Theorem curfv 37640

Description: Value of currying. (Contributed by Brendan Leahy, 2-Jun-2021.)

Assertion

Ref	Expression
curfv	⊢ (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝑊 ∈ 𝑋) → ((curry 𝐹‘𝐴)‘𝐵) = (𝐴𝐹𝐵))

Proof of Theorem curfv

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffn5 6875	. . . . . . . . . 10 ⊢ (𝐹 Fn (𝑉 × 𝑊) ↔ 𝐹 = (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹‘𝑧)))
2		cureq 37636	. . . . . . . . . 10 ⊢ (𝐹 = (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹‘𝑧)) → curry 𝐹 = curry (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹‘𝑧)))
3	1, 2	sylbi 217	. . . . . . . . 9 ⊢ (𝐹 Fn (𝑉 × 𝑊) → curry 𝐹 = curry (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹‘𝑧)))
4	3	adantr 480	. . . . . . . 8 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵 ∈ 𝑊) → curry 𝐹 = curry (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹‘𝑧)))
5		fveq2 6817	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
6	5	mpompt 7455	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹‘𝑧)) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))
7		fvex 6830	. . . . . . . . . . 11 ⊢ (𝐹‘⟨𝑥, 𝑦⟩) ∈ V
8	7	rgen2w 3052	. . . . . . . . . 10 ⊢ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑊 (𝐹‘⟨𝑥, 𝑦⟩) ∈ V
9	8	a1i 11	. . . . . . . . 9 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵 ∈ 𝑊) → ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑊 (𝐹‘⟨𝑥, 𝑦⟩) ∈ V)
10		ne0i 4286	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑊 → 𝑊 ≠ ∅)
11	10	adantl 481	. . . . . . . . 9 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵 ∈ 𝑊) → 𝑊 ≠ ∅)
12	6, 9, 11	mpocurryd 8194	. . . . . . . 8 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵 ∈ 𝑊) → curry (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹‘𝑧)) = (𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
13	4, 12	eqtrd 2766	. . . . . . 7 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵 ∈ 𝑊) → curry 𝐹 = (𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
14	13	3adant2 1131	. . . . . 6 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → curry 𝐹 = (𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
15	14	fveq1d 6819	. . . . 5 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (curry 𝐹‘𝐴) = ((𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))‘𝐴))
16	15	adantr 480	. . . 4 ⊢ (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝑊 ∈ 𝑋) → (curry 𝐹‘𝐴) = ((𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))‘𝐴))
17		mptexg 7150	. . . . . 6 ⊢ (𝑊 ∈ 𝑋 → (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)) ∈ V)
18		opeq1 4820	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
19	18	fveq2d 6821	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐹‘⟨𝑥, 𝑦⟩) = (𝐹‘⟨𝐴, 𝑦⟩))
20	19	mpteq2dv 5180	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) = (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
21		eqid 2731	. . . . . . 7 ⊢ (𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))) = (𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))
22	20, 21	fvmptg 6922	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)) ∈ V) → ((𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))‘𝐴) = (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
23	17, 22	sylan2 593	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑊 ∈ 𝑋) → ((𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))‘𝐴) = (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
24	23	3ad2antl2 1187	. . . 4 ⊢ (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝑊 ∈ 𝑋) → ((𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))‘𝐴) = (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
25	16, 24	eqtrd 2766	. . 3 ⊢ (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝑊 ∈ 𝑋) → (curry 𝐹‘𝐴) = (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
26	25	fveq1d 6819	. 2 ⊢ (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝑊 ∈ 𝑋) → ((curry 𝐹‘𝐴)‘𝐵) = ((𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵))
27		opeq2 4821	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
28	27	fveq2d 6821	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐹‘⟨𝐴, 𝑦⟩) = (𝐹‘⟨𝐴, 𝐵⟩))
29		eqid 2731	. . . . . 6 ⊢ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)) = (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))
30		fvex 6830	. . . . . 6 ⊢ (𝐹‘⟨𝐴, 𝐵⟩) ∈ V
31	28, 29, 30	fvmpt 6924	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → ((𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵) = (𝐹‘⟨𝐴, 𝐵⟩))
32		df-ov 7344	. . . . 5 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
33	31, 32	eqtr4di 2784	. . . 4 ⊢ (𝐵 ∈ 𝑊 → ((𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵) = (𝐴𝐹𝐵))
34	33	3ad2ant3 1135	. . 3 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵) = (𝐴𝐹𝐵))
35	34	adantr 480	. 2 ⊢ (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝑊 ∈ 𝑋) → ((𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵) = (𝐴𝐹𝐵))
36	26, 35	eqtrd 2766	1 ⊢ (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝑊 ∈ 𝑋) → ((curry 𝐹‘𝐴)‘𝐵) = (𝐴𝐹𝐵))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∀wral 3047  Vcvv 3436  ∅c0 4278  ⟨cop 4577   ↦ cmpt 5167   × cxp 5609   Fn wfn 6471  ‘cfv 6476  (class class class)co 7341  curry ccur 8190

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-1st 7916  df-2nd 7917  df-cur 8192

This theorem is referenced by:  unccur  37643  matunitlindflem1  37656  matunitlindflem2  37657