Theorem curunc 38618

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 6953	. 2 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝐵 ≠ ∅) → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
3		uncf 8894	. . . . . . . 8 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶)
4	3	fdmd 6720	. . . . . . 7 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → dom uncurry 𝐹 = (𝐴 × 𝐵))
5	4	dmeqd 5886	. . . . . 6 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → dom dom uncurry 𝐹 = dom (𝐴 × 𝐵))
6		dmxp 5910	. . . . . 6 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
7	5, 6	sylan9eq 2815	. . . . 5 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝐵 ≠ ∅) → dom dom uncurry 𝐹 = 𝐴)
8	7	eqcomd 2766	. . . 4 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝐵 ≠ ∅) → 𝐴 = dom dom uncurry 𝐹)
9		df-mpt 5186	. . . . . 6 ⊢ (𝑦 ∈ 𝐵 ↦ ((𝐹‘𝑥)‘𝑦)) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐵 ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦))}
10		ffvelcdm 7081	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐶 ↑m 𝐵))
11		elmapi 8872	. . . . . . . 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 6953	. . . . . 6 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝑦 ∈ 𝐵 ↦ ((𝐹‘𝑥)‘𝑦)))
14		ffun 6712	. . . . . . . . . 10 ⊢ (uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶 → Fun uncurry 𝐹)
15		funbrfv2b 6942	. . . . . . . . . 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 2846	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → (⟨𝑥, 𝑦⟩ ∈ dom uncurry 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
19		opelxp 5686	. . . . . . . . . . 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 7423	. . . . . . . . . . . . 13 ⊢ (𝑥uncurry 𝐹𝑦) = (uncurry 𝐹‘⟨𝑥, 𝑦⟩)
23		uncov 8896	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥uncurry 𝐹𝑦) = ((𝐹‘𝑥)‘𝑦))
24	23	el2v 3457	. . . . . . . . . . . . 13 ⊢ (𝑥uncurry 𝐹𝑦) = ((𝐹‘𝑥)‘𝑦)
25	22, 24	eqtr3i 2785	. . . . . . . . . . . 12 ⊢ (uncurry 𝐹‘⟨𝑥, 𝑦⟩) = ((𝐹‘𝑥)‘𝑦)
26	25	eqeq1i 2765	. . . . . . . . . . 11 ⊢ ((uncurry 𝐹‘⟨𝑥, 𝑦⟩) = 𝑧 ↔ ((𝐹‘𝑥)‘𝑦) = 𝑧)
27		eqcom 2767	. . . . . . . . . . 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 5170	. . . . . 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 2821	. . . . 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 5190	. . 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 8287	. . 3 ⊢ curry uncurry 𝐹 = (𝑥 ∈ dom dom uncurry 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩uncurry 𝐹𝑧})
37	35, 36	eqtr4di 2813	. 2 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝐵 ≠ ∅) → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = curry uncurry 𝐹)
38	2, 37	eqtr2d 2796	1 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝐵 ≠ ∅) → curry uncurry 𝐹 = 𝐹)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  Vcvv 3450  ∅c0 4278  ⟨cop 4834   class class class wbr 5102  {copab 5166   ↦ cmpt 5185   × cxp 5648  dom cdm 5650  Fun wfun 6532  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  curry ccur 8285  uncurry cunc 8286   ↑_m cmap 8850

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-cur 8287  df-unc 8288  df-map 8852

This theorem is used by: (None)