Theorem curunc 38717

Description: Currying of uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021.)

Assertion

Ref	Expression
curunc	⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝐵 ≠ ∅) → curry uncurry 𝐹 = 𝐹)

Proof of Theorem curunc

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

Step	Hyp	Ref	Expression
1		simpl 488	. . 3 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝐵 ≠ ∅) → 𝐹:𝐴⟶(𝐶 ↑m 𝐵))
2	1	feqmptd 6976	. 2 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝐵 ≠ ∅) → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
3		uncf 8919	. . . . . . . 8 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶)
4	3	fdmd 6742	. . . . . . 7 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → dom uncurry 𝐹 = (𝐴 × 𝐵))
5	4	dmeqd 5894	. . . . . 6 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → dom dom uncurry 𝐹 = dom (𝐴 × 𝐵))
6		dmxp 5918	. . . . . 6 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
7	5, 6	sylan9eq 2818	. . . . 5 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝐵 ≠ ∅) → dom dom uncurry 𝐹 = 𝐴)
8	7	eqcomd 2769	. . . 4 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝐵 ≠ ∅) → 𝐴 = dom dom uncurry 𝐹)
9		df-mpt 5190	. . . . . 6 ⊢ (𝑦 ∈ 𝐵 ↦ ((𝐹‘𝑥)‘𝑦)) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐵 ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦))}
10		ffvelcdm 7105	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐶 ↑m 𝐵))
11		elmapi 8897	. . . . . . . 8 ⊢ ((𝐹‘𝑥) ∈ (𝐶 ↑m 𝐵) → (𝐹‘𝑥):𝐵⟶𝐶)
12	10, 11	syl 18	. . . . . . 7 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥):𝐵⟶𝐶)
13	12	feqmptd 6976	. . . . . 6 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝑦 ∈ 𝐵 ↦ ((𝐹‘𝑥)‘𝑦)))
14		ffun 6734	. . . . . . . . . 10 ⊢ (uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶 → Fun uncurry 𝐹)
15		funbrfv2b 6965	. . . . . . . . . 10 ⊢ (Fun uncurry 𝐹 → (⟨𝑥, 𝑦⟩uncurry 𝐹𝑧 ↔ (⟨𝑥, 𝑦⟩ ∈ dom uncurry 𝐹 ∧ (uncurry 𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
16	3, 14, 15	3syl 19	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → (⟨𝑥, 𝑦⟩uncurry 𝐹𝑧 ↔ (⟨𝑥, 𝑦⟩ ∈ dom uncurry 𝐹 ∧ (uncurry 𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
17	16	adantr 486	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → (⟨𝑥, 𝑦⟩uncurry 𝐹𝑧 ↔ (⟨𝑥, 𝑦⟩ ∈ dom uncurry 𝐹 ∧ (uncurry 𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
18	4	eleq2d 2850	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → (⟨𝑥, 𝑦⟩ ∈ dom uncurry 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
19		opelxp 5691	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
20	19	baib 545	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ 𝑦 ∈ 𝐵))
21	18, 20	sylan9bb 519	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → (⟨𝑥, 𝑦⟩ ∈ dom uncurry 𝐹 ↔ 𝑦 ∈ 𝐵))
22		df-ov 7446	. . . . . . . . . . . . 13 ⊢ (𝑥uncurry 𝐹𝑦) = (uncurry 𝐹‘⟨𝑥, 𝑦⟩)
23		uncov 8921	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥uncurry 𝐹𝑦) = ((𝐹‘𝑥)‘𝑦))
24	23	el2v 3461	. . . . . . . . . . . . 13 ⊢ (𝑥uncurry 𝐹𝑦) = ((𝐹‘𝑥)‘𝑦)
25	22, 24	eqtr3i 2788	. . . . . . . . . . . 12 ⊢ (uncurry 𝐹‘⟨𝑥, 𝑦⟩) = ((𝐹‘𝑥)‘𝑦)
26	25	eqeq1i 2768	. . . . . . . . . . 11 ⊢ ((uncurry 𝐹‘⟨𝑥, 𝑦⟩) = 𝑧 ↔ ((𝐹‘𝑥)‘𝑦) = 𝑧)
27		eqcom 2770	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥)‘𝑦) = 𝑧 ↔ 𝑧 = ((𝐹‘𝑥)‘𝑦))
28	26, 27	bitri 278	. . . . . . . . . 10 ⊢ ((uncurry 𝐹‘⟨𝑥, 𝑦⟩) = 𝑧 ↔ 𝑧 = ((𝐹‘𝑥)‘𝑦))
29	28	a1i 11	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → ((uncurry 𝐹‘⟨𝑥, 𝑦⟩) = 𝑧 ↔ 𝑧 = ((𝐹‘𝑥)‘𝑦)))
30	21, 29	anbi12d 644	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → ((⟨𝑥, 𝑦⟩ ∈ dom uncurry 𝐹 ∧ (uncurry 𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦))))
31	17, 30	bitrd 282	. . . . . . 7 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → (⟨𝑥, 𝑦⟩uncurry 𝐹𝑧 ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦))))
32	31	opabbidv 5174	. . . . . 6 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩uncurry 𝐹𝑧} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐵 ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦))})
33	9, 13, 32	3eqtr4a 2824	. . . . 5 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩uncurry 𝐹𝑧})
34	33	adantlr 728	. . . 4 ⊢ (((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝐵 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩uncurry 𝐹𝑧})
35	8, 34	mpteq12dva 5194	. . 3 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝐵 ≠ ∅) → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = (𝑥 ∈ dom dom uncurry 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩uncurry 𝐹𝑧}))
36		df-cur 8312	. . 3 ⊢ curry uncurry 𝐹 = (𝑥 ∈ dom dom uncurry 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩uncurry 𝐹𝑧})
37	35, 36	eqtr4di 2816	. 2 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝐵 ≠ ∅) → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = curry uncurry 𝐹)
38	2, 37	eqtr2d 2799	1 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝐵 ≠ ∅) → curry uncurry 𝐹 = 𝐹)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  Vcvv 3454  ∅c0 4281  ⟨cop 4838   class class class wbr 5106  {copab 5170   ↦ cmpt 5189   × cxp 5653  dom cdm 5655  Fun wfun 6552  ⟶wf 6554  ‘cfv 6558  (class class class)co 7443  curry ccur 8310  uncurry cunc 8311   ↑_m cmap 8875

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-1st 8025  df-2nd 8026  df-cur 8312  df-unc 8313  df-map 8877

This theorem is used by: (None)