Theorem curunc 36470

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 484	. . 3 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝐵 ≠ ∅) → 𝐹:𝐴⟶(𝐶 ↑m 𝐵))
2	1	feqmptd 6961	. 2 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝐵 ≠ ∅) → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
3		uncf 36467	. . . . . . . 8 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶)
4	3	fdmd 6729	. . . . . . 7 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → dom uncurry 𝐹 = (𝐴 × 𝐵))
5	4	dmeqd 5906	. . . . . 6 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → dom dom uncurry 𝐹 = dom (𝐴 × 𝐵))
6		dmxp 5929	. . . . . 6 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
7	5, 6	sylan9eq 2793	. . . . 5 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝐵 ≠ ∅) → dom dom uncurry 𝐹 = 𝐴)
8	7	eqcomd 2739	. . . 4 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝐵 ≠ ∅) → 𝐴 = dom dom uncurry 𝐹)
9		df-mpt 5233	. . . . . 6 ⊢ (𝑦 ∈ 𝐵 ↦ ((𝐹‘𝑥)‘𝑦)) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐵 ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦))}
10		ffvelcdm 7084	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐶 ↑m 𝐵))
11		elmapi 8843	. . . . . . . 8 ⊢ ((𝐹‘𝑥) ∈ (𝐶 ↑m 𝐵) → (𝐹‘𝑥):𝐵⟶𝐶)
12	10, 11	syl 17	. . . . . . 7 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥):𝐵⟶𝐶)
13	12	feqmptd 6961	. . . . . 6 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝑦 ∈ 𝐵 ↦ ((𝐹‘𝑥)‘𝑦)))
14		ffun 6721	. . . . . . . . . 10 ⊢ (uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶 → Fun uncurry 𝐹)
15		funbrfv2b 6950	. . . . . . . . . 10 ⊢ (Fun uncurry 𝐹 → (⟨𝑥, 𝑦⟩uncurry 𝐹𝑧 ↔ (⟨𝑥, 𝑦⟩ ∈ dom uncurry 𝐹 ∧ (uncurry 𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
16	3, 14, 15	3syl 18	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → (⟨𝑥, 𝑦⟩uncurry 𝐹𝑧 ↔ (⟨𝑥, 𝑦⟩ ∈ dom uncurry 𝐹 ∧ (uncurry 𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
17	16	adantr 482	. . . . . . . 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 2820	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → (⟨𝑥, 𝑦⟩ ∈ dom uncurry 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
19		opelxp 5713	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
20	19	baib 537	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ 𝑦 ∈ 𝐵))
21	18, 20	sylan9bb 511	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → (⟨𝑥, 𝑦⟩ ∈ dom uncurry 𝐹 ↔ 𝑦 ∈ 𝐵))
22		df-ov 7412	. . . . . . . . . . . . 13 ⊢ (𝑥uncurry 𝐹𝑦) = (uncurry 𝐹‘⟨𝑥, 𝑦⟩)
23		uncov 36469	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥uncurry 𝐹𝑦) = ((𝐹‘𝑥)‘𝑦))
24	23	el2v 3483	. . . . . . . . . . . . 13 ⊢ (𝑥uncurry 𝐹𝑦) = ((𝐹‘𝑥)‘𝑦)
25	22, 24	eqtr3i 2763	. . . . . . . . . . . 12 ⊢ (uncurry 𝐹‘⟨𝑥, 𝑦⟩) = ((𝐹‘𝑥)‘𝑦)
26	25	eqeq1i 2738	. . . . . . . . . . 11 ⊢ ((uncurry 𝐹‘⟨𝑥, 𝑦⟩) = 𝑧 ↔ ((𝐹‘𝑥)‘𝑦) = 𝑧)
27		eqcom 2740	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥)‘𝑦) = 𝑧 ↔ 𝑧 = ((𝐹‘𝑥)‘𝑦))
28	26, 27	bitri 275	. . . . . . . . . 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 632	. . . . . . . 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 279	. . . . . . 7 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → (⟨𝑥, 𝑦⟩uncurry 𝐹𝑧 ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦))))
32	31	opabbidv 5215	. . . . . 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 2799	. . . . 5 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩uncurry 𝐹𝑧})
34	33	adantlr 714	. . . 4 ⊢ (((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝐵 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩uncurry 𝐹𝑧})
35	8, 34	mpteq12dva 5238	. . 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 8252	. . 3 ⊢ curry uncurry 𝐹 = (𝑥 ∈ dom dom uncurry 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩uncurry 𝐹𝑧})
37	35, 36	eqtr4di 2791	. 2 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝐵 ≠ ∅) → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = curry uncurry 𝐹)
38	2, 37	eqtr2d 2774	1 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝐵 ≠ ∅) → curry uncurry 𝐹 = 𝐹)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   ≠ wne 2941  Vcvv 3475  ∅c0 4323  ⟨cop 4635   class class class wbr 5149  {copab 5211   ↦ cmpt 5232   × cxp 5675  dom cdm 5677  Fun wfun 6538  ⟶wf 6540  ‘cfv 6544  (class class class)co 7409  curry ccur 8250  uncurry cunc 8251   ↑_m cmap 8820

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-cur 8252  df-unc 8253  df-map 8822

This theorem is referenced by: (None)