Theorem curry1 8129

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 6668	. . . . 5 ⊢ (𝐹 Fn (𝐴 × 𝐵) → Fun 𝐹)
2		2ndconst 8126	. . . . . 6 ⊢ (𝐶 ∈ 𝐴 → (2nd ↾ ({𝐶} × V)):({𝐶} × V)–1-1-onto→V)
3		dff1o3 6854	. . . . . . 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 6606	. . . . 5 ⊢ ((Fun 𝐹 ∧ Fun ◡(2nd ↾ ({𝐶} × V))) → Fun (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))))
7	1, 5, 6	syl2an 596	. . . 4 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → Fun (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))))
8		dmco 6274	. . . . 5 ⊢ dom (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) = (◡◡(2nd ↾ ({𝐶} × V)) “ dom 𝐹)
9		fndm 6671	. . . . . . . 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 6078	. . . . . 6 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (◡◡(2nd ↾ ({𝐶} × V)) “ dom 𝐹) = (◡◡(2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)))
12		imacnvcnv 6226	. . . . . . . . 9 ⊢ (◡◡(2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = ((2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵))
13		df-ima 5698	. . . . . . . . 9 ⊢ ((2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = ran ((2nd ↾ ({𝐶} × V)) ↾ (𝐴 × 𝐵))
14		resres 6010	. . . . . . . . . 10 ⊢ ((2nd ↾ ({𝐶} × V)) ↾ (𝐴 × 𝐵)) = (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵)))
15	14	rneqi 5948	. . . . . . . . 9 ⊢ ran ((2nd ↾ ({𝐶} × V)) ↾ (𝐴 × 𝐵)) = ran (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵)))
16	12, 13, 15	3eqtri 2769	. . . . . . . 8 ⊢ (◡◡(2nd ↾ ({𝐶} × V)) “ (𝐴 × 𝐵)) = ran (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵)))
17		inxp 5842	. . . . . . . . . . . . 13 ⊢ (({𝐶} × V) ∩ (𝐴 × 𝐵)) = (({𝐶} ∩ 𝐴) × (V ∩ 𝐵))
18		incom 4209	. . . . . . . . . . . . . . 15 ⊢ (V ∩ 𝐵) = (𝐵 ∩ V)
19		inv1 4398	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∩ V) = 𝐵
20	18, 19	eqtri 2765	. . . . . . . . . . . . . 14 ⊢ (V ∩ 𝐵) = 𝐵
21	20	xpeq2i 5712	. . . . . . . . . . . . 13 ⊢ (({𝐶} ∩ 𝐴) × (V ∩ 𝐵)) = (({𝐶} ∩ 𝐴) × 𝐵)
22	17, 21	eqtri 2765	. . . . . . . . . . . 12 ⊢ (({𝐶} × V) ∩ (𝐴 × 𝐵)) = (({𝐶} ∩ 𝐴) × 𝐵)
23		snssi 4808	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ 𝐴 → {𝐶} ⊆ 𝐴)
24		dfss2 3969	. . . . . . . . . . . . . 14 ⊢ ({𝐶} ⊆ 𝐴 ↔ ({𝐶} ∩ 𝐴) = {𝐶})
25	23, 24	sylib 218	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ 𝐴 → ({𝐶} ∩ 𝐴) = {𝐶})
26	25	xpeq1d 5714	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ 𝐴 → (({𝐶} ∩ 𝐴) × 𝐵) = ({𝐶} × 𝐵))
27	22, 26	eqtrid 2789	. . . . . . . . . . 11 ⊢ (𝐶 ∈ 𝐴 → (({𝐶} × V) ∩ (𝐴 × 𝐵)) = ({𝐶} × 𝐵))
28	27	reseq2d 5997	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝐴 → (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵))) = (2nd ↾ ({𝐶} × 𝐵)))
29	28	rneqd 5949	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝐴 → ran (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵))) = ran (2nd ↾ ({𝐶} × 𝐵)))
30		2ndconst 8126	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝐴 → (2nd ↾ ({𝐶} × 𝐵)):({𝐶} × 𝐵)–1-1-onto→𝐵)
31		f1ofo 6855	. . . . . . . . . 10 ⊢ ((2nd ↾ ({𝐶} × 𝐵)):({𝐶} × 𝐵)–1-1-onto→𝐵 → (2nd ↾ ({𝐶} × 𝐵)):({𝐶} × 𝐵)–onto→𝐵)
32		forn 6823	. . . . . . . . . 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 2777	. . . . . . . 8 ⊢ (𝐶 ∈ 𝐴 → ran (2nd ↾ (({𝐶} × V) ∩ (𝐴 × 𝐵))) = 𝐵)
35	16, 34	eqtrid 2789	. . . . . . 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 2777	. . . . 5 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (◡◡(2nd ↾ ({𝐶} × V)) “ dom 𝐹) = 𝐵)
38	8, 37	eqtrid 2789	. . . 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 6665	. . . . 5 ⊢ (𝐺 Fn 𝐵 ↔ (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) Fn 𝐵)
41		df-fn 6564	. . . . 5 ⊢ ((𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) Fn 𝐵 ↔ (Fun (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) ∧ dom (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) = 𝐵))
42	40, 41	bitri 275	. . . 4 ⊢ (𝐺 Fn 𝐵 ↔ (Fun (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) ∧ dom (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) = 𝐵))
43	7, 38, 42	sylanbrc 583	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → 𝐺 Fn 𝐵)
44		dffn5 6967	. . 3 ⊢ (𝐺 Fn 𝐵 ↔ 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐺‘𝑥)))
45	43, 44	sylib 218	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐺‘𝑥)))
46	39	fveq1i 6907	. . . . 5 ⊢ (𝐺‘𝑥) = ((𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))‘𝑥)
47		dff1o4 6856	. . . . . . . 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 218	. . . . . . 7 ⊢ (𝐶 ∈ 𝐴 → ((2nd ↾ ({𝐶} × V)) Fn ({𝐶} × V) ∧ ◡(2nd ↾ ({𝐶} × V)) Fn V))
49		fvco2 7006	. . . . . . . 8 ⊢ ((◡(2nd ↾ ({𝐶} × V)) Fn V ∧ 𝑥 ∈ V) → ((𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))‘𝑥) = (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥)))
50	49	elvd 3486	. . . . . . 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 727	. . . . 5 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → ((𝐹 ∘ ◡(2nd ↾ ({𝐶} × V)))‘𝑥) = (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥)))
53	46, 52	eqtrid 2789	. . . 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 4660	. . . . . . . . . . 11 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ {𝐶})
56		vex 3484	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
57		opelxp 5721	. . . . . . . . . . 11 ⊢ (⟨𝐶, 𝑥⟩ ∈ ({𝐶} × V) ↔ (𝐶 ∈ {𝐶} ∧ 𝑥 ∈ V))
58	55, 56, 57	sylanblrc 590	. . . . . . . . . 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 6926	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝐴 → ((2nd ↾ ({𝐶} × V))‘⟨𝐶, 𝑥⟩) = (2nd ‘⟨𝐶, 𝑥⟩))
62		op2ndg 8027	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑥 ∈ V) → (2nd ‘⟨𝐶, 𝑥⟩) = 𝑥)
63	62	elvd 3486	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝐴 → (2nd ‘⟨𝐶, 𝑥⟩) = 𝑥)
64	61, 63	eqtrd 2777	. . . . . . . . 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 7298	. . . . . . . 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 6910	. . . . . 6 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥)) = (𝐹‘⟨𝐶, 𝑥⟩))
69	68	adantll 714	. . . . 5 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥)) = (𝐹‘⟨𝐶, 𝑥⟩))
70		df-ov 7434	. . . . 5 ⊢ (𝐶𝐹𝑥) = (𝐹‘⟨𝐶, 𝑥⟩)
71	69, 70	eqtr4di 2795	. . . 4 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡(2nd ↾ ({𝐶} × V))‘𝑥)) = (𝐶𝐹𝑥))
72	53, 71	eqtrd 2777	. . 3 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐶𝐹𝑥))
73	72	mpteq2dva 5242	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ 𝐵 ↦ (𝐺‘𝑥)) = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥)))
74	45, 73	eqtrd 2777	1 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3480   ∩ cin 3950   ⊆ wss 3951  {csn 4626  ⟨cop 4632   ↦ cmpt 5225   × cxp 5683  ^◡ccnv 5684  dom cdm 5685  ran crn 5686   ↾ cres 5687   “ cima 5688   ∘ ccom 5689  Fun wfun 6555   Fn wfn 6556  –onto→wfo 6559  –1-1-onto→wf1o 6560  ‘cfv 6561  (class class class)co 7431  2^nd c2nd 8013

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-1st 8014  df-2nd 8015

This theorem is referenced by:  curry1val  8130  curry1f  8131