
31		f1ofo 6818	. . . . . . . . . 10 ⊢ ((2nd ↾ ({𝐶} × 𝐵)):({𝐶} × 𝐵)–1-1-onto→𝐵 → (2nd ↾ ({𝐶} × 𝐵)):({𝐶} × 𝐵)–onto→𝐵)
32		forn 6785	. . . . . . . . . 10 ⊢ ((2nd ↾ ({𝐶} × 𝐵)):({𝐶} × 𝐵)–onto→𝐵 → ran (2nd ↾ ({𝐶} × 𝐵)) = 𝐵)
33	30, 31, 32	3syl 19	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝐴 → ran (2nd ↾ ({𝐶} × 𝐵)) = 𝐵)
34	29, 33	eqtrd 2800	. . . . . . . 8 ⊢ (𝐶 ∈ 𝐴 → ran (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵))) = 𝐵)
35	16, 34	eqtrid 2812	. . . . . . 7 ⊢ (𝐶 ∈ 𝐴 → (◡◡(2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = 𝐵)
36	35	adantl 486	. . . . . 6 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (◡◡(2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = 𝐵)
37	11, 36	eqtrd 2800	. . . . 5 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (◡◡(2nd ↾ ({𝐶} × V)) “ dom 𝐹) = 𝐵)
38	8, 37	eqtrid 2812	. . . 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 6622	. . . . 5 ⊢ (𝐺 Fn 𝐵 ↔ (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) Fn 𝐵)
41		df-fn 6528	. . . . 5 ⊢ ((𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) Fn 𝐵 ↔ (Fun (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) ∧ dom (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) = 𝐵))
42	40, 41	bitri 278	. . . 4 ⊢ (𝐺 Fn 𝐵 ↔ (Fun (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) ∧ dom (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) = 𝐵))
43	7, 38, 42	sylanbrc 594	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → 𝐺 Fn 𝐵)
44		dffn5 6929	. . 3 ⊢ (𝐺 Fn 𝐵 ↔ 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐺‘𝑥)))
45	43, 44	sylib 221	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐺‘𝑥)))
46	39	fveq1i 6872	. . . . 5 ⊢ (𝐺‘𝑥) = ((𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))‘𝑥)
47		dff1o4 6819	. . . . . . . 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 221	. . . . . . 7 ⊢ (𝐶 ∈ 𝐴 → ((2nd ↾ ({𝐶} × V)) Fn ({𝐶} × V) ∧ ◡(2nd ↾ ({𝐶} × V)) Fn V))
49		fvco2 6968	. . . . . . . 8 ⊢ ((◡(2nd ↾ ({𝐶} × V)) Fn V ∧ 𝑥 ∈ V) → ((𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))‘𝑥) = (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥)))
50	49	elvd 3463	. . . . . . 7 ⊢ (◡(2nd ↾ ({𝐶} × V)) Fn V → ((𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))‘𝑥) = (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥)))
51	48, 50	simpl2im 512	. . . . . 6 ⊢ (𝐶 ∈ 𝐴 → ((𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))‘𝑥) = (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥)))
52	51	ad2antlr 739	. . . . 5 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → ((𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))‘𝑥) = (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥)))
53	46, 52	eqtrid 2812	. . . 4 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥)))
54	2	adantr 485	. . . . . . . . 9 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → (2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V)
55		snidg 4622	. . . . . . . . . . 11 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ {𝐶})
56		vex 3461	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
57		opelxp 5687	. . . . . . . . . . 11 ⊢ (⟨𝐶, 𝑥⟩ ∈ ({𝐶} × V) ↔ (𝐶 ∈ {𝐶} ∧ 𝑥 ∈ V))
58	55, 56, 57	sylanblrc 601	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝐴 → ⟨𝐶, 𝑥⟩ ∈ ({𝐶} × V))
59	58	adantr 485	. . . . . . . . 9 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ⟨𝐶, 𝑥⟩ ∈ ({𝐶} × V))
60	54, 59	jca 520	. . . . . . . 8 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ((2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V ∧ ⟨𝐶, 𝑥⟩ ∈ ({𝐶} × V)))
61	58	fvresd 6891	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝐴 → ((2nd ↾ ({𝐶} × V))‘⟨𝐶, 𝑥⟩) = (2nd ‘⟨𝐶, 𝑥⟩))
62		op2ndg 7987	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑥 ∈ V) → (2nd ‘⟨𝐶, 𝑥⟩) = 𝑥)
63	62	elvd 3463	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝐴 → (2nd ‘⟨𝐶, 𝑥⟩) = 𝑥)
64	61, 63	eqtrd 2800	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝐴 → ((2nd ↾ ({𝐶} × V))‘⟨𝐶, 𝑥⟩) = 𝑥)
65	64	adantr 485	. . . . . . . 8 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → ((2nd ↾ ({𝐶} × V))‘⟨𝐶, 𝑥⟩) = 𝑥)
66		f1ocnvfv 7266	. . . . . . . 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 66	. . . . . . 7 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → (◡(2nd ↾ ({𝐶} × V))‘𝑥) = ⟨𝐶, 𝑥⟩)
68	67	fveq2d 6875	. . . . . 6 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥)) = (𝐹‘⟨𝐶, 𝑥⟩))
69	68	adantll 726	. . . . 5 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥)) = (𝐹‘⟨𝐶, 𝑥⟩))
70		df-ov 7403	. . . . 5 ⊢ (𝐶𝐹𝑥) = (𝐹‘⟨𝐶, 𝑥⟩)
71	69, 70	eqtr4di 2818	. . . 4 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥)) = (𝐶𝐹𝑥))
72	53, 71	eqtrd 2800	. . 3 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐶𝐹𝑥))
73	72	mpteq2dva 5197	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ 𝐵 ↦ (𝐺‘𝑥)) = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥)))
74	45, 73	eqtrd 2800	1 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥)))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  Vcvv 3457   ∩ cin 3906   ⊆ wss 3907  {csn 4585  ⟨cop 4591   ↦ cmpt 5185   × cxp 5649  ^◡ccnv 5650  dom cdm 5651  ran crn 5652   ↾ cres 5653   “ cima 5654   ∘ ccom 5655  Fun wfun 6519   Fn wfn 6520  –onto→wfo 6523  –1-1-onto→wf1o 6524  ‘cfv 6525  (class class class)co 7400  2^nd c2nd 7973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-nul 5260  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-1st 7974  df-2nd 7975

This theorem is referenced by:  curry1val  8088  curry1f  8089