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Theorem curunc 37589
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 482 . . 3 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝐵 ≠ ∅) → 𝐹:𝐴⟶(𝐶m 𝐵))
21feqmptd 6977 . 2 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝐵 ≠ ∅) → 𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
3 uncf 37586 . . . . . . . 8 (𝐹:𝐴⟶(𝐶m 𝐵) → uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶)
43fdmd 6747 . . . . . . 7 (𝐹:𝐴⟶(𝐶m 𝐵) → dom uncurry 𝐹 = (𝐴 × 𝐵))
54dmeqd 5919 . . . . . 6 (𝐹:𝐴⟶(𝐶m 𝐵) → dom dom uncurry 𝐹 = dom (𝐴 × 𝐵))
6 dmxp 5942 . . . . . 6 (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
75, 6sylan9eq 2795 . . . . 5 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝐵 ≠ ∅) → dom dom uncurry 𝐹 = 𝐴)
87eqcomd 2741 . . . 4 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝐵 ≠ ∅) → 𝐴 = dom dom uncurry 𝐹)
9 df-mpt 5232 . . . . . 6 (𝑦𝐵 ↦ ((𝐹𝑥)‘𝑦)) = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐵𝑧 = ((𝐹𝑥)‘𝑦))}
10 ffvelcdm 7101 . . . . . . . 8 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → (𝐹𝑥) ∈ (𝐶m 𝐵))
11 elmapi 8888 . . . . . . . 8 ((𝐹𝑥) ∈ (𝐶m 𝐵) → (𝐹𝑥):𝐵𝐶)
1210, 11syl 17 . . . . . . 7 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → (𝐹𝑥):𝐵𝐶)
1312feqmptd 6977 . . . . . 6 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → (𝐹𝑥) = (𝑦𝐵 ↦ ((𝐹𝑥)‘𝑦)))
14 ffun 6740 . . . . . . . . . 10 (uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶 → Fun uncurry 𝐹)
15 funbrfv2b 6966 . . . . . . . . . 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 480 . . . . . . . 8 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → (⟨𝑥, 𝑦⟩uncurry 𝐹𝑧 ↔ (⟨𝑥, 𝑦⟩ ∈ dom uncurry 𝐹 ∧ (uncurry 𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
184eleq2d 2825 . . . . . . . . . 10 (𝐹:𝐴⟶(𝐶m 𝐵) → (⟨𝑥, 𝑦⟩ ∈ dom uncurry 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
19 opelxp 5725 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
2019baib 535 . . . . . . . . . 10 (𝑥𝐴 → (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ 𝑦𝐵))
2118, 20sylan9bb 509 . . . . . . . . 9 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → (⟨𝑥, 𝑦⟩ ∈ dom uncurry 𝐹𝑦𝐵))
22 df-ov 7434 . . . . . . . . . . . . 13 (𝑥uncurry 𝐹𝑦) = (uncurry 𝐹‘⟨𝑥, 𝑦⟩)
23 uncov 37588 . . . . . . . . . . . . . 14 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥uncurry 𝐹𝑦) = ((𝐹𝑥)‘𝑦))
2423el2v 3485 . . . . . . . . . . . . 13 (𝑥uncurry 𝐹𝑦) = ((𝐹𝑥)‘𝑦)
2522, 24eqtr3i 2765 . . . . . . . . . . . 12 (uncurry 𝐹‘⟨𝑥, 𝑦⟩) = ((𝐹𝑥)‘𝑦)
2625eqeq1i 2740 . . . . . . . . . . 11 ((uncurry 𝐹‘⟨𝑥, 𝑦⟩) = 𝑧 ↔ ((𝐹𝑥)‘𝑦) = 𝑧)
27 eqcom 2742 . . . . . . . . . . 11 (((𝐹𝑥)‘𝑦) = 𝑧𝑧 = ((𝐹𝑥)‘𝑦))
2826, 27bitri 275 . . . . . . . . . 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 632 . . . . . . . 8 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → ((⟨𝑥, 𝑦⟩ ∈ dom uncurry 𝐹 ∧ (uncurry 𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ (𝑦𝐵𝑧 = ((𝐹𝑥)‘𝑦))))
3117, 30bitrd 279 . . . . . . 7 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → (⟨𝑥, 𝑦⟩uncurry 𝐹𝑧 ↔ (𝑦𝐵𝑧 = ((𝐹𝑥)‘𝑦))))
3231opabbidv 5214 . . . . . 6 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩uncurry 𝐹𝑧} = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐵𝑧 = ((𝐹𝑥)‘𝑦))})
339, 13, 323eqtr4a 2801 . . . . 5 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → (𝐹𝑥) = {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩uncurry 𝐹𝑧})
3433adantlr 715 . . . 4 (((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝐵 ≠ ∅) ∧ 𝑥𝐴) → (𝐹𝑥) = {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩uncurry 𝐹𝑧})
358, 34mpteq12dva 5237 . . 3 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝐵 ≠ ∅) → (𝑥𝐴 ↦ (𝐹𝑥)) = (𝑥 ∈ dom dom uncurry 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩uncurry 𝐹𝑧}))
36 df-cur 8291 . . 3 curry uncurry 𝐹 = (𝑥 ∈ dom dom uncurry 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩uncurry 𝐹𝑧})
3735, 36eqtr4di 2793 . 2 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝐵 ≠ ∅) → (𝑥𝐴 ↦ (𝐹𝑥)) = curry uncurry 𝐹)
382, 37eqtr2d 2776 1 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝐵 ≠ ∅) → curry uncurry 𝐹 = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wne 2938  Vcvv 3478  c0 4339  cop 4637   class class class wbr 5148  {copab 5210  cmpt 5231   × cxp 5687  dom cdm 5689  Fun wfun 6557  wf 6559  cfv 6563  (class class class)co 7431  curry ccur 8289  uncurry cunc 8290  m cmap 8865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-cur 8291  df-unc 8292  df-map 8867
This theorem is referenced by: (None)
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