Theorem curf 8894

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

Assertion

Ref	Expression
curf	⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → curry 𝐹:𝐴⟶(𝐶 ↑m 𝐵))

Proof of Theorem curf

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

Step	Hyp	Ref	Expression
1		opelxpi 5687	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
2		ffvelcdm 7082	. . . . . . . 8 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝐶)
3	1, 2	sylan2 605	. . . . . . 7 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝐶)
4	3	anassrs 473	. . . . . 6 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝐶)
5	4	fmpttd 7116	. . . . 5 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)):𝐵⟶𝐶)
6	5	3ad2antl1 1204	. . . 4 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)):𝐵⟶𝐶)
7		elmapg 8863	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑊 ∧ 𝐵 ∈ (𝑉 ∖ {∅})) → ((𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) ∈ (𝐶 ↑m 𝐵) ↔ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)):𝐵⟶𝐶))
8	7	ancoms 464	. . . . . 6 ⊢ ((𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → ((𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) ∈ (𝐶 ↑m 𝐵) ↔ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)):𝐵⟶𝐶))
9	8	3adant1 1148	. . . . 5 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → ((𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) ∈ (𝐶 ↑m 𝐵) ↔ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)):𝐵⟶𝐶))
10	9	adantr 486	. . . 4 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) ∧ 𝑥 ∈ 𝐴) → ((𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) ∈ (𝐶 ↑m 𝐵) ↔ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)):𝐵⟶𝐶))
11	6, 10	mpbird 260	. . 3 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) ∈ (𝐶 ↑m 𝐵))
12	11	fmpttd 7116	. 2 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → (𝑥 ∈ 𝐴 ↦ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))):𝐴⟶(𝐶 ↑m 𝐵))
13		eldifsni 4749	. . . 4 ⊢ (𝐵 ∈ (𝑉 ∖ {∅}) → 𝐵 ≠ ∅)
14		df-cur 8288	. . . . . 6 ⊢ curry 𝐹 = (𝑥 ∈ dom dom 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧})
15		fdm 6720	. . . . . . . . . 10 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → dom 𝐹 = (𝐴 × 𝐵))
16	15	dmeqd 5886	. . . . . . . . 9 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → dom dom 𝐹 = dom (𝐴 × 𝐵))
17		dmxp 5910	. . . . . . . . 9 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
18	16, 17	sylan9eq 2815	. . . . . . . 8 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ≠ ∅) → dom dom 𝐹 = 𝐴)
19	18	mpteq1d 5194	. . . . . . 7 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ≠ ∅) → (𝑥 ∈ dom dom 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧}) = (𝑥 ∈ 𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧}))
20		ffun 6713	. . . . . . . . . . . . . 14 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → Fun 𝐹)
21		funbrfv2b 6943	. . . . . . . . . . . . . 14 ⊢ (Fun 𝐹 → (⟨𝑥, 𝑦⟩𝐹𝑧 ↔ (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
22	20, 21	syl 18	. . . . . . . . . . . . 13 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → (⟨𝑥, 𝑦⟩𝐹𝑧 ↔ (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
23	15	eleq2d 2846	. . . . . . . . . . . . . . 15 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
24		opelxp 5686	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
25	23, 24	bitrdi 290	. . . . . . . . . . . . . 14 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
26	25	anbi1d 643	. . . . . . . . . . . . 13 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → ((⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
27	22, 26	bitrd 282	. . . . . . . . . . . 12 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → (⟨𝑥, 𝑦⟩𝐹𝑧 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
28		ibar 538	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 → ((𝑦 ∈ 𝐵 ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩)))))
29		anass 474	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))))
30		eqcom 2767	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝐹‘⟨𝑥, 𝑦⟩) ↔ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)
31	30	anbi2i 635	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧))
32	29, 31	bitr3i 280	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧))
33	28, 32	bitr2di 291	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐴 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))))
34	27, 33	sylan9bb 519	. . . . . . . . . . 11 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝑥 ∈ 𝐴) → (⟨𝑥, 𝑦⟩𝐹𝑧 ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))))
35	34	opabbidv 5170	. . . . . . . . . 10 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝑥 ∈ 𝐴) → {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐵 ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))})
36		df-mpt 5186	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐵 ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))}
37	35, 36	eqtr4di 2813	. . . . . . . . 9 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝑥 ∈ 𝐴) → {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧} = (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))
38	37	mpteq2dva 5197	. . . . . . . 8 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → (𝑥 ∈ 𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧}) = (𝑥 ∈ 𝐴 ↦ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
39	38	adantr 486	. . . . . . 7 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ≠ ∅) → (𝑥 ∈ 𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧}) = (𝑥 ∈ 𝐴 ↦ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
40	19, 39	eqtrd 2795	. . . . . 6 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ≠ ∅) → (𝑥 ∈ dom dom 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧}) = (𝑥 ∈ 𝐴 ↦ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
41	14, 40	eqtrid 2807	. . . . 5 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ≠ ∅) → curry 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
42	41	feq1d 6692	. . . 4 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ≠ ∅) → (curry 𝐹:𝐴⟶(𝐶 ↑m 𝐵) ↔ (𝑥 ∈ 𝐴 ↦ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))):𝐴⟶(𝐶 ↑m 𝐵)))
43	13, 42	sylan2 605	. . 3 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅})) → (curry 𝐹:𝐴⟶(𝐶 ↑m 𝐵) ↔ (𝑥 ∈ 𝐴 ↦ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))):𝐴⟶(𝐶 ↑m 𝐵)))
44	43	3adant3 1150	. 2 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → (curry 𝐹:𝐴⟶(𝐶 ↑m 𝐵) ↔ (𝑥 ∈ 𝐴 ↦ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))):𝐴⟶(𝐶 ↑m 𝐵)))
45	12, 44	mpbird 260	1 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → curry 𝐹:𝐴⟶(𝐶 ↑m 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955   ∖ cdif 4045  ∅c0 4278  {csn 4584  ⟨cop 4834   class class class wbr 5102  {copab 5166   ↦ cmpt 5185   × cxp 5648  dom cdm 5650  Fun wfun 6533  ⟶wf 6535  ‘cfv 6539  (class class class)co 7421  curry ccur 8286   ↑_m cmap 8851

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-cur 8288  df-map 8853

This theorem is used by:  matunitlindflem1  23000  matunitlindflem2  23001  unccur  38620  veroquadmodzerod  51153  veroquadnolindfd  51154