Theorem curf 37846

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 5669	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
2		ffvelcdm 7035	. . . . . . . 8 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝐶)
3	1, 2	sylan2 594	. . . . . . 7 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝐶)
4	3	anassrs 467	. . . . . 6 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝐶)
5	4	fmpttd 7069	. . . . 5 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)):𝐵⟶𝐶)
6	5	3ad2antl1 1187	. . . 4 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)):𝐵⟶𝐶)
7		elmapg 8788	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑊 ∧ 𝐵 ∈ (𝑉 ∖ {∅})) → ((𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) ∈ (𝐶 ↑m 𝐵) ↔ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)):𝐵⟶𝐶))
8	7	ancoms 458	. . . . . 6 ⊢ ((𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → ((𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) ∈ (𝐶 ↑m 𝐵) ↔ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)):𝐵⟶𝐶))
9	8	3adant1 1131	. . . . 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 480	. . . 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 257	. . 3 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) ∈ (𝐶 ↑m 𝐵))
12	11	fmpttd 7069	. 2 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → (𝑥 ∈ 𝐴 ↦ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))):𝐴⟶(𝐶 ↑m 𝐵))
13		eldifsni 4748	. . . 4 ⊢ (𝐵 ∈ (𝑉 ∖ {∅}) → 𝐵 ≠ ∅)
14		df-cur 8219	. . . . . 6 ⊢ curry 𝐹 = (𝑥 ∈ dom dom 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧})
15		fdm 6679	. . . . . . . . . 10 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → dom 𝐹 = (𝐴 × 𝐵))
16	15	dmeqd 5862	. . . . . . . . 9 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → dom dom 𝐹 = dom (𝐴 × 𝐵))
17		dmxp 5886	. . . . . . . . 9 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
18	16, 17	sylan9eq 2792	. . . . . . . 8 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ≠ ∅) → dom dom 𝐹 = 𝐴)
19	18	mpteq1d 5190	. . . . . . 7 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ≠ ∅) → (𝑥 ∈ dom dom 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧}) = (𝑥 ∈ 𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧}))
20		ffun 6673	. . . . . . . . . . . . . 14 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → Fun 𝐹)
21		funbrfv2b 6899	. . . . . . . . . . . . . 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 2823	. . . . . . . . . . . . . . 15 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
24		opelxp 5668	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
25	23, 24	bitrdi 287	. . . . . . . . . . . . . 14 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
26	25	anbi1d 632	. . . . . . . . . . . . 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 279	. . . . . . . . . . . 12 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → (⟨𝑥, 𝑦⟩𝐹𝑧 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
28		ibar 528	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 → ((𝑦 ∈ 𝐵 ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩)))))
29		anass 468	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))))
30		eqcom 2744	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝐹‘⟨𝑥, 𝑦⟩) ↔ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)
31	30	anbi2i 624	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧))
32	29, 31	bitr3i 277	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧))
33	28, 32	bitr2di 288	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐴 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))))
34	27, 33	sylan9bb 509	. . . . . . . . . . 11 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝑥 ∈ 𝐴) → (⟨𝑥, 𝑦⟩𝐹𝑧 ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))))
35	34	opabbidv 5166	. . . . . . . . . 10 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝑥 ∈ 𝐴) → {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐵 ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))})
36		df-mpt 5182	. . . . . . . . . 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 5193	. . . . . . . 8 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → (𝑥 ∈ 𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧}) = (𝑥 ∈ 𝐴 ↦ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
39	38	adantr 480	. . . . . . 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 6652	. . . 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 594	. . 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 1133	. 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 257	1 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → curry 𝐹:𝐴⟶(𝐶 ↑m 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933   ∖ cdif 3900  ∅c0 4287  {csn 4582  ⟨cop 4588   class class class wbr 5100  {copab 5162   ↦ cmpt 5181   × cxp 5630  dom cdm 5632  Fun wfun 6494  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368  curry ccur 8217   ↑_m cmap 8775

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-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

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-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-cur 8219  df-map 8777

This theorem is referenced by:  unccur  37851  matunitlindflem1  37864  matunitlindflem2  37865