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Theorem curunc 32561
 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 472 . . 3 ((𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝐵 ≠ ∅) → 𝐹:𝐴⟶(𝐶𝑚 𝐵))
21feqmptd 6159 . 2 ((𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝐵 ≠ ∅) → 𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
3 uncf 32558 . . . . . . . 8 (𝐹:𝐴⟶(𝐶𝑚 𝐵) → uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶)
4 fdm 5964 . . . . . . . 8 (uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶 → dom uncurry 𝐹 = (𝐴 × 𝐵))
53, 4syl 17 . . . . . . 7 (𝐹:𝐴⟶(𝐶𝑚 𝐵) → dom uncurry 𝐹 = (𝐴 × 𝐵))
65dmeqd 5248 . . . . . 6 (𝐹:𝐴⟶(𝐶𝑚 𝐵) → dom dom uncurry 𝐹 = dom (𝐴 × 𝐵))
7 dmxp 5265 . . . . . 6 (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
86, 7sylan9eq 2664 . . . . 5 ((𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝐵 ≠ ∅) → dom dom uncurry 𝐹 = 𝐴)
98eqcomd 2616 . . . 4 ((𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝐵 ≠ ∅) → 𝐴 = dom dom uncurry 𝐹)
10 df-mpt 4645 . . . . . 6 (𝑦𝐵 ↦ ((𝐹𝑥)‘𝑦)) = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐵𝑧 = ((𝐹𝑥)‘𝑦))}
11 ffvelrn 6265 . . . . . . . 8 ((𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝑥𝐴) → (𝐹𝑥) ∈ (𝐶𝑚 𝐵))
12 elmapi 7765 . . . . . . . 8 ((𝐹𝑥) ∈ (𝐶𝑚 𝐵) → (𝐹𝑥):𝐵𝐶)
1311, 12syl 17 . . . . . . 7 ((𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝑥𝐴) → (𝐹𝑥):𝐵𝐶)
1413feqmptd 6159 . . . . . 6 ((𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝑥𝐴) → (𝐹𝑥) = (𝑦𝐵 ↦ ((𝐹𝑥)‘𝑦)))
15 ffun 5961 . . . . . . . . . 10 (uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶 → Fun uncurry 𝐹)
16 funbrfv2b 6150 . . . . . . . . . 10 (Fun uncurry 𝐹 → (⟨𝑥, 𝑦⟩uncurry 𝐹𝑧 ↔ (⟨𝑥, 𝑦⟩ ∈ dom uncurry 𝐹 ∧ (uncurry 𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
173, 15, 163syl 18 . . . . . . . . 9 (𝐹:𝐴⟶(𝐶𝑚 𝐵) → (⟨𝑥, 𝑦⟩uncurry 𝐹𝑧 ↔ (⟨𝑥, 𝑦⟩ ∈ dom uncurry 𝐹 ∧ (uncurry 𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
1817adantr 480 . . . . . . . 8 ((𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝑥𝐴) → (⟨𝑥, 𝑦⟩uncurry 𝐹𝑧 ↔ (⟨𝑥, 𝑦⟩ ∈ dom uncurry 𝐹 ∧ (uncurry 𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
195eleq2d 2673 . . . . . . . . . 10 (𝐹:𝐴⟶(𝐶𝑚 𝐵) → (⟨𝑥, 𝑦⟩ ∈ dom uncurry 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
20 opelxp 5070 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
2120baib 942 . . . . . . . . . 10 (𝑥𝐴 → (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ 𝑦𝐵))
2219, 21sylan9bb 732 . . . . . . . . 9 ((𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝑥𝐴) → (⟨𝑥, 𝑦⟩ ∈ dom uncurry 𝐹𝑦𝐵))
23 df-ov 6552 . . . . . . . . . . . . 13 (𝑥uncurry 𝐹𝑦) = (uncurry 𝐹‘⟨𝑥, 𝑦⟩)
24 vex 3176 . . . . . . . . . . . . . 14 𝑥 ∈ V
25 vex 3176 . . . . . . . . . . . . . 14 𝑦 ∈ V
26 uncov 32560 . . . . . . . . . . . . . 14 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥uncurry 𝐹𝑦) = ((𝐹𝑥)‘𝑦))
2724, 25, 26mp2an 704 . . . . . . . . . . . . 13 (𝑥uncurry 𝐹𝑦) = ((𝐹𝑥)‘𝑦)
2823, 27eqtr3i 2634 . . . . . . . . . . . 12 (uncurry 𝐹‘⟨𝑥, 𝑦⟩) = ((𝐹𝑥)‘𝑦)
2928eqeq1i 2615 . . . . . . . . . . 11 ((uncurry 𝐹‘⟨𝑥, 𝑦⟩) = 𝑧 ↔ ((𝐹𝑥)‘𝑦) = 𝑧)
30 eqcom 2617 . . . . . . . . . . 11 (((𝐹𝑥)‘𝑦) = 𝑧𝑧 = ((𝐹𝑥)‘𝑦))
3129, 30bitri 263 . . . . . . . . . 10 ((uncurry 𝐹‘⟨𝑥, 𝑦⟩) = 𝑧𝑧 = ((𝐹𝑥)‘𝑦))
3231a1i 11 . . . . . . . . 9 ((𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝑥𝐴) → ((uncurry 𝐹‘⟨𝑥, 𝑦⟩) = 𝑧𝑧 = ((𝐹𝑥)‘𝑦)))
3322, 32anbi12d 743 . . . . . . . 8 ((𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝑥𝐴) → ((⟨𝑥, 𝑦⟩ ∈ dom uncurry 𝐹 ∧ (uncurry 𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ (𝑦𝐵𝑧 = ((𝐹𝑥)‘𝑦))))
3418, 33bitrd 267 . . . . . . 7 ((𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝑥𝐴) → (⟨𝑥, 𝑦⟩uncurry 𝐹𝑧 ↔ (𝑦𝐵𝑧 = ((𝐹𝑥)‘𝑦))))
3534opabbidv 4648 . . . . . 6 ((𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝑥𝐴) → {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩uncurry 𝐹𝑧} = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐵𝑧 = ((𝐹𝑥)‘𝑦))})
3610, 14, 353eqtr4a 2670 . . . . 5 ((𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝑥𝐴) → (𝐹𝑥) = {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩uncurry 𝐹𝑧})
3736adantlr 747 . . . 4 (((𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝐵 ≠ ∅) ∧ 𝑥𝐴) → (𝐹𝑥) = {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩uncurry 𝐹𝑧})
389, 37mpteq12dva 4662 . . 3 ((𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝐵 ≠ ∅) → (𝑥𝐴 ↦ (𝐹𝑥)) = (𝑥 ∈ dom dom uncurry 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩uncurry 𝐹𝑧}))
39 df-cur 7280 . . 3 curry uncurry 𝐹 = (𝑥 ∈ dom dom uncurry 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩uncurry 𝐹𝑧})
4038, 39syl6eqr 2662 . 2 ((𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝐵 ≠ ∅) → (𝑥𝐴 ↦ (𝐹𝑥)) = curry uncurry 𝐹)
412, 40eqtr2d 2645 1 ((𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝐵 ≠ ∅) → curry uncurry 𝐹 = 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173  ∅c0 3874  ⟨cop 4131   class class class wbr 4583  {copab 4642   ↦ cmpt 4643   × cxp 5036  dom cdm 5038  Fun wfun 5798  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  curry ccur 7278  uncurry cunc 7279   ↑𝑚 cmap 7744 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-cur 7280  df-unc 7281  df-map 7746 This theorem is referenced by: (None)
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