Theorem unccur 35687

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 6584	. . . . . . . . 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 1130	. . . . . . 7 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → (𝐹 Fn (𝐴 × 𝐵) ∧ 𝐵 ∈ (𝑉 ∖ {∅})))
4		3anass 1093	. . . . . . . . . . 11 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
5		curfv 35684	. . . . . . . . . . 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 648	. . . . . . . . 9 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐵 ∈ (𝑉 ∖ {∅})) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ((curry 𝐹‘𝑥)‘𝑦) = (𝑥𝐹𝑦))
8	7	eqeq1d 2740	. . . . . . . 8 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐵 ∈ (𝑉 ∖ {∅})) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (((curry 𝐹‘𝑥)‘𝑦) = 𝑧 ↔ (𝑥𝐹𝑦) = 𝑧))
9		eqcom 2745	. . . . . . . 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 579	. . . . . 6 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (((curry 𝐹‘𝑥)‘𝑦) = 𝑧 ↔ 𝑧 = (𝑥𝐹𝑦)))
12		curf 35682	. . . . . . . . . 10 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → curry 𝐹:𝐴⟶(𝐶 ↑m 𝐵))
13	12	ffvelrnda 6943	. . . . . . . . 9 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) ∧ 𝑥 ∈ 𝐴) → (curry 𝐹‘𝑥) ∈ (𝐶 ↑m 𝐵))
14		elmapfn 8611	. . . . . . . . 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 6804	. . . . . . . 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 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 5071	. . . . . . . . . . 11 ⊢ (𝑦(curry 𝐹‘𝑥)𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ (curry 𝐹‘𝑥))
23		elfvdm 6788	. . . . . . . . . . 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 6593	. . . . . . . . . . . 12 ⊢ (curry 𝐹:𝐴⟶(𝐶 ↑m 𝐵) → dom curry 𝐹 = 𝐴)
26	25	eleq2d 2824	. . . . . . . . . . 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		ffvelrn 6941	. . . . . . . . . . . . 13 ⊢ ((curry 𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → (curry 𝐹‘𝑥) ∈ (𝐶 ↑m 𝐵))
30		elmapi 8595	. . . . . . . . . . . . 13 ⊢ ((curry 𝐹‘𝑥) ∈ (𝐶 ↑m 𝐵) → (curry 𝐹‘𝑥):𝐵⟶𝐶)
31		fdm 6593	. . . . . . . . . . . . 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 3426	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
34		vex 3426	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
35	33, 34	breldm 5806	. . . . . . . . . . . 12 ⊢ (𝑦(curry 𝐹‘𝑥)𝑧 → 𝑦 ∈ dom (curry 𝐹‘𝑥))
36		eleq2 2827	. . . . . . . . . . . . 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 648	. . . . . . . . . 10 ⊢ (((curry 𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑦(curry 𝐹‘𝑥)𝑧) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵)
40	28, 39	mpdan 683	. . . . . . . . 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 1776	. . . . . 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 809	. . . 4 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → (𝑦(curry 𝐹‘𝑥)𝑧 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = (𝑥𝐹𝑦))))
49	48	oprabbidv 7319	. . 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 8055	. . 3 ⊢ uncurry curry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(curry 𝐹‘𝑥)𝑧}
51		df-mpo 7260	. . 3 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = (𝑥𝐹𝑦))}
52	49, 50, 51	3eqtr4g 2804	. 2 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → uncurry curry 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦)))
53		fnov 7383	. . . 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 1131	. 2 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦)))
56	52, 55	eqtr4d 2781	1 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ 𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶 ∈ 𝑊) → uncurry curry 𝐹 = 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ∖ cdif 3880  ∅c0 4253  {csn 4558  ⟨cop 4564   class class class wbr 5070   × cxp 5578  dom cdm 5580   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255  {coprab 7256   ∈ cmpo 7257  curry ccur 8052  uncurry cunc 8053   ↑_m cmap 8573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-cur 8054  df-unc 8055  df-map 8575

This theorem is referenced by: (None)