Theorem curfv 37967

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 6885	. . . . . . . . . 10 ⊢ (𝐹 Fn (𝑉 × 𝑊) ↔ 𝐹 = (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹‘𝑧)))
2		cureq 37963	. . . . . . . . . 10 ⊢ (𝐹 = (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹‘𝑧)) → curry 𝐹 = curry (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹‘𝑧)))
3	1, 2	sylbi 218	. . . . . . . . 9 ⊢ (𝐹 Fn (𝑉 × 𝑊) → curry 𝐹 = curry (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹‘𝑧)))
4	3	adantr 481	. . . . . . . 8 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵 ∈ 𝑊) → curry 𝐹 = curry (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹‘𝑧)))
5		fveq2 6827	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
6	5	mpompt 7470	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹‘𝑧)) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))
7		fvex 6840	. . . . . . . . . . 11 ⊢ (𝐹‘⟨𝑥, 𝑦⟩) ∈ V
8	7	rgen2w 3058	. . . . . . . . . 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 4269	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑊 → 𝑊 ≠ ∅)
11	10	adantl 482	. . . . . . . . 9 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵 ∈ 𝑊) → 𝑊 ≠ ∅)
12	6, 9, 11	mpocurryd 8209	. . . . . . . 8 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵 ∈ 𝑊) → curry (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹‘𝑧)) = (𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
13	4, 12	eqtrd 2774	. . . . . . 7 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵 ∈ 𝑊) → curry 𝐹 = (𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
14	13	3adant2 1137	. . . . . 6 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → curry 𝐹 = (𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
15	14	fveq1d 6829	. . . . 5 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (curry 𝐹‘𝐴) = ((𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))‘𝐴))
16	15	adantr 481	. . . 4 ⊢ (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝑊 ∈ 𝑋) → (curry 𝐹‘𝐴) = ((𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))‘𝐴))
17		mptexg 7165	. . . . . 6 ⊢ (𝑊 ∈ 𝑋 → (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)) ∈ V)
18		opeq1 4804	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
19	18	fveq2d 6831	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐹‘⟨𝑥, 𝑦⟩) = (𝐹‘⟨𝐴, 𝑦⟩))
20	19	mpteq2dv 5166	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) = (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
21		eqid 2739	. . . . . . 7 ⊢ (𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))) = (𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))
22	20, 21	fvmptg 6933	. . . . . 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 599	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑊 ∈ 𝑋) → ((𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))‘𝐴) = (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
24	23	3ad2antl2 1193	. . . 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 2774	. . 3 ⊢ (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝑊 ∈ 𝑋) → (curry 𝐹‘𝐴) = (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
26	25	fveq1d 6829	. 2 ⊢ (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝑊 ∈ 𝑋) → ((curry 𝐹‘𝐴)‘𝐵) = ((𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵))
27		opeq2 4805	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
28	27	fveq2d 6831	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐹‘⟨𝐴, 𝑦⟩) = (𝐹‘⟨𝐴, 𝐵⟩))
29		eqid 2739	. . . . . 6 ⊢ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)) = (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))
30		fvex 6840	. . . . . 6 ⊢ (𝐹‘⟨𝐴, 𝐵⟩) ∈ V
31	28, 29, 30	fvmpt 6935	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → ((𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵) = (𝐹‘⟨𝐴, 𝐵⟩))
32		df-ov 7359	. . . . 5 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
33	31, 32	eqtr4di 2792	. . . 4 ⊢ (𝐵 ∈ 𝑊 → ((𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵) = (𝐴𝐹𝐵))
34	33	3ad2ant3 1141	. . 3 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵) = (𝐴𝐹𝐵))
35	34	adantr 481	. 2 ⊢ (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝑊 ∈ 𝑋) → ((𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵) = (𝐴𝐹𝐵))
36	26, 35	eqtrd 2774	1 ⊢ (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝑊 ∈ 𝑋) → ((curry 𝐹‘𝐴)‘𝐵) = (𝐴𝐹𝐵))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  ∀wral 3053  Vcvv 3431  ∅c0 4261  ⟨cop 4561   ↦ cmpt 5153   × cxp 5616   Fn wfn 6480  ‘cfv 6485  (class class class)co 7356  curry ccur 8205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-cur 8207

This theorem is referenced by:  unccur  37970  matunitlindflem1  37983  matunitlindflem2  37984