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Theorem curunc 36089
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 484 . . 3 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝐵 ≠ ∅) → 𝐹:𝐴⟶(𝐶m 𝐵))
21feqmptd 6915 . 2 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝐵 ≠ ∅) → 𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
3 uncf 36086 . . . . . . . 8 (𝐹:𝐴⟶(𝐶m 𝐵) → uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶)
43fdmd 6684 . . . . . . 7 (𝐹:𝐴⟶(𝐶m 𝐵) → dom uncurry 𝐹 = (𝐴 × 𝐵))
54dmeqd 5866 . . . . . 6 (𝐹:𝐴⟶(𝐶m 𝐵) → dom dom uncurry 𝐹 = dom (𝐴 × 𝐵))
6 dmxp 5889 . . . . . 6 (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
75, 6sylan9eq 2797 . . . . 5 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝐵 ≠ ∅) → dom dom uncurry 𝐹 = 𝐴)
87eqcomd 2743 . . . 4 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝐵 ≠ ∅) → 𝐴 = dom dom uncurry 𝐹)
9 df-mpt 5194 . . . . . 6 (𝑦𝐵 ↦ ((𝐹𝑥)‘𝑦)) = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐵𝑧 = ((𝐹𝑥)‘𝑦))}
10 ffvelcdm 7037 . . . . . . . 8 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → (𝐹𝑥) ∈ (𝐶m 𝐵))
11 elmapi 8794 . . . . . . . 8 ((𝐹𝑥) ∈ (𝐶m 𝐵) → (𝐹𝑥):𝐵𝐶)
1210, 11syl 17 . . . . . . 7 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → (𝐹𝑥):𝐵𝐶)
1312feqmptd 6915 . . . . . 6 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → (𝐹𝑥) = (𝑦𝐵 ↦ ((𝐹𝑥)‘𝑦)))
14 ffun 6676 . . . . . . . . . 10 (uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶 → Fun uncurry 𝐹)
15 funbrfv2b 6905 . . . . . . . . . 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 482 . . . . . . . 8 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → (⟨𝑥, 𝑦⟩uncurry 𝐹𝑧 ↔ (⟨𝑥, 𝑦⟩ ∈ dom uncurry 𝐹 ∧ (uncurry 𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
184eleq2d 2824 . . . . . . . . . 10 (𝐹:𝐴⟶(𝐶m 𝐵) → (⟨𝑥, 𝑦⟩ ∈ dom uncurry 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
19 opelxp 5674 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
2019baib 537 . . . . . . . . . 10 (𝑥𝐴 → (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ 𝑦𝐵))
2118, 20sylan9bb 511 . . . . . . . . 9 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → (⟨𝑥, 𝑦⟩ ∈ dom uncurry 𝐹𝑦𝐵))
22 df-ov 7365 . . . . . . . . . . . . 13 (𝑥uncurry 𝐹𝑦) = (uncurry 𝐹‘⟨𝑥, 𝑦⟩)
23 uncov 36088 . . . . . . . . . . . . . 14 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥uncurry 𝐹𝑦) = ((𝐹𝑥)‘𝑦))
2423el2v 3456 . . . . . . . . . . . . 13 (𝑥uncurry 𝐹𝑦) = ((𝐹𝑥)‘𝑦)
2522, 24eqtr3i 2767 . . . . . . . . . . . 12 (uncurry 𝐹‘⟨𝑥, 𝑦⟩) = ((𝐹𝑥)‘𝑦)
2625eqeq1i 2742 . . . . . . . . . . 11 ((uncurry 𝐹‘⟨𝑥, 𝑦⟩) = 𝑧 ↔ ((𝐹𝑥)‘𝑦) = 𝑧)
27 eqcom 2744 . . . . . . . . . . 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 5176 . . . . . 6 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩uncurry 𝐹𝑧} = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐵𝑧 = ((𝐹𝑥)‘𝑦))})
339, 13, 323eqtr4a 2803 . . . . 5 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → (𝐹𝑥) = {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩uncurry 𝐹𝑧})
3433adantlr 714 . . . 4 (((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝐵 ≠ ∅) ∧ 𝑥𝐴) → (𝐹𝑥) = {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩uncurry 𝐹𝑧})
358, 34mpteq12dva 5199 . . 3 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝐵 ≠ ∅) → (𝑥𝐴 ↦ (𝐹𝑥)) = (𝑥 ∈ dom dom uncurry 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩uncurry 𝐹𝑧}))
36 df-cur 8203 . . 3 curry uncurry 𝐹 = (𝑥 ∈ dom dom uncurry 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦⟩uncurry 𝐹𝑧})
3735, 36eqtr4di 2795 . 2 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝐵 ≠ ∅) → (𝑥𝐴 ↦ (𝐹𝑥)) = curry uncurry 𝐹)
382, 37eqtr2d 2778 1 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝐵 ≠ ∅) → curry uncurry 𝐹 = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wne 2944  Vcvv 3448  c0 4287  cop 4597   class class class wbr 5110  {copab 5172  cmpt 5193   × cxp 5636  dom cdm 5638  Fun wfun 6495  wf 6497  cfv 6501  (class class class)co 7362  curry ccur 8201  uncurry cunc 8202  m cmap 8772
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 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7926  df-2nd 7927  df-cur 8203  df-unc 8204  df-map 8774
This theorem is referenced by: (None)
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