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Theorem curf 34738
 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 5590 . . . . . . . 8 ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
2 ffvelrn 6844 . . . . . . . 8 ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝐶)
31, 2sylan2 592 . . . . . . 7 ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ (𝑥𝐴𝑦𝐵)) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝐶)
43anassrs 468 . . . . . 6 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝑥𝐴) ∧ 𝑦𝐵) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝐶)
54fmpttd 6874 . . . . 5 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝑥𝐴) → (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)):𝐵𝐶)
653ad2antl1 1179 . . . 4 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ 𝑥𝐴) → (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)):𝐵𝐶)
7 elmapg 8412 . . . . . . 7 ((𝐶𝑊𝐵 ∈ (𝑉 ∖ {∅})) → ((𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) ∈ (𝐶m 𝐵) ↔ (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)):𝐵𝐶))
87ancoms 459 . . . . . 6 ((𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → ((𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) ∈ (𝐶m 𝐵) ↔ (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)):𝐵𝐶))
983adant1 1124 . . . . 5 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → ((𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) ∈ (𝐶m 𝐵) ↔ (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)):𝐵𝐶))
109adantr 481 . . . 4 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ 𝑥𝐴) → ((𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) ∈ (𝐶m 𝐵) ↔ (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)):𝐵𝐶))
116, 10mpbird 258 . . 3 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ 𝑥𝐴) → (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) ∈ (𝐶m 𝐵))
1211fmpttd 6874 . 2 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → (𝑥𝐴 ↦ (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))):𝐴⟶(𝐶m 𝐵))
13 eldifsni 4720 . . . 4 (𝐵 ∈ (𝑉 ∖ {∅}) → 𝐵 ≠ ∅)
14 df-cur 7927 . . . . . 6 curry 𝐹 = (𝑥 ∈ dom dom 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐹𝑧})
15 fdm 6518 . . . . . . . . . 10 (𝐹:(𝐴 × 𝐵)⟶𝐶 → dom 𝐹 = (𝐴 × 𝐵))
1615dmeqd 5772 . . . . . . . . 9 (𝐹:(𝐴 × 𝐵)⟶𝐶 → dom dom 𝐹 = dom (𝐴 × 𝐵))
17 dmxp 5797 . . . . . . . . 9 (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
1816, 17sylan9eq 2880 . . . . . . . 8 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ≠ ∅) → dom dom 𝐹 = 𝐴)
1918mpteq1d 5151 . . . . . . 7 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ≠ ∅) → (𝑥 ∈ dom dom 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐹𝑧}) = (𝑥𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐹𝑧}))
20 ffun 6513 . . . . . . . . . . . . . 14 (𝐹:(𝐴 × 𝐵)⟶𝐶 → Fun 𝐹)
21 funbrfv2b 6719 . . . . . . . . . . . . . 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 2902 . . . . . . . . . . . . . . 15 (𝐹:(𝐴 × 𝐵)⟶𝐶 → (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
24 opelxp 5589 . . . . . . . . . . . . . . 15 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
2523, 24syl6bb 288 . . . . . . . . . . . . . 14 (𝐹:(𝐴 × 𝐵)⟶𝐶 → (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ (𝑥𝐴𝑦𝐵)))
2625anbi1d 629 . . . . . . . . . . . . 13 (𝐹:(𝐴 × 𝐵)⟶𝐶 → ((⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ ((𝑥𝐴𝑦𝐵) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
2722, 26bitrd 280 . . . . . . . . . . . 12 (𝐹:(𝐴 × 𝐵)⟶𝐶 → (⟨𝑥, 𝑦𝐹𝑧 ↔ ((𝑥𝐴𝑦𝐵) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
28 ibar 529 . . . . . . . . . . . . 13 (𝑥𝐴 → ((𝑦𝐵𝑧 = (𝐹‘⟨𝑥, 𝑦⟩)) ↔ (𝑥𝐴 ∧ (𝑦𝐵𝑧 = (𝐹‘⟨𝑥, 𝑦⟩)))))
29 anass 469 . . . . . . . . . . . . . 14 (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩)) ↔ (𝑥𝐴 ∧ (𝑦𝐵𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))))
30 eqcom 2831 . . . . . . . . . . . . . . 15 (𝑧 = (𝐹‘⟨𝑥, 𝑦⟩) ↔ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)
3130anbi2i 622 . . . . . . . . . . . . . 14 (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩)) ↔ ((𝑥𝐴𝑦𝐵) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧))
3229, 31bitr3i 278 . . . . . . . . . . . . 13 ((𝑥𝐴 ∧ (𝑦𝐵𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))) ↔ ((𝑥𝐴𝑦𝐵) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧))
3328, 32syl6rbb 289 . . . . . . . . . . . 12 (𝑥𝐴 → (((𝑥𝐴𝑦𝐵) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ (𝑦𝐵𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))))
3427, 33sylan9bb 510 . . . . . . . . . . 11 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝑥𝐴) → (⟨𝑥, 𝑦𝐹𝑧 ↔ (𝑦𝐵𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))))
3534opabbidv 5128 . . . . . . . . . 10 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝑥𝐴) → {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐹𝑧} = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐵𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))})
36 df-mpt 5143 . . . . . . . . . 10 (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐵𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))}
3735, 36syl6eqr 2878 . . . . . . . . 9 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝑥𝐴) → {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐹𝑧} = (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))
3837mpteq2dva 5157 . . . . . . . 8 (𝐹:(𝐴 × 𝐵)⟶𝐶 → (𝑥𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐹𝑧}) = (𝑥𝐴 ↦ (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
3938adantr 481 . . . . . . 7 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ≠ ∅) → (𝑥𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐹𝑧}) = (𝑥𝐴 ↦ (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
4019, 39eqtrd 2860 . . . . . 6 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ≠ ∅) → (𝑥 ∈ dom dom 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐹𝑧}) = (𝑥𝐴 ↦ (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
4114, 40syl5eq 2872 . . . . 5 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ≠ ∅) → curry 𝐹 = (𝑥𝐴 ↦ (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
4241feq1d 6495 . . . 4 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ≠ ∅) → (curry 𝐹:𝐴⟶(𝐶m 𝐵) ↔ (𝑥𝐴 ↦ (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))):𝐴⟶(𝐶m 𝐵)))
4313, 42sylan2 592 . . 3 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅})) → (curry 𝐹:𝐴⟶(𝐶m 𝐵) ↔ (𝑥𝐴 ↦ (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))):𝐴⟶(𝐶m 𝐵)))
44433adant3 1126 . 2 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → (curry 𝐹:𝐴⟶(𝐶m 𝐵) ↔ (𝑥𝐴 ↦ (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))):𝐴⟶(𝐶m 𝐵)))
4512, 44mpbird 258 1 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → curry 𝐹:𝐴⟶(𝐶m 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1081   = wceq 1530   ∈ wcel 2106   ≠ wne 3020   ∖ cdif 3936  ∅c0 4294  {csn 4563  ⟨cop 4569   class class class wbr 5062  {copab 5124   ↦ cmpt 5142   × cxp 5551  dom cdm 5553  Fun wfun 6345  ⟶wf 6347  ‘cfv 6351  (class class class)co 7151  curry ccur 7925   ↑m cmap 8399 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-cur 7927  df-map 8401 This theorem is referenced by:  unccur  34743  matunitlindflem1  34756  matunitlindflem2  34757
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