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Theorem curf 37585
Description: Functional property of currying. (Contributed by Brendan Leahy, 2-Jun-2021.)
Assertion
Ref Expression
curf ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → curry 𝐹:𝐴⟶(𝐶m 𝐵))

Proof of Theorem curf
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opelxpi 5726 . . . . . . . 8 ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
2 ffvelcdm 7101 . . . . . . . 8 ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝐶)
31, 2sylan2 593 . . . . . . 7 ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ (𝑥𝐴𝑦𝐵)) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝐶)
43anassrs 467 . . . . . 6 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝑥𝐴) ∧ 𝑦𝐵) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝐶)
54fmpttd 7135 . . . . 5 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝑥𝐴) → (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)):𝐵𝐶)
653ad2antl1 1184 . . . 4 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ 𝑥𝐴) → (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)):𝐵𝐶)
7 elmapg 8878 . . . . . . 7 ((𝐶𝑊𝐵 ∈ (𝑉 ∖ {∅})) → ((𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) ∈ (𝐶m 𝐵) ↔ (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)):𝐵𝐶))
87ancoms 458 . . . . . 6 ((𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → ((𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) ∈ (𝐶m 𝐵) ↔ (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)):𝐵𝐶))
983adant1 1129 . . . . 5 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → ((𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) ∈ (𝐶m 𝐵) ↔ (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)):𝐵𝐶))
109adantr 480 . . . 4 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ 𝑥𝐴) → ((𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) ∈ (𝐶m 𝐵) ↔ (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)):𝐵𝐶))
116, 10mpbird 257 . . 3 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ 𝑥𝐴) → (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) ∈ (𝐶m 𝐵))
1211fmpttd 7135 . 2 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → (𝑥𝐴 ↦ (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))):𝐴⟶(𝐶m 𝐵))
13 eldifsni 4795 . . . 4 (𝐵 ∈ (𝑉 ∖ {∅}) → 𝐵 ≠ ∅)
14 df-cur 8291 . . . . . 6 curry 𝐹 = (𝑥 ∈ dom dom 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐹𝑧})
15 fdm 6746 . . . . . . . . . 10 (𝐹:(𝐴 × 𝐵)⟶𝐶 → dom 𝐹 = (𝐴 × 𝐵))
1615dmeqd 5919 . . . . . . . . 9 (𝐹:(𝐴 × 𝐵)⟶𝐶 → dom dom 𝐹 = dom (𝐴 × 𝐵))
17 dmxp 5942 . . . . . . . . 9 (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
1816, 17sylan9eq 2795 . . . . . . . 8 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ≠ ∅) → dom dom 𝐹 = 𝐴)
1918mpteq1d 5243 . . . . . . 7 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ≠ ∅) → (𝑥 ∈ dom dom 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐹𝑧}) = (𝑥𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐹𝑧}))
20 ffun 6740 . . . . . . . . . . . . . 14 (𝐹:(𝐴 × 𝐵)⟶𝐶 → Fun 𝐹)
21 funbrfv2b 6966 . . . . . . . . . . . . . 14 (Fun 𝐹 → (⟨𝑥, 𝑦𝐹𝑧 ↔ (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
2220, 21syl 17 . . . . . . . . . . . . 13 (𝐹:(𝐴 × 𝐵)⟶𝐶 → (⟨𝑥, 𝑦𝐹𝑧 ↔ (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
2315eleq2d 2825 . . . . . . . . . . . . . . 15 (𝐹:(𝐴 × 𝐵)⟶𝐶 → (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
24 opelxp 5725 . . . . . . . . . . . . . . 15 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
2523, 24bitrdi 287 . . . . . . . . . . . . . 14 (𝐹:(𝐴 × 𝐵)⟶𝐶 → (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ (𝑥𝐴𝑦𝐵)))
2625anbi1d 631 . . . . . . . . . . . . 13 (𝐹:(𝐴 × 𝐵)⟶𝐶 → ((⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ ((𝑥𝐴𝑦𝐵) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
2722, 26bitrd 279 . . . . . . . . . . . 12 (𝐹:(𝐴 × 𝐵)⟶𝐶 → (⟨𝑥, 𝑦𝐹𝑧 ↔ ((𝑥𝐴𝑦𝐵) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
28 ibar 528 . . . . . . . . . . . . 13 (𝑥𝐴 → ((𝑦𝐵𝑧 = (𝐹‘⟨𝑥, 𝑦⟩)) ↔ (𝑥𝐴 ∧ (𝑦𝐵𝑧 = (𝐹‘⟨𝑥, 𝑦⟩)))))
29 anass 468 . . . . . . . . . . . . . 14 (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩)) ↔ (𝑥𝐴 ∧ (𝑦𝐵𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))))
30 eqcom 2742 . . . . . . . . . . . . . . 15 (𝑧 = (𝐹‘⟨𝑥, 𝑦⟩) ↔ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)
3130anbi2i 623 . . . . . . . . . . . . . 14 (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩)) ↔ ((𝑥𝐴𝑦𝐵) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧))
3229, 31bitr3i 277 . . . . . . . . . . . . 13 ((𝑥𝐴 ∧ (𝑦𝐵𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))) ↔ ((𝑥𝐴𝑦𝐵) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧))
3328, 32bitr2di 288 . . . . . . . . . . . 12 (𝑥𝐴 → (((𝑥𝐴𝑦𝐵) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ (𝑦𝐵𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))))
3427, 33sylan9bb 509 . . . . . . . . . . 11 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝑥𝐴) → (⟨𝑥, 𝑦𝐹𝑧 ↔ (𝑦𝐵𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))))
3534opabbidv 5214 . . . . . . . . . 10 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝑥𝐴) → {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐹𝑧} = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐵𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))})
36 df-mpt 5232 . . . . . . . . . 10 (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐵𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))}
3735, 36eqtr4di 2793 . . . . . . . . 9 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝑥𝐴) → {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐹𝑧} = (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))
3837mpteq2dva 5248 . . . . . . . 8 (𝐹:(𝐴 × 𝐵)⟶𝐶 → (𝑥𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐹𝑧}) = (𝑥𝐴 ↦ (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
3938adantr 480 . . . . . . 7 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ≠ ∅) → (𝑥𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐹𝑧}) = (𝑥𝐴 ↦ (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
4019, 39eqtrd 2775 . . . . . 6 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ≠ ∅) → (𝑥 ∈ dom dom 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐹𝑧}) = (𝑥𝐴 ↦ (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
4114, 40eqtrid 2787 . . . . 5 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ≠ ∅) → curry 𝐹 = (𝑥𝐴 ↦ (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
4241feq1d 6721 . . . 4 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ≠ ∅) → (curry 𝐹:𝐴⟶(𝐶m 𝐵) ↔ (𝑥𝐴 ↦ (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))):𝐴⟶(𝐶m 𝐵)))
4313, 42sylan2 593 . . 3 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅})) → (curry 𝐹:𝐴⟶(𝐶m 𝐵) ↔ (𝑥𝐴 ↦ (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))):𝐴⟶(𝐶m 𝐵)))
44433adant3 1131 . 2 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → (curry 𝐹:𝐴⟶(𝐶m 𝐵) ↔ (𝑥𝐴 ↦ (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))):𝐴⟶(𝐶m 𝐵)))
4512, 44mpbird 257 1 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → curry 𝐹:𝐴⟶(𝐶m 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  cdif 3960  c0 4339  {csn 4631  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  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-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-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-cur 8291  df-map 8867
This theorem is referenced by:  unccur  37590  matunitlindflem1  37603  matunitlindflem2  37604
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