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Theorem curf 33832
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 5316 . . . . . . . 8 ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
2 ffvelrn 6551 . . . . . . . 8 ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝐶)
31, 2sylan2 586 . . . . . . 7 ((𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ (𝑥𝐴𝑦𝐵)) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝐶)
43anassrs 459 . . . . . 6 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝑥𝐴) ∧ 𝑦𝐵) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝐶)
54fmpttd 6579 . . . . 5 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝑥𝐴) → (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)):𝐵𝐶)
653ad2antl1 1236 . . . 4 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ 𝑥𝐴) → (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)):𝐵𝐶)
7 elmapg 8077 . . . . . . 7 ((𝐶𝑊𝐵 ∈ (𝑉 ∖ {∅})) → ((𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) ∈ (𝐶𝑚 𝐵) ↔ (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)):𝐵𝐶))
87ancoms 450 . . . . . 6 ((𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → ((𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) ∈ (𝐶𝑚 𝐵) ↔ (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)):𝐵𝐶))
983adant1 1160 . . . . 5 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → ((𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) ∈ (𝐶𝑚 𝐵) ↔ (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)):𝐵𝐶))
109adantr 472 . . . 4 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ 𝑥𝐴) → ((𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) ∈ (𝐶𝑚 𝐵) ↔ (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)):𝐵𝐶))
116, 10mpbird 248 . . 3 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ 𝑥𝐴) → (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) ∈ (𝐶𝑚 𝐵))
1211fmpttd 6579 . 2 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → (𝑥𝐴 ↦ (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))):𝐴⟶(𝐶𝑚 𝐵))
13 eldifsni 4478 . . . 4 (𝐵 ∈ (𝑉 ∖ {∅}) → 𝐵 ≠ ∅)
14 df-cur 7600 . . . . . 6 curry 𝐹 = (𝑥 ∈ dom dom 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐹𝑧})
15 fdm 6233 . . . . . . . . . 10 (𝐹:(𝐴 × 𝐵)⟶𝐶 → dom 𝐹 = (𝐴 × 𝐵))
1615dmeqd 5496 . . . . . . . . 9 (𝐹:(𝐴 × 𝐵)⟶𝐶 → dom dom 𝐹 = dom (𝐴 × 𝐵))
17 dmxp 5514 . . . . . . . . 9 (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
1816, 17sylan9eq 2819 . . . . . . . 8 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ≠ ∅) → dom dom 𝐹 = 𝐴)
1918mpteq1d 4899 . . . . . . 7 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ≠ ∅) → (𝑥 ∈ dom dom 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐹𝑧}) = (𝑥𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐹𝑧}))
20 ffun 6228 . . . . . . . . . . . . . 14 (𝐹:(𝐴 × 𝐵)⟶𝐶 → Fun 𝐹)
21 funbrfv2b 6433 . . . . . . . . . . . . . 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 2830 . . . . . . . . . . . . . . 15 (𝐹:(𝐴 × 𝐵)⟶𝐶 → (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
24 opelxp 5315 . . . . . . . . . . . . . . 15 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
2523, 24syl6bb 278 . . . . . . . . . . . . . 14 (𝐹:(𝐴 × 𝐵)⟶𝐶 → (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ (𝑥𝐴𝑦𝐵)))
2625anbi1d 623 . . . . . . . . . . . . 13 (𝐹:(𝐴 × 𝐵)⟶𝐶 → ((⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ ((𝑥𝐴𝑦𝐵) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
2722, 26bitrd 270 . . . . . . . . . . . 12 (𝐹:(𝐴 × 𝐵)⟶𝐶 → (⟨𝑥, 𝑦𝐹𝑧 ↔ ((𝑥𝐴𝑦𝐵) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)))
28 ibar 524 . . . . . . . . . . . . 13 (𝑥𝐴 → ((𝑦𝐵𝑧 = (𝐹‘⟨𝑥, 𝑦⟩)) ↔ (𝑥𝐴 ∧ (𝑦𝐵𝑧 = (𝐹‘⟨𝑥, 𝑦⟩)))))
29 anass 460 . . . . . . . . . . . . . 14 (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩)) ↔ (𝑥𝐴 ∧ (𝑦𝐵𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))))
30 eqcom 2772 . . . . . . . . . . . . . . 15 (𝑧 = (𝐹‘⟨𝑥, 𝑦⟩) ↔ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧)
3130anbi2i 616 . . . . . . . . . . . . . 14 (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = (𝐹‘⟨𝑥, 𝑦⟩)) ↔ ((𝑥𝐴𝑦𝐵) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧))
3229, 31bitr3i 268 . . . . . . . . . . . . 13 ((𝑥𝐴 ∧ (𝑦𝐵𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))) ↔ ((𝑥𝐴𝑦𝐵) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧))
3328, 32syl6rbb 279 . . . . . . . . . . . 12 (𝑥𝐴 → (((𝑥𝐴𝑦𝐵) ∧ (𝐹‘⟨𝑥, 𝑦⟩) = 𝑧) ↔ (𝑦𝐵𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))))
3427, 33sylan9bb 505 . . . . . . . . . . 11 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝑥𝐴) → (⟨𝑥, 𝑦𝐹𝑧 ↔ (𝑦𝐵𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))))
3534opabbidv 4877 . . . . . . . . . 10 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝑥𝐴) → {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐹𝑧} = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐵𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))})
36 df-mpt 4891 . . . . . . . . . 10 (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)) = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐵𝑧 = (𝐹‘⟨𝑥, 𝑦⟩))}
3735, 36syl6eqr 2817 . . . . . . . . 9 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝑥𝐴) → {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐹𝑧} = (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩)))
3837mpteq2dva 4905 . . . . . . . 8 (𝐹:(𝐴 × 𝐵)⟶𝐶 → (𝑥𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐹𝑧}) = (𝑥𝐴 ↦ (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
3938adantr 472 . . . . . . 7 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ≠ ∅) → (𝑥𝐴 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐹𝑧}) = (𝑥𝐴 ↦ (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
4019, 39eqtrd 2799 . . . . . 6 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ≠ ∅) → (𝑥 ∈ dom dom 𝐹 ↦ {⟨𝑦, 𝑧⟩ ∣ ⟨𝑥, 𝑦𝐹𝑧}) = (𝑥𝐴 ↦ (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
4114, 40syl5eq 2811 . . . . 5 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ≠ ∅) → curry 𝐹 = (𝑥𝐴 ↦ (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
4241feq1d 6210 . . . 4 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ≠ ∅) → (curry 𝐹:𝐴⟶(𝐶𝑚 𝐵) ↔ (𝑥𝐴 ↦ (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))):𝐴⟶(𝐶𝑚 𝐵)))
4313, 42sylan2 586 . . 3 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅})) → (curry 𝐹:𝐴⟶(𝐶𝑚 𝐵) ↔ (𝑥𝐴 ↦ (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))):𝐴⟶(𝐶𝑚 𝐵)))
44433adant3 1162 . 2 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → (curry 𝐹:𝐴⟶(𝐶𝑚 𝐵) ↔ (𝑥𝐴 ↦ (𝑦𝐵 ↦ (𝐹‘⟨𝑥, 𝑦⟩))):𝐴⟶(𝐶𝑚 𝐵)))
4512, 44mpbird 248 1 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → curry 𝐹:𝐴⟶(𝐶𝑚 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2937  cdif 3731  c0 4081  {csn 4336  cop 4342   class class class wbr 4811  {copab 4873  cmpt 4890   × cxp 5277  dom cdm 5279  Fun wfun 6064  wf 6066  cfv 6070  (class class class)co 6846  curry ccur 7598  𝑚 cmap 8064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-fv 6078  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-cur 7600  df-map 8066
This theorem is referenced by:  unccur  33837  matunitlindflem1  33850  matunitlindflem2  33851
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