Theorem unccur 36061

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 6668	. . . . . . . . 9 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → 𝐹 Fn (𝐴 × 𝐵))
2	1	anim1i 615	. . . . . . . 8 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅})) → (𝐹 Fn (𝐴 × 𝐵) ∧ 𝐵 ∈ (𝑉 ∖ {∅})))
3	2	3adant3 1132	. . . . . . 7 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → (𝐹 Fn (𝐴 × 𝐵) ∧ 𝐵 ∈ (𝑉 ∖ {∅})))
4		3anass 1095	. . . . . . . . . . 11 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
5		curfv 36058	. . . . . . . . . . 11 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝐵 ∈ (𝑉 ∖ {∅})) → ((curry 𝐹‘𝑥)‘𝑦) = (𝑥𝐹𝑦))
6	4, 5	sylanbr 582	. . . . . . . . . 10 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝐵 ∈ (𝑉 ∖ {∅})) → ((curry 𝐹‘𝑥)‘𝑦) = (𝑥𝐹𝑦))
7	6	an32s 650	. . . . . . . . 9 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐵 ∈ (𝑉 ∖ {∅})) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ((curry 𝐹‘𝑥)‘𝑦) = (𝑥𝐹𝑦))
8	7	eqeq1d 2738	. . . . . . . 8 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐵 ∈ (𝑉 ∖ {∅})) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (((curry 𝐹‘𝑥)‘𝑦) = 𝑧 ↔ (𝑥𝐹𝑦) = 𝑧))
9		eqcom 2743	. . . . . . . 8 ⊢ ((𝑥𝐹𝑦) = 𝑧 ↔ 𝑧 = (𝑥𝐹𝑦))
10	8, 9	bitrdi 286	. . . . . . 7 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐵 ∈ (𝑉 ∖ {∅})) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (((curry 𝐹‘𝑥)‘𝑦) = 𝑧 ↔ 𝑧 = (𝑥𝐹𝑦)))
11	3, 10	sylan 580	. . . . . 6 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (((curry 𝐹‘𝑥)‘𝑦) = 𝑧 ↔ 𝑧 = (𝑥𝐹𝑦)))
12		curf 36056	. . . . . . . . . 10 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → curry 𝐹:𝐴⟶(𝐶 ↑m 𝐵))
13	12	ffvelcdmda 7035	. . . . . . . . 9 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) ∧ 𝑥 ∈ 𝐴) → (curry 𝐹‘𝑥) ∈ (𝐶 ↑m 𝐵))
14		elmapfn 8803	. . . . . . . . 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 6895	. . . . . . . 8 ⊢ (((curry 𝐹‘𝑥) Fn 𝐵 ∧ 𝑦 ∈ 𝐵) → (((curry 𝐹‘𝑥)‘𝑦) = 𝑧 ↔ 𝑦(curry 𝐹‘𝑥)𝑧))
17	15, 16	sylan 580	. . . . . . 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 467	. . . . . 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 529	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑧 = (𝑥𝐹𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = (𝑥𝐹𝑦))))
20	19	adantl 482	. . . . . 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 308	. . . . 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 5106	. . . . . . . . . . 11 ⊢ (𝑦(curry 𝐹‘𝑥)𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ (curry 𝐹‘𝑥))
23		elfvdm 6879	. . . . . . . . . . 11 ⊢ (⟨𝑦, 𝑧⟩ ∈ (curry 𝐹‘𝑥) → 𝑥 ∈ dom curry 𝐹)
24	22, 23	sylbi 216	. . . . . . . . . 10 ⊢ (𝑦(curry 𝐹‘𝑥)𝑧 → 𝑥 ∈ dom curry 𝐹)
25		fdm 6677	. . . . . . . . . . . 12 ⊢ (curry 𝐹:𝐴⟶(𝐶 ↑m 𝐵) → dom curry 𝐹 = 𝐴)
26	25	eleq2d 2823	. . . . . . . . . . 11 ⊢ (curry 𝐹:𝐴⟶(𝐶 ↑m 𝐵) → (𝑥 ∈ dom curry 𝐹 ↔ 𝑥 ∈ 𝐴))
27	26	biimpa 477	. . . . . . . . . 10 ⊢ ((curry 𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ dom curry 𝐹) → 𝑥 ∈ 𝐴)
28	24, 27	sylan2 593	. . . . . . . . 9 ⊢ ((curry 𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑦(curry 𝐹‘𝑥)𝑧) → 𝑥 ∈ 𝐴)
29		ffvelcdm 7032	. . . . . . . . . . . . 13 ⊢ ((curry 𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → (curry 𝐹‘𝑥) ∈ (𝐶 ↑m 𝐵))
30		elmapi 8787	. . . . . . . . . . . . 13 ⊢ ((curry 𝐹‘𝑥) ∈ (𝐶 ↑m 𝐵) → (curry 𝐹‘𝑥):𝐵⟶𝐶)
31		fdm 6677	. . . . . . . . . . . . 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 3449	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
34		vex 3449	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
35	33, 34	breldm 5864	. . . . . . . . . . . 12 ⊢ (𝑦(curry 𝐹‘𝑥)𝑧 → 𝑦 ∈ dom (curry 𝐹‘𝑥))
36		eleq2 2826	. . . . . . . . . . . . 13 ⊢ (dom (curry 𝐹‘𝑥) = 𝐵 → (𝑦 ∈ dom (curry 𝐹‘𝑥) ↔ 𝑦 ∈ 𝐵))
37	36	biimpa 477	. . . . . . . . . . . 12 ⊢ ((dom (curry 𝐹‘𝑥) = 𝐵 ∧ 𝑦 ∈ dom (curry 𝐹‘𝑥)) → 𝑦 ∈ 𝐵)
38	32, 35, 37	syl2an 596	. . . . . . . . . . 11 ⊢ (((curry 𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦(curry 𝐹‘𝑥)𝑧) → 𝑦 ∈ 𝐵)
39	38	an32s 650	. . . . . . . . . 10 ⊢ (((curry 𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑦(curry 𝐹‘𝑥)𝑧) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵)
40	28, 39	mpdan 685	. . . . . . . . 9 ⊢ ((curry 𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑦(curry 𝐹‘𝑥)𝑧) → 𝑦 ∈ 𝐵)
41	28, 40	jca 512	. . . . . . . 8 ⊢ ((curry 𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑦(curry 𝐹‘𝑥)𝑧) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
42	12, 41	sylan 580	. . . . . . 7 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) ∧ 𝑦(curry 𝐹‘𝑥)𝑧) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
43	42	stoic1a 1774	. . . . . 6 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ¬ 𝑦(curry 𝐹‘𝑥)𝑧)
44		simpl 483	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = (𝑥𝐹𝑦)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
45	44	con3i 154	. . . . . . 7 ⊢ (¬ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ¬ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = (𝑥𝐹𝑦)))
46	45	adantl 482	. . . . . 6 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ¬ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = (𝑥𝐹𝑦)))
47	43, 46	2falsed 376	. . . . 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 811	. . . 4 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → (𝑦(curry 𝐹‘𝑥)𝑧 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = (𝑥𝐹𝑦))))
49	48	oprabbidv 7423	. . 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 8199	. . 3 ⊢ uncurry curry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(curry 𝐹‘𝑥)𝑧}
51		df-mpo 7362	. . 3 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = (𝑥𝐹𝑦))}
52	49, 50, 51	3eqtr4g 2801	. 2 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → uncurry curry 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦)))
53		fnov 7487	. . . 4 ⊢ (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦)))
54	1, 53	sylib 217	. . 3 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦)))
55	54	3ad2ant1 1133	. 2 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦)))
56	52, 55	eqtr4d 2779	1 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → uncurry curry 𝐹 = 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ∖ cdif 3907  ∅c0 4282  {csn 4586  ⟨cop 4592   class class class wbr 5105   × cxp 5631  dom cdm 5633   Fn wfn 6491  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  {coprab 7358   ∈ cmpo 7359  curry ccur 8196  uncurry cunc 8197   ↑_m cmap 8765

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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-cur 8198  df-unc 8199  df-map 8767

This theorem is referenced by: (None)