Theorem unccur 33703

Description: Uncurrying of currying. (Contributed by Brendan Leahy, 5-Jun-2021.)

Assertion

Ref	Expression
unccur	⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → uncurry curry 𝐹 = 𝐹)

Proof of Theorem unccur

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

Step	Hyp	Ref	Expression
1		ffn 6252	. . . . . . . . 9 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → 𝐹 Fn (𝐴 × 𝐵))
2	1	anim1i 604	. . . . . . . 8 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅})) → (𝐹 Fn (𝐴 × 𝐵) ∧ 𝐵 ∈ (𝑉 ∖ {∅})))
3	2	3adant3 1155	. . . . . . 7 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → (𝐹 Fn (𝐴 × 𝐵) ∧ 𝐵 ∈ (𝑉 ∖ {∅})))
4		3anass 1109	. . . . . . . . . . 11 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
5		curfv 33700	. . . . . . . . . . 11 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝐵 ∈ (𝑉 ∖ {∅})) → ((curry 𝐹‘𝑥)‘𝑦) = (𝑥𝐹𝑦))
6	4, 5	sylanbr 573	. . . . . . . . . 10 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝐵 ∈ (𝑉 ∖ {∅})) → ((curry 𝐹‘𝑥)‘𝑦) = (𝑥𝐹𝑦))
7	6	an32s 634	. . . . . . . . 9 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐵 ∈ (𝑉 ∖ {∅})) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ((curry 𝐹‘𝑥)‘𝑦) = (𝑥𝐹𝑦))
8	7	eqeq1d 2808	. . . . . . . 8 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐵 ∈ (𝑉 ∖ {∅})) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (((curry 𝐹‘𝑥)‘𝑦) = 𝑧 ↔ (𝑥𝐹𝑦) = 𝑧))
9		eqcom 2813	. . . . . . . 8 ⊢ ((𝑥𝐹𝑦) = 𝑧 ↔ 𝑧 = (𝑥𝐹𝑦))
10	8, 9	syl6bb 278	. . . . . . 7 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐵 ∈ (𝑉 ∖ {∅})) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (((curry 𝐹‘𝑥)‘𝑦) = 𝑧 ↔ 𝑧 = (𝑥𝐹𝑦)))
11	3, 10	sylan 571	. . . . . 6 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (((curry 𝐹‘𝑥)‘𝑦) = 𝑧 ↔ 𝑧 = (𝑥𝐹𝑦)))
12		curf 33698	. . . . . . . . . 10 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → curry 𝐹:𝐴⟶(𝐶 ↑𝑚 𝐵))
13	12	ffvelrnda 6577	. . . . . . . . 9 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) ∧ 𝑥 ∈ 𝐴) → (curry 𝐹‘𝑥) ∈ (𝐶 ↑𝑚 𝐵))
14		elmapfn 8111	. . . . . . . . 9 ⊢ ((curry 𝐹‘𝑥) ∈ (𝐶 ↑𝑚 𝐵) → (curry 𝐹‘𝑥) Fn 𝐵)
15	13, 14	syl 17	. . . . . . . 8 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) ∧ 𝑥 ∈ 𝐴) → (curry 𝐹‘𝑥) Fn 𝐵)
16		fnbrfvb 6452	. . . . . . . 8 ⊢ (((curry 𝐹‘𝑥) Fn 𝐵 ∧ 𝑦 ∈ 𝐵) → (((curry 𝐹‘𝑥)‘𝑦) = 𝑧 ↔ 𝑦(curry 𝐹‘𝑥)𝑧))
17	15, 16	sylan 571	. . . . . . 7 ⊢ ((((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (((curry 𝐹‘𝑥)‘𝑦) = 𝑧 ↔ 𝑦(curry 𝐹‘𝑥)𝑧))
18	17	anasss 454	. . . . . 6 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (((curry 𝐹‘𝑥)‘𝑦) = 𝑧 ↔ 𝑦(curry 𝐹‘𝑥)𝑧))
19		ibar 520	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑧 = (𝑥𝐹𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = (𝑥𝐹𝑦))))
20	19	adantl 469	. . . . . 6 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑧 = (𝑥𝐹𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = (𝑥𝐹𝑦))))
21	11, 18, 20	3bitr3d 300	. . . . 5 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑦(curry 𝐹‘𝑥)𝑧 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = (𝑥𝐹𝑦))))
22		df-br 4845	. . . . . . . . . . 11 ⊢ (𝑦(curry 𝐹‘𝑥)𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ (curry 𝐹‘𝑥))
23		elfvdm 6436	. . . . . . . . . . 11 ⊢ (⟨𝑦, 𝑧⟩ ∈ (curry 𝐹‘𝑥) → 𝑥 ∈ dom curry 𝐹)
24	22, 23	sylbi 208	. . . . . . . . . 10 ⊢ (𝑦(curry 𝐹‘𝑥)𝑧 → 𝑥 ∈ dom curry 𝐹)
25		fdm 6260	. . . . . . . . . . . 12 ⊢ (curry 𝐹:𝐴⟶(𝐶 ↑𝑚 𝐵) → dom curry 𝐹 = 𝐴)
26	25	eleq2d 2871	. . . . . . . . . . 11 ⊢ (curry 𝐹:𝐴⟶(𝐶 ↑𝑚 𝐵) → (𝑥 ∈ dom curry 𝐹 ↔ 𝑥 ∈ 𝐴))
27	26	biimpa 464	. . . . . . . . . 10 ⊢ ((curry 𝐹:𝐴⟶(𝐶 ↑𝑚 𝐵) ∧ 𝑥 ∈ dom curry 𝐹) → 𝑥 ∈ 𝐴)
28	24, 27	sylan2 582	. . . . . . . . 9 ⊢ ((curry 𝐹:𝐴⟶(𝐶 ↑𝑚 𝐵) ∧ 𝑦(curry 𝐹‘𝑥)𝑧) → 𝑥 ∈ 𝐴)
29		ffvelrn 6575	. . . . . . . . . . . . 13 ⊢ ((curry 𝐹:𝐴⟶(𝐶 ↑𝑚 𝐵) ∧ 𝑥 ∈ 𝐴) → (curry 𝐹‘𝑥) ∈ (𝐶 ↑𝑚 𝐵))
30		elmapi 8110	. . . . . . . . . . . . 13 ⊢ ((curry 𝐹‘𝑥) ∈ (𝐶 ↑𝑚 𝐵) → (curry 𝐹‘𝑥):𝐵⟶𝐶)
31		fdm 6260	. . . . . . . . . . . . 13 ⊢ ((curry 𝐹‘𝑥):𝐵⟶𝐶 → dom (curry 𝐹‘𝑥) = 𝐵)
32	29, 30, 31	3syl 18	. . . . . . . . . . . 12 ⊢ ((curry 𝐹:𝐴⟶(𝐶 ↑𝑚 𝐵) ∧ 𝑥 ∈ 𝐴) → dom (curry 𝐹‘𝑥) = 𝐵)
33		vex 3394	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
34		vex 3394	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
35	33, 34	breldm 5530	. . . . . . . . . . . 12 ⊢ (𝑦(curry 𝐹‘𝑥)𝑧 → 𝑦 ∈ dom (curry 𝐹‘𝑥))
36		eleq2 2874	. . . . . . . . . . . . 13 ⊢ (dom (curry 𝐹‘𝑥) = 𝐵 → (𝑦 ∈ dom (curry 𝐹‘𝑥) ↔ 𝑦 ∈ 𝐵))
37	36	biimpa 464	. . . . . . . . . . . 12 ⊢ ((dom (curry 𝐹‘𝑥) = 𝐵 ∧ 𝑦 ∈ dom (curry 𝐹‘𝑥)) → 𝑦 ∈ 𝐵)
38	32, 35, 37	syl2an 585	. . . . . . . . . . 11 ⊢ (((curry 𝐹:𝐴⟶(𝐶 ↑𝑚 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦(curry 𝐹‘𝑥)𝑧) → 𝑦 ∈ 𝐵)
39	38	an32s 634	. . . . . . . . . 10 ⊢ (((curry 𝐹:𝐴⟶(𝐶 ↑𝑚 𝐵) ∧ 𝑦(curry 𝐹‘𝑥)𝑧) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵)
40	28, 39	mpdan 670	. . . . . . . . 9 ⊢ ((curry 𝐹:𝐴⟶(𝐶 ↑𝑚 𝐵) ∧ 𝑦(curry 𝐹‘𝑥)𝑧) → 𝑦 ∈ 𝐵)
41	28, 40	jca 503	. . . . . . . 8 ⊢ ((curry 𝐹:𝐴⟶(𝐶 ↑𝑚 𝐵) ∧ 𝑦(curry 𝐹‘𝑥)𝑧) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
42	12, 41	sylan 571	. . . . . . 7 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) ∧ 𝑦(curry 𝐹‘𝑥)𝑧) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
43	42	stoic1a 1852	. . . . . 6 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ¬ 𝑦(curry 𝐹‘𝑥)𝑧)
44		simpl 470	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = (𝑥𝐹𝑦)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
45	44	con3i 151	. . . . . . 7 ⊢ (¬ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ¬ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = (𝑥𝐹𝑦)))
46	45	adantl 469	. . . . . 6 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ¬ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = (𝑥𝐹𝑦)))
47	43, 46	2falsed 367	. . . . 5 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑦(curry 𝐹‘𝑥)𝑧 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = (𝑥𝐹𝑦))))
48	21, 47	pm2.61dan 838	. . . 4 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → (𝑦(curry 𝐹‘𝑥)𝑧 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = (𝑥𝐹𝑦))))
49	48	oprabbidv 6935	. . 3 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(curry 𝐹‘𝑥)𝑧} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = (𝑥𝐹𝑦))})
50		df-unc 7625	. . 3 ⊢ uncurry curry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(curry 𝐹‘𝑥)𝑧}
51		df-mpt2 6875	. . 3 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = (𝑥𝐹𝑦))}
52	49, 50, 51	3eqtr4g 2865	. 2 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → uncurry curry 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦)))
53		fnov 6994	. . . 4 ⊢ (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦)))
54	1, 53	sylib 209	. . 3 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦)))
55	54	3ad2ant1 1156	. 2 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦)))
56	52, 55	eqtr4d 2843	1 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → uncurry curry 𝐹 = 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 197   ∧ wa 384   ∧ w3a 1100   = wceq 1637   ∈ wcel 2156   ∖ cdif 3766  ∅c0 4116  {csn 4370  ⟨cop 4376   class class class wbr 4844   × cxp 5309  dom cdm 5311   Fn wfn 6092  ⟶wf 6093  ‘cfv 6097  (class class class)co 6870  {coprab 6871   ↦ cmpt2 6872  curry ccur 7622  uncurry cunc 7623   ↑_𝑚 cmap 8088

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105  df-ov 6873  df-oprab 6874  df-mpt2 6875  df-1st 7394  df-2nd 7395  df-cur 7624  df-unc 7625  df-map 8090

This theorem is referenced by: (None)