Theorem curfv 8885

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 6941	. . . . . . . . . 10 ⊢ (𝐹 Fn (𝑉 × 𝑊) ↔ 𝐹 = (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹‘𝑧)))
2		cureq 8882	. . . . . . . . . 10 ⊢ (𝐹 = (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹‘𝑧)) → curry 𝐹 = curry (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹‘𝑧)))
3	1, 2	sylbi 220	. . . . . . . . 9 ⊢ (𝐹 Fn (𝑉 × 𝑊) → curry 𝐹 = curry (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹‘𝑧)))
4	3	adantr 486	. . . . . . . 8 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵 ∈ 𝑊) → curry 𝐹 = curry (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹‘𝑧)))
5		fveq2 6883	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
6	5	mpompt 7532	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹‘𝑧)) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))
7		fvex 6896	. . . . . . . . . . 11 ⊢ (𝐹‘⟨𝑥, 𝑦⟩) ∈ V
8	7	rgen2w 3082	. . . . . . . . . 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 4287	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑊 → 𝑊 ≠ ∅)
11	10	adantl 487	. . . . . . . . 9 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵 ∈ 𝑊) → 𝑊 ≠ ∅)
12	6, 9, 11	mpocurryd 8279	. . . . . . . 8 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵 ∈ 𝑊) → curry (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹‘𝑧)) = (𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
13	4, 12	eqtrd 2796	. . . . . . 7 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵 ∈ 𝑊) → curry 𝐹 = (𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
14	13	3adant2 1149	. . . . . 6 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → curry 𝐹 = (𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
15	14	fveq1d 6885	. . . . 5 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (curry 𝐹‘𝐴) = ((𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))‘𝐴))
16	15	adantr 486	. . . 4 ⊢ (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝑊 ∈ 𝑋) → (curry 𝐹‘𝐴) = ((𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))‘𝐴))
17		mptexg 7225	. . . . . 6 ⊢ (𝑊 ∈ 𝑋 → (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)) ∈ V)
18		opeq1 4833	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
19	18	fveq2d 6887	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐹‘⟨𝑥, 𝑦⟩) = (𝐹‘⟨𝐴, 𝑦⟩))
20	19	mpteq2dv 5199	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) = (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
21		eqid 2761	. . . . . . 7 ⊢ (𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))) = (𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))
22	20, 21	fvmptg 6989	. . . . . 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 605	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑊 ∈ 𝑋) → ((𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))‘𝐴) = (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
24	23	3ad2antl2 1205	. . . 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 2796	. . 3 ⊢ (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝑊 ∈ 𝑋) → (curry 𝐹‘𝐴) = (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
26	25	fveq1d 6885	. 2 ⊢ (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝑊 ∈ 𝑋) → ((curry 𝐹‘𝐴)‘𝐵) = ((𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵))
27		opeq2 4834	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
28	27	fveq2d 6887	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐹‘⟨𝐴, 𝑦⟩) = (𝐹‘⟨𝐴, 𝐵⟩))
29		eqid 2761	. . . . . 6 ⊢ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)) = (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))
30		fvex 6896	. . . . . 6 ⊢ (𝐹‘⟨𝐴, 𝐵⟩) ∈ V
31	28, 29, 30	fvmpt 6991	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → ((𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵) = (𝐹‘⟨𝐴, 𝐵⟩))
32		df-ov 7421	. . . . 5 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
33	31, 32	eqtr4di 2814	. . . 4 ⊢ (𝐵 ∈ 𝑊 → ((𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵) = (𝐴𝐹𝐵))
34	33	3ad2ant3 1153	. . 3 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵) = (𝐴𝐹𝐵))
35	34	adantr 486	. 2 ⊢ (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝑊 ∈ 𝑋) → ((𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵) = (𝐴𝐹𝐵))
36	26, 35	eqtrd 2796	1 ⊢ (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝑊 ∈ 𝑋) → ((curry 𝐹‘𝐴)‘𝐵) = (𝐴𝐹𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  Vcvv 3451  ∅c0 4279  ⟨cop 4590   ↦ cmpt 5186   × cxp 5649   Fn wfn 6532  ‘cfv 6537  (class class class)co 7418  curry ccur 8275

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  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 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-1st 7999  df-2nd 8000  df-cur 8277

This theorem is used by:  matunitlindflem1  22987  matunitlindflem2  22988  unccur  38506