Theorem unccur 37563

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 6747	. . . . . . . . 9 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 → 𝐹 Fn (𝐴 × 𝐵))
2	1	anim1i 614	. . . . . . . 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 37560	. . . . . . . . . . 11 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝐵 ∈ (𝑉 ∖ {∅})) → ((curry 𝐹‘𝑥)‘𝑦) = (𝑥𝐹𝑦))
6	4, 5	sylanbr 581	. . . . . . . . . 10 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝐵 ∈ (𝑉 ∖ {∅})) → ((curry 𝐹‘𝑥)‘𝑦) = (𝑥𝐹𝑦))
7	6	an32s 651	. . . . . . . . 9 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐵 ∈ (𝑉 ∖ {∅})) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ((curry 𝐹‘𝑥)‘𝑦) = (𝑥𝐹𝑦))
8	7	eqeq1d 2742	. . . . . . . 8 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐵 ∈ (𝑉 ∖ {∅})) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (((curry 𝐹‘𝑥)‘𝑦) = 𝑧 ↔ (𝑥𝐹𝑦) = 𝑧))
9		eqcom 2747	. . . . . . . 8 ⊢ ((𝑥𝐹𝑦) = 𝑧 ↔ 𝑧 = (𝑥𝐹𝑦))
10	8, 9	bitrdi 287	. . . . . . 7 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐵 ∈ (𝑉 ∖ {∅})) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (((curry 𝐹‘𝑥)‘𝑦) = 𝑧 ↔ 𝑧 = (𝑥𝐹𝑦)))
11	3, 10	sylan 579	. . . . . 6 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (((curry 𝐹‘𝑥)‘𝑦) = 𝑧 ↔ 𝑧 = (𝑥𝐹𝑦)))
12		curf 37558	. . . . . . . . . 10 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → curry 𝐹:𝐴⟶(𝐶 ↑m 𝐵))
13	12	ffvelcdmda 7118	. . . . . . . . 9 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) ∧ 𝑥 ∈ 𝐴) → (curry 𝐹‘𝑥) ∈ (𝐶 ↑m 𝐵))
14		elmapfn 8923	. . . . . . . . 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 6973	. . . . . . . 8 ⊢ (((curry 𝐹‘𝑥) Fn 𝐵 ∧ 𝑦 ∈ 𝐵) → (((curry 𝐹‘𝑥)‘𝑦) = 𝑧 ↔ 𝑦(curry 𝐹‘𝑥)𝑧))
17	15, 16	sylan 579	. . . . . . 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 466	. . . . . 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 528	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑧 = (𝑥𝐹𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = (𝑥𝐹𝑦))))
20	19	adantl 481	. . . . . 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 309	. . . . 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 5167	. . . . . . . . . . 11 ⊢ (𝑦(curry 𝐹‘𝑥)𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ (curry 𝐹‘𝑥))
23		elfvdm 6957	. . . . . . . . . . 11 ⊢ (⟨𝑦, 𝑧⟩ ∈ (curry 𝐹‘𝑥) → 𝑥 ∈ dom curry 𝐹)
24	22, 23	sylbi 217	. . . . . . . . . 10 ⊢ (𝑦(curry 𝐹‘𝑥)𝑧 → 𝑥 ∈ dom curry 𝐹)
25		fdm 6756	. . . . . . . . . . . 12 ⊢ (curry 𝐹:𝐴⟶(𝐶 ↑m 𝐵) → dom curry 𝐹 = 𝐴)
26	25	eleq2d 2830	. . . . . . . . . . 11 ⊢ (curry 𝐹:𝐴⟶(𝐶 ↑m 𝐵) → (𝑥 ∈ dom curry 𝐹 ↔ 𝑥 ∈ 𝐴))
27	26	biimpa 476	. . . . . . . . . 10 ⊢ ((curry 𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ dom curry 𝐹) → 𝑥 ∈ 𝐴)
28	24, 27	sylan2 592	. . . . . . . . 9 ⊢ ((curry 𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑦(curry 𝐹‘𝑥)𝑧) → 𝑥 ∈ 𝐴)
29		ffvelcdm 7115	. . . . . . . . . . . . 13 ⊢ ((curry 𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → (curry 𝐹‘𝑥) ∈ (𝐶 ↑m 𝐵))
30		elmapi 8907	. . . . . . . . . . . . 13 ⊢ ((curry 𝐹‘𝑥) ∈ (𝐶 ↑m 𝐵) → (curry 𝐹‘𝑥):𝐵⟶𝐶)
31		fdm 6756	. . . . . . . . . . . . 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 3492	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
34		vex 3492	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
35	33, 34	breldm 5933	. . . . . . . . . . . 12 ⊢ (𝑦(curry 𝐹‘𝑥)𝑧 → 𝑦 ∈ dom (curry 𝐹‘𝑥))
36		eleq2 2833	. . . . . . . . . . . . 13 ⊢ (dom (curry 𝐹‘𝑥) = 𝐵 → (𝑦 ∈ dom (curry 𝐹‘𝑥) ↔ 𝑦 ∈ 𝐵))
37	36	biimpa 476	. . . . . . . . . . . 12 ⊢ ((dom (curry 𝐹‘𝑥) = 𝐵 ∧ 𝑦 ∈ dom (curry 𝐹‘𝑥)) → 𝑦 ∈ 𝐵)
38	32, 35, 37	syl2an 595	. . . . . . . . . . 11 ⊢ (((curry 𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦(curry 𝐹‘𝑥)𝑧) → 𝑦 ∈ 𝐵)
39	38	an32s 651	. . . . . . . . . 10 ⊢ (((curry 𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑦(curry 𝐹‘𝑥)𝑧) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵)
40	28, 39	mpdan 686	. . . . . . . . 9 ⊢ ((curry 𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑦(curry 𝐹‘𝑥)𝑧) → 𝑦 ∈ 𝐵)
41	28, 40	jca 511	. . . . . . . 8 ⊢ ((curry 𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑦(curry 𝐹‘𝑥)𝑧) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
42	12, 41	sylan 579	. . . . . . 7 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) ∧ 𝑦(curry 𝐹‘𝑥)𝑧) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
43	42	stoic1a 1770	. . . . . 6 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ¬ 𝑦(curry 𝐹‘𝑥)𝑧)
44		simpl 482	. . . . . . . 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 481	. . . . . 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 812	. . . 4 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → (𝑦(curry 𝐹‘𝑥)𝑧 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = (𝑥𝐹𝑦))))
49	48	oprabbidv 7516	. . 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 8309	. . 3 ⊢ uncurry curry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(curry 𝐹‘𝑥)𝑧}
51		df-mpo 7453	. . 3 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = (𝑥𝐹𝑦))}
52	49, 50, 51	3eqtr4g 2805	. 2 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → uncurry curry 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦)))
53		fnov 7581	. . . 4 ⊢ (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦)))
54	1, 53	sylib 218	. . 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 2783	1 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → uncurry curry 𝐹 = 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ∖ cdif 3973  ∅c0 4352  {csn 4648  ⟨cop 4654   class class class wbr 5166   × cxp 5698  dom cdm 5700   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  {coprab 7449   ∈ cmpo 7450  curry ccur 8306  uncurry cunc 8307   ↑_m cmap 8884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-cur 8308  df-unc 8309  df-map 8886

This theorem is referenced by: (None)