Theorem curfv 34864

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 6717	. . . . . . . . . 10 ⊢ (𝐹 Fn (𝑉 × 𝑊) ↔ 𝐹 = (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹‘𝑧)))
2		cureq 34860	. . . . . . . . . 10 ⊢ (𝐹 = (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹‘𝑧)) → curry 𝐹 = curry (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹‘𝑧)))
3	1, 2	sylbi 219	. . . . . . . . 9 ⊢ (𝐹 Fn (𝑉 × 𝑊) → curry 𝐹 = curry (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹‘𝑧)))
4	3	adantr 483	. . . . . . . 8 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵 ∈ 𝑊) → curry 𝐹 = curry (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹‘𝑧)))
5		fveq2 6663	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
6	5	mpompt 7258	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹‘𝑧)) = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))
7		fvex 6676	. . . . . . . . . . 11 ⊢ (𝐹‘⟨𝑥, 𝑦⟩) ∈ V
8	7	rgen2w 3149	. . . . . . . . . 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 4298	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑊 → 𝑊 ≠ ∅)
11	10	adantl 484	. . . . . . . . 9 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵 ∈ 𝑊) → 𝑊 ≠ ∅)
12	6, 9, 11	mpocurryd 7927	. . . . . . . 8 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵 ∈ 𝑊) → curry (𝑧 ∈ (𝑉 × 𝑊) ↦ (𝐹‘𝑧)) = (𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
13	4, 12	eqtrd 2854	. . . . . . 7 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐵 ∈ 𝑊) → curry 𝐹 = (𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
14	13	3adant2 1126	. . . . . 6 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → curry 𝐹 = (𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
15	14	fveq1d 6665	. . . . 5 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (curry 𝐹‘𝐴) = ((𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))‘𝐴))
16	15	adantr 483	. . . 4 ⊢ (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝑊 ∈ 𝑋) → (curry 𝐹‘𝐴) = ((𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))‘𝐴))
17		mptexg 6976	. . . . . 6 ⊢ (𝑊 ∈ 𝑋 → (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)) ∈ V)
18		opeq1 4795	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
19	18	fveq2d 6667	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐹‘⟨𝑥, 𝑦⟩) = (𝐹‘⟨𝐴, 𝑦⟩))
20	19	mpteq2dv 5153	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) = (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
21		eqid 2819	. . . . . . 7 ⊢ (𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩))) = (𝑥 ∈ 𝑉 ↦ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))
22	20, 21	fvmptg 6759	. . . . . 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 1181	. . . 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 2854	. . 3 ⊢ (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝑊 ∈ 𝑋) → (curry 𝐹‘𝐴) = (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)))
26	25	fveq1d 6665	. 2 ⊢ (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝑊 ∈ 𝑋) → ((curry 𝐹‘𝐴)‘𝐵) = ((𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵))
27		opeq2 4796	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
28	27	fveq2d 6667	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐹‘⟨𝐴, 𝑦⟩) = (𝐹‘⟨𝐴, 𝐵⟩))
29		eqid 2819	. . . . . 6 ⊢ (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩)) = (𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))
30		fvex 6676	. . . . . 6 ⊢ (𝐹‘⟨𝐴, 𝐵⟩) ∈ V
31	28, 29, 30	fvmpt 6761	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → ((𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵) = (𝐹‘⟨𝐴, 𝐵⟩))
32		df-ov 7151	. . . . 5 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
33	31, 32	syl6eqr 2872	. . . 4 ⊢ (𝐵 ∈ 𝑊 → ((𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵) = (𝐴𝐹𝐵))
34	33	3ad2ant3 1130	. . 3 ⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵) = (𝐴𝐹𝐵))
35	34	adantr 483	. 2 ⊢ (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝑊 ∈ 𝑋) → ((𝑦 ∈ 𝑊 ↦ (𝐹‘⟨𝐴, 𝑦⟩))‘𝐵) = (𝐴𝐹𝐵))
36	26, 35	eqtrd 2854	1 ⊢ (((𝐹 Fn (𝑉 × 𝑊) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝑊 ∈ 𝑋) → ((curry 𝐹‘𝐴)‘𝐵) = (𝐴𝐹𝐵))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1082   = wceq 1531   ∈ wcel 2108   ≠ wne 3014  ∀wral 3136  Vcvv 3493  ∅c0 4289  ⟨cop 4565   ↦ cmpt 5137   × cxp 5546   Fn wfn 6343  ‘cfv 6348  (class class class)co 7148  curry ccur 7923

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7681  df-2nd 7682  df-cur 7925

This theorem is referenced by:  unccur  34867  matunitlindflem1  34880  matunitlindflem2  34881