Theorem curf 33387

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 5148	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
2		ffvelrn 6357	. . . . . . . 8 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝐶)
3	1, 2	sylan2 491	. . . . . . 7 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝐶)
4	3	anassrs 680	. . . . . 6 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝐶)
5		eqid 2622	. . . . . 6 ⊢ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) = (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))
6	4, 5	fmptd 6385	. . . . 5 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)):𝐵⟶𝐶)
7	6	3ad2antl1 1223	. . . 4 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)):𝐵⟶𝐶)
8		elmapg 7870	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑊 ∧ 𝐵 ∈ (𝑉 ∖ {∅})) → ((𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) ∈ (𝐶 ↑𝑚 𝐵) ↔ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)):𝐵⟶𝐶))
9	8	ancoms 469	. . . . . 6 ⊢ ((𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → ((𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) ∈ (𝐶 ↑𝑚 𝐵) ↔ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)):𝐵⟶𝐶))
10	9	3adant1 1079	. . . . 5 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → ((𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) ∈ (𝐶 ↑𝑚 𝐵) ↔ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)):𝐵⟶𝐶))
11	10	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))
12	7, 11	mpbird 247	. . 3 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) ∈ (𝐶 ↑𝑚 𝐵))
13		eqid 2622	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))) = (𝑥 ∈ 𝐴 ↦ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))
14	12, 13	fmptd 6385	. 2 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → (𝑥 ∈ 𝐴 ↦ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))):𝐴⟶(𝐶 ↑𝑚 𝐵))
15		eldifsni 4320	. . . 4 ⊢ (𝐵 ∈ (𝑉 ∖ {∅}) → 𝐵 ≠ ∅)
16		df-cur 7393	. . . . . 6 ⊢ curry 𝐹 = (𝑥 ∈ dom dom 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧})
17		fdm 6051	. . . . . . . . . 10 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → dom 𝐹 = (𝐴 × 𝐵))
18	17	dmeqd 5326	. . . . . . . . 9 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → dom dom 𝐹 = dom (𝐴 × 𝐵))
19		dmxp 5344	. . . . . . . . 9 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
20	18, 19	sylan9eq 2676	. . . . . . . 8 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ≠ ∅) → dom dom 𝐹 = 𝐴)
21	20	mpteq1d 4738	. . . . . . 7 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ≠ ∅) → (𝑥 ∈ dom dom 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧}) = (𝑥 ∈ 𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧}))
22		ffun 6048	. . . . . . . . . . . . . 14 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → Fun 𝐹)
23		funbrfv2b 6240	. . . . . . . . . . . . . 14 ⊢ (Fun 𝐹 → (⟨𝑥, 𝑦⟩𝐹𝑧 ↔ (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
24	22, 23	syl 17	. . . . . . . . . . . . 13 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → (⟨𝑥, 𝑦⟩𝐹𝑧 ↔ (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
25	17	eleq2d 2687	. . . . . . . . . . . . . . 15 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
26		opelxp 5146	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
27	25, 26	syl6bb 276	. . . . . . . . . . . . . 14 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
28	27	anbi1d 741	. . . . . . . . . . . . 13 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → ((⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
29	24, 28	bitrd 268	. . . . . . . . . . . 12 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → (⟨𝑥, 𝑦⟩𝐹𝑧 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
30		ibar 525	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 → ((𝑦 ∈ 𝐵 ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩)))))
31		anass 681	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))))
32		eqcom 2629	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝐹‘⟨𝑥, 𝑦⟩) ↔ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)
33	32	anbi2i 730	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧))
34	31, 33	bitr3i 266	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧))
35	30, 34	syl6rbb 277	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐴 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))))
36	29, 35	sylan9bb 736	. . . . . . . . . . 11 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝑥 ∈ 𝐴) → (⟨𝑥, 𝑦⟩𝐹𝑧 ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))))
37	36	opabbidv 4716	. . . . . . . . . 10 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝑥 ∈ 𝐴) → {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐵 ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))})
38		df-mpt 4730	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐵 ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))}
39	37, 38	syl6eqr 2674	. . . . . . . . 9 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝑥 ∈ 𝐴) → {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧} = (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))
40	39	mpteq2dva 4744	. . . . . . . 8 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → (𝑥 ∈ 𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧}) = (𝑥 ∈ 𝐴 ↦ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
41	40	adantr 481	. . . . . . 7 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ≠ ∅) → (𝑥 ∈ 𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧}) = (𝑥 ∈ 𝐴 ↦ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
42	21, 41	eqtrd 2656	. . . . . 6 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ≠ ∅) → (𝑥 ∈ dom dom 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩𝐹𝑧}) = (𝑥 ∈ 𝐴 ↦ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
43	16, 42	syl5eq 2668	. . . . 5 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ≠ ∅) → curry 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
44	43	feq1d 6030	. . . 4 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ≠ ∅) → (curry 𝐹:𝐴⟶(𝐶 ↑𝑚 𝐵) ↔ (𝑥 ∈ 𝐴 ↦ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))):𝐴⟶(𝐶 ↑𝑚 𝐵)))
45	15, 44	sylan2 491	. . 3 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅})) → (curry 𝐹:𝐴⟶(𝐶 ↑𝑚 𝐵) ↔ (𝑥 ∈ 𝐴 ↦ (𝑦 ∈ 𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))):𝐴⟶(𝐶 ↑𝑚 𝐵)))
46	45	3adant3 1081	. 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)))
47	14, 46	mpbird 247	1 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → curry 𝐹:𝐴⟶(𝐶 ↑𝑚 𝐵))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794   ∖ cdif 3571  ∅c0 3915  {csn 4177  ⟨cop 4183   class class class wbr 4653  {copab 4712   ↦ cmpt 4729   × cxp 5112  dom cdm 5114  Fun wfun 5882  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  curry ccur 7391   ↑_𝑚 cmap 7857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-cur 7393  df-map 7859

This theorem is referenced by:  unccur  33392  matunitlindflem1  33405  matunitlindflem2  33406