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Theorem curunc 36060
Description: Currying of uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021.)
Assertion
Ref Expression
curunc ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝐵 ≠ ∅) → curry uncurry 𝐹 = 𝐹)

Proof of Theorem curunc
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 483 . . 3 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝐵 ≠ ∅) → 𝐹:𝐴⟶(𝐶m 𝐵))
21feqmptd 6910 . 2 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝐵 ≠ ∅) → 𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
3 uncf 36057 . . . . . . . 8 (𝐹:𝐴⟶(𝐶m 𝐵) → uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶)
43fdmd 6679 . . . . . . 7 (𝐹:𝐴⟶(𝐶m 𝐵) → dom uncurry 𝐹 = (𝐴 × 𝐵))
54dmeqd 5861 . . . . . 6 (𝐹:𝐴⟶(𝐶m 𝐵) → dom dom uncurry 𝐹 = dom (𝐴 × 𝐵))
6 dmxp 5884 . . . . . 6 (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
75, 6sylan9eq 2796 . . . . 5 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝐵 ≠ ∅) → dom dom uncurry 𝐹 = 𝐴)
87eqcomd 2742 . . . 4 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝐵 ≠ ∅) → 𝐴 = dom dom uncurry 𝐹)
9 df-mpt 5189 . . . . . 6 (𝑦𝐵 ↦ ((𝐹𝑥)‘𝑦)) = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐵𝑧 = ((𝐹𝑥)‘𝑦))}
10 ffvelcdm 7032 . . . . . . . 8 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → (𝐹𝑥) ∈ (𝐶m 𝐵))
11 elmapi 8787 . . . . . . . 8 ((𝐹𝑥) ∈ (𝐶m 𝐵) → (𝐹𝑥):𝐵𝐶)
1210, 11syl 17 . . . . . . 7 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → (𝐹𝑥):𝐵𝐶)
1312feqmptd 6910 . . . . . 6 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → (𝐹𝑥) = (𝑦𝐵 ↦ ((𝐹𝑥)‘𝑦)))
14 ffun 6671 . . . . . . . . . 10 (uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶 → Fun uncurry 𝐹)
15 funbrfv2b 6900 . . . . . . . . . 10 (Fun uncurry 𝐹 → (⟨𝑥, 𝑦⟩uncurry 𝐹𝑧 ↔ (⟨𝑥, 𝑦⟩ ∈ dom uncurry 𝐹 ∧ (uncurry 𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
163, 14, 153syl 18 . . . . . . . . 9 (𝐹:𝐴⟶(𝐶m 𝐵) → (⟨𝑥, 𝑦⟩uncurry 𝐹𝑧 ↔ (⟨𝑥, 𝑦⟩ ∈ dom uncurry 𝐹 ∧ (uncurry 𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
1716adantr 481 . . . . . . . 8 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → (⟨𝑥, 𝑦⟩uncurry 𝐹𝑧 ↔ (⟨𝑥, 𝑦⟩ ∈ dom uncurry 𝐹 ∧ (uncurry 𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
184eleq2d 2823 . . . . . . . . . 10 (𝐹:𝐴⟶(𝐶m 𝐵) → (⟨𝑥, 𝑦⟩ ∈ dom uncurry 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
19 opelxp 5669 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
2019baib 536 . . . . . . . . . 10 (𝑥𝐴 → (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ 𝑦𝐵))
2118, 20sylan9bb 510 . . . . . . . . 9 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → (⟨𝑥, 𝑦⟩ ∈ dom uncurry 𝐹𝑦𝐵))
22 df-ov 7360 . . . . . . . . . . . . 13 (𝑥uncurry 𝐹𝑦) = (uncurry 𝐹‘⟨𝑥, 𝑦⟩)
23 uncov 36059 . . . . . . . . . . . . . 14 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥uncurry 𝐹𝑦) = ((𝐹𝑥)‘𝑦))
2423el2v 3453 . . . . . . . . . . . . 13 (𝑥uncurry 𝐹𝑦) = ((𝐹𝑥)‘𝑦)
2522, 24eqtr3i 2766 . . . . . . . . . . . 12 (uncurry 𝐹‘⟨𝑥, 𝑦⟩) = ((𝐹𝑥)‘𝑦)
2625eqeq1i 2741 . . . . . . . . . . 11 ((uncurry 𝐹‘⟨𝑥, 𝑦⟩) = 𝑧 ↔ ((𝐹𝑥)‘𝑦) = 𝑧)
27 eqcom 2743 . . . . . . . . . . 11 (((𝐹𝑥)‘𝑦) = 𝑧𝑧 = ((𝐹𝑥)‘𝑦))
2826, 27bitri 274 . . . . . . . . . 10 ((uncurry 𝐹‘⟨𝑥, 𝑦⟩) = 𝑧𝑧 = ((𝐹𝑥)‘𝑦))
2928a1i 11 . . . . . . . . 9 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → ((uncurry 𝐹‘⟨𝑥, 𝑦⟩) = 𝑧𝑧 = ((𝐹𝑥)‘𝑦)))
3021, 29anbi12d 631 . . . . . . . 8 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → ((⟨𝑥, 𝑦⟩ ∈ dom uncurry 𝐹 ∧ (uncurry 𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ (𝑦𝐵𝑧 = ((𝐹𝑥)‘𝑦))))
3117, 30bitrd 278 . . . . . . 7 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → (⟨𝑥, 𝑦⟩uncurry 𝐹𝑧 ↔ (𝑦𝐵𝑧 = ((𝐹𝑥)‘𝑦))))
3231opabbidv 5171 . . . . . 6 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩uncurry 𝐹𝑧} = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐵𝑧 = ((𝐹𝑥)‘𝑦))})
339, 13, 323eqtr4a 2802 . . . . 5 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → (𝐹𝑥) = {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩uncurry 𝐹𝑧})
3433adantlr 713 . . . 4 (((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝐵 ≠ ∅) ∧ 𝑥𝐴) → (𝐹𝑥) = {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩uncurry 𝐹𝑧})
358, 34mpteq12dva 5194 . . 3 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝐵 ≠ ∅) → (𝑥𝐴 ↦ (𝐹𝑥)) = (𝑥 ∈ dom dom uncurry 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩uncurry 𝐹𝑧}))
36 df-cur 8198 . . 3 curry uncurry 𝐹 = (𝑥 ∈ dom dom uncurry 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩uncurry 𝐹𝑧})
3735, 36eqtr4di 2794 . 2 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝐵 ≠ ∅) → (𝑥𝐴 ↦ (𝐹𝑥)) = curry uncurry 𝐹)
382, 37eqtr2d 2777 1 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝐵 ≠ ∅) → curry uncurry 𝐹 = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2943  Vcvv 3445  c0 4282  cop 4592   class class class wbr 5105  {copab 5167  cmpt 5188   × cxp 5631  dom cdm 5633  Fun wfun 6490  wf 6492  cfv 6496  (class class class)co 7357  curry ccur 8196  uncurry cunc 8197  m cmap 8765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-cur 8198  df-unc 8199  df-map 8767
This theorem is referenced by: (None)
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