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Theorem curunc 37610
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 6976 . 2 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝐵 ≠ ∅) → 𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
3 uncf 37607 . . . . . . . 8 (𝐹:𝐴⟶(𝐶m 𝐵) → uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶)
43fdmd 6745 . . . . . . 7 (𝐹:𝐴⟶(𝐶m 𝐵) → dom uncurry 𝐹 = (𝐴 × 𝐵))
54dmeqd 5915 . . . . . 6 (𝐹:𝐴⟶(𝐶m 𝐵) → dom dom uncurry 𝐹 = dom (𝐴 × 𝐵))
6 dmxp 5938 . . . . . 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 5225 . . . . . 6 (𝑦𝐵 ↦ ((𝐹𝑥)‘𝑦)) = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐵𝑧 = ((𝐹𝑥)‘𝑦))}
10 ffvelcdm 7100 . . . . . . . 8 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → (𝐹𝑥) ∈ (𝐶m 𝐵))
11 elmapi 8890 . . . . . . . 8 ((𝐹𝑥) ∈ (𝐶m 𝐵) → (𝐹𝑥):𝐵𝐶)
1210, 11syl 17 . . . . . . 7 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → (𝐹𝑥):𝐵𝐶)
1312feqmptd 6976 . . . . . 6 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → (𝐹𝑥) = (𝑦𝐵 ↦ ((𝐹𝑥)‘𝑦)))
14 ffun 6738 . . . . . . . . . 10 (uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶 → Fun uncurry 𝐹)
15 funbrfv2b 6965 . . . . . . . . . 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 2826 . . . . . . . . . 10 (𝐹:𝐴⟶(𝐶m 𝐵) → (⟨𝑥, 𝑦⟩ ∈ dom uncurry 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
19 opelxp 5720 . . . . . . . . . . 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 7435 . . . . . . . . . . . . 13 (𝑥uncurry 𝐹𝑦) = (uncurry 𝐹‘⟨𝑥, 𝑦⟩)
23 uncov 37609 . . . . . . . . . . . . . 14 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥uncurry 𝐹𝑦) = ((𝐹𝑥)‘𝑦))
2423el2v 3486 . . . . . . . . . . . . 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 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 5208 . . . . . 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 715 . . . 4 (((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝐵 ≠ ∅) ∧ 𝑥𝐴) → (𝐹𝑥) = {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩uncurry 𝐹𝑧})
358, 34mpteq12dva 5230 . . 3 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝐵 ≠ ∅) → (𝑥𝐴 ↦ (𝐹𝑥)) = (𝑥 ∈ dom dom uncurry 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩uncurry 𝐹𝑧}))
36 df-cur 8293 . . 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 206  wa 395   = wceq 1539  wcel 2107  wne 2939  Vcvv 3479  c0 4332  cop 4631   class class class wbr 5142  {copab 5204  cmpt 5224   × cxp 5682  dom cdm 5684  Fun wfun 6554  wf 6556  cfv 6560  (class class class)co 7432  curry ccur 8291  uncurry cunc 8292  m cmap 8867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-cur 8293  df-unc 8294  df-map 8869
This theorem is referenced by: (None)
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