Theorem curfv 37935

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 6892	. . . . . . . . . 10 ⊢ (𝐹 Fn (𝑉 × 𝑊) ↔ 𝐹 = (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹‘𝑧)))
2		cureq 37931	. . . . . . . . . 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 6834	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
6	5	mpompt 7474	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹‘𝑧)) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))
7		fvex 6847	. . . . . . . . . . 11 ⊢ (𝐹‘⟨𝑥, 𝑦⟩) ∈ V
8	7	rgen2w 3057	. . . . . . . . . 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 4282	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑊 → 𝑊 ≠ ∅)
11	10	adantl 481	. . . . . . . . 9 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵 ∈ 𝑊) → 𝑊 ≠ ∅)
12	6, 9, 11	mpocurryd 8212	. . . . . . . 8 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵 ∈ 𝑊) → curry (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹‘𝑧)) = (𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
13	4, 12	eqtrd 2772	. . . . . . 7 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵 ∈ 𝑊) → curry 𝐹 = (𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
14	13	3adant2 1132	. . . . . 6 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → curry 𝐹 = (𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
15	14	fveq1d 6836	. . . . 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 7169	. . . . . 6 ⊢ (𝑊 ∈ 𝑋 → (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)) ∈ V)
18		opeq1 4817	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
19	18	fveq2d 6838	. . . . . . . 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 2737	. . . . . . 7 ⊢ (𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))) = (𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))
22	20, 21	fvmptg 6939	. . . . . 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 594	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑊 ∈ 𝑋) → ((𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))‘𝐴) = (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
24	23	3ad2antl2 1188	. . . 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 2772	. . 3 ⊢ (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝑊 ∈ 𝑋) → (curry 𝐹‘𝐴) = (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
26	25	fveq1d 6836	. 2 ⊢ (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝑊 ∈ 𝑋) → ((curry 𝐹‘𝐴)‘𝐵) = ((𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵))
27		opeq2 4818	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
28	27	fveq2d 6838	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐹‘⟨𝐴, 𝑦⟩) = (𝐹‘⟨𝐴, 𝐵⟩))
29		eqid 2737	. . . . . 6 ⊢ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)) = (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))
30		fvex 6847	. . . . . 6 ⊢ (𝐹‘⟨𝐴, 𝐵⟩) ∈ V
31	28, 29, 30	fvmpt 6941	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → ((𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵) = (𝐹‘⟨𝐴, 𝐵⟩))
32		df-ov 7363	. . . . 5 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
33	31, 32	eqtr4di 2790	. . . 4 ⊢ (𝐵 ∈ 𝑊 → ((𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵) = (𝐴𝐹𝐵))
34	33	3ad2ant3 1136	. . 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 2772	1 ⊢ (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝑊 ∈ 𝑋) → ((curry 𝐹‘𝐴)‘𝐵) = (𝐴𝐹𝐵))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  Vcvv 3430  ∅c0 4274  ⟨cop 4574   ↦ cmpt 5167   × cxp 5622   Fn wfn 6487  ‘cfv 6492  (class class class)co 7360  curry ccur 8208

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-cur 8210

This theorem is referenced by:  unccur  37938  matunitlindflem1  37951  matunitlindflem2  37952