Theorem curf 36454

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 5712	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
2		ffvelcdm 7080	. . . . . . . 8 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝐶)
3	1, 2	sylan2 593	. . . . . . 7 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝐶)
4	3	anassrs 468	. . . . . 6 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝐶)
5	4	fmpttd 7111	. . . . 5 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)):𝐵⟶𝐶)
6	5	3ad2antl1 1185	. . . 4 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)):𝐵⟶𝐶)
7		elmapg 8829	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑊 ∧ 𝐵 ∈ (𝑉 ∖ {∅})) → ((𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) ∈ (𝐶 ↑m 𝐵) ↔ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)):𝐵⟶𝐶))
8	7	ancoms 459	. . . . . 6 ⊢ ((𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → ((𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) ∈ (𝐶 ↑m 𝐵) ↔ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)):𝐵⟶𝐶))
9	8	3adant1 1130	. . . . 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 481	. . . 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 256	. . 3 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) ∈ (𝐶 ↑m 𝐵))
12	11	fmpttd 7111	. 2 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → (𝑥 ∈ 𝐴 ↦ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))):𝐴⟶(𝐶 ↑m 𝐵))
13		eldifsni 4792	. . . 4 ⊢ (𝐵 ∈ (𝑉 ∖ {∅}) → 𝐵 ≠ ∅)
14		df-cur 8248	. . . . . 6 ⊢ curry 𝐹 = (𝑥 ∈ dom dom 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧})
15		fdm 6723	. . . . . . . . . 10 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → dom 𝐹 = (𝐴 × 𝐵))
16	15	dmeqd 5903	. . . . . . . . 9 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → dom dom 𝐹 = dom (𝐴 × 𝐵))
17		dmxp 5926	. . . . . . . . 9 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
18	16, 17	sylan9eq 2792	. . . . . . . 8 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ≠ ∅) → dom dom 𝐹 = 𝐴)
19	18	mpteq1d 5242	. . . . . . 7 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ≠ ∅) → (𝑥 ∈ dom dom 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧}) = (𝑥 ∈ 𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧}))
20		ffun 6717	. . . . . . . . . . . . . 14 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → Fun 𝐹)
21		funbrfv2b 6946	. . . . . . . . . . . . . 14 ⊢ (Fun 𝐹 → (⟨𝑥, 𝑦⟩𝐹𝑧 ↔ (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
22	20, 21	syl 17	. . . . . . . . . . . . 13 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → (⟨𝑥, 𝑦⟩𝐹𝑧 ↔ (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
23	15	eleq2d 2819	. . . . . . . . . . . . . . 15 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
24		opelxp 5711	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
25	23, 24	bitrdi 286	. . . . . . . . . . . . . 14 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
26	25	anbi1d 630	. . . . . . . . . . . . 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 278	. . . . . . . . . . . 12 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → (⟨𝑥, 𝑦⟩𝐹𝑧 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
28		ibar 529	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 → ((𝑦 ∈ 𝐵 ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩)))))
29		anass 469	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))))
30		eqcom 2739	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝐹‘⟨𝑥, 𝑦⟩) ↔ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)
31	30	anbi2i 623	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧))
32	29, 31	bitr3i 276	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧))
33	28, 32	bitr2di 287	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐴 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))))
34	27, 33	sylan9bb 510	. . . . . . . . . . 11 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝑥 ∈ 𝐴) → (⟨𝑥, 𝑦⟩𝐹𝑧 ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))))
35	34	opabbidv 5213	. . . . . . . . . 10 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝑥 ∈ 𝐴) → {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐵 ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))})
36		df-mpt 5231	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐵 ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))}
37	35, 36	eqtr4di 2790	. . . . . . . . 9 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝑥 ∈ 𝐴) → {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧} = (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))
38	37	mpteq2dva 5247	. . . . . . . 8 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → (𝑥 ∈ 𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧}) = (𝑥 ∈ 𝐴 ↦ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
39	38	adantr 481	. . . . . . 7 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ≠ ∅) → (𝑥 ∈ 𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧}) = (𝑥 ∈ 𝐴 ↦ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
40	19, 39	eqtrd 2772	. . . . . 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 2784	. . . . 5 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ≠ ∅) → curry 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
42	41	feq1d 6699	. . . 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 593	. . 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 1132	. 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 256	1 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → curry 𝐹:𝐴⟶(𝐶 ↑m 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2940   ∖ cdif 3944  ∅c0 4321  {csn 4627  ⟨cop 4633   class class class wbr 5147  {copab 5209   ↦ cmpt 5230   × cxp 5673  dom cdm 5675  Fun wfun 6534  ⟶wf 6536  ‘cfv 6540  (class class class)co 7405  curry ccur 8246   ↑_m cmap 8816

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-cur 8248  df-map 8818

This theorem is referenced by:  unccur  36459  matunitlindflem1  36472  matunitlindflem2  36473