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Theorem unccur 33718
Description: Uncurrying of currying. (Contributed by Brendan Leahy, 5-Jun-2021.)
Assertion
Ref Expression
unccur ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → uncurry curry 𝐹 = 𝐹)

Proof of Theorem unccur
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ffn 6183 . . . . . . . . 9 (𝐹:(𝐴 × 𝐵)⟶𝐶𝐹 Fn (𝐴 × 𝐵))
21anim1i 602 . . . . . . . 8 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅})) → (𝐹 Fn (𝐴 × 𝐵) ∧ 𝐵 ∈ (𝑉 ∖ {∅})))
323adant3 1126 . . . . . . 7 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → (𝐹 Fn (𝐴 × 𝐵) ∧ 𝐵 ∈ (𝑉 ∖ {∅})))
4 3anass 1080 . . . . . . . . . . 11 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝑥𝐴𝑦𝐵) ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ (𝑥𝐴𝑦𝐵)))
5 curfv 33715 . . . . . . . . . . 11 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝑥𝐴𝑦𝐵) ∧ 𝐵 ∈ (𝑉 ∖ {∅})) → ((curry 𝐹𝑥)‘𝑦) = (𝑥𝐹𝑦))
64, 5sylanbr 571 . . . . . . . . . 10 (((𝐹 Fn (𝐴 × 𝐵) ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝐵 ∈ (𝑉 ∖ {∅})) → ((curry 𝐹𝑥)‘𝑦) = (𝑥𝐹𝑦))
76an32s 631 . . . . . . . . 9 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐵 ∈ (𝑉 ∖ {∅})) ∧ (𝑥𝐴𝑦𝐵)) → ((curry 𝐹𝑥)‘𝑦) = (𝑥𝐹𝑦))
87eqeq1d 2773 . . . . . . . 8 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐵 ∈ (𝑉 ∖ {∅})) ∧ (𝑥𝐴𝑦𝐵)) → (((curry 𝐹𝑥)‘𝑦) = 𝑧 ↔ (𝑥𝐹𝑦) = 𝑧))
9 eqcom 2778 . . . . . . . 8 ((𝑥𝐹𝑦) = 𝑧𝑧 = (𝑥𝐹𝑦))
108, 9syl6bb 276 . . . . . . 7 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐵 ∈ (𝑉 ∖ {∅})) ∧ (𝑥𝐴𝑦𝐵)) → (((curry 𝐹𝑥)‘𝑦) = 𝑧𝑧 = (𝑥𝐹𝑦)))
113, 10sylan 569 . . . . . 6 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ (𝑥𝐴𝑦𝐵)) → (((curry 𝐹𝑥)‘𝑦) = 𝑧𝑧 = (𝑥𝐹𝑦)))
12 curf 33713 . . . . . . . . . 10 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → curry 𝐹:𝐴⟶(𝐶𝑚 𝐵))
1312ffvelrnda 6500 . . . . . . . . 9 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ 𝑥𝐴) → (curry 𝐹𝑥) ∈ (𝐶𝑚 𝐵))
14 elmapfn 8030 . . . . . . . . 9 ((curry 𝐹𝑥) ∈ (𝐶𝑚 𝐵) → (curry 𝐹𝑥) Fn 𝐵)
1513, 14syl 17 . . . . . . . 8 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ 𝑥𝐴) → (curry 𝐹𝑥) Fn 𝐵)
16 fnbrfvb 6375 . . . . . . . 8 (((curry 𝐹𝑥) Fn 𝐵𝑦𝐵) → (((curry 𝐹𝑥)‘𝑦) = 𝑧𝑦(curry 𝐹𝑥)𝑧))
1715, 16sylan 569 . . . . . . 7 ((((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ 𝑥𝐴) ∧ 𝑦𝐵) → (((curry 𝐹𝑥)‘𝑦) = 𝑧𝑦(curry 𝐹𝑥)𝑧))
1817anasss 452 . . . . . 6 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ (𝑥𝐴𝑦𝐵)) → (((curry 𝐹𝑥)‘𝑦) = 𝑧𝑦(curry 𝐹𝑥)𝑧))
19 ibar 518 . . . . . . 7 ((𝑥𝐴𝑦𝐵) → (𝑧 = (𝑥𝐹𝑦) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = (𝑥𝐹𝑦))))
2019adantl 467 . . . . . 6 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ (𝑥𝐴𝑦𝐵)) → (𝑧 = (𝑥𝐹𝑦) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = (𝑥𝐹𝑦))))
2111, 18, 203bitr3d 298 . . . . 5 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ (𝑥𝐴𝑦𝐵)) → (𝑦(curry 𝐹𝑥)𝑧 ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = (𝑥𝐹𝑦))))
22 df-br 4787 . . . . . . . . . . 11 (𝑦(curry 𝐹𝑥)𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ (curry 𝐹𝑥))
23 elfvdm 6359 . . . . . . . . . . 11 (⟨𝑦, 𝑧⟩ ∈ (curry 𝐹𝑥) → 𝑥 ∈ dom curry 𝐹)
2422, 23sylbi 207 . . . . . . . . . 10 (𝑦(curry 𝐹𝑥)𝑧𝑥 ∈ dom curry 𝐹)
25 fdm 6189 . . . . . . . . . . . 12 (curry 𝐹:𝐴⟶(𝐶𝑚 𝐵) → dom curry 𝐹 = 𝐴)
2625eleq2d 2836 . . . . . . . . . . 11 (curry 𝐹:𝐴⟶(𝐶𝑚 𝐵) → (𝑥 ∈ dom curry 𝐹𝑥𝐴))
2726biimpa 462 . . . . . . . . . 10 ((curry 𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝑥 ∈ dom curry 𝐹) → 𝑥𝐴)
2824, 27sylan2 580 . . . . . . . . 9 ((curry 𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝑦(curry 𝐹𝑥)𝑧) → 𝑥𝐴)
29 ffvelrn 6498 . . . . . . . . . . . . 13 ((curry 𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝑥𝐴) → (curry 𝐹𝑥) ∈ (𝐶𝑚 𝐵))
30 elmapi 8029 . . . . . . . . . . . . 13 ((curry 𝐹𝑥) ∈ (𝐶𝑚 𝐵) → (curry 𝐹𝑥):𝐵𝐶)
31 fdm 6189 . . . . . . . . . . . . 13 ((curry 𝐹𝑥):𝐵𝐶 → dom (curry 𝐹𝑥) = 𝐵)
3229, 30, 313syl 18 . . . . . . . . . . . 12 ((curry 𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝑥𝐴) → dom (curry 𝐹𝑥) = 𝐵)
33 vex 3354 . . . . . . . . . . . . 13 𝑦 ∈ V
34 vex 3354 . . . . . . . . . . . . 13 𝑧 ∈ V
3533, 34breldm 5465 . . . . . . . . . . . 12 (𝑦(curry 𝐹𝑥)𝑧𝑦 ∈ dom (curry 𝐹𝑥))
36 eleq2 2839 . . . . . . . . . . . . 13 (dom (curry 𝐹𝑥) = 𝐵 → (𝑦 ∈ dom (curry 𝐹𝑥) ↔ 𝑦𝐵))
3736biimpa 462 . . . . . . . . . . . 12 ((dom (curry 𝐹𝑥) = 𝐵𝑦 ∈ dom (curry 𝐹𝑥)) → 𝑦𝐵)
3832, 35, 37syl2an 583 . . . . . . . . . . 11 (((curry 𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝑥𝐴) ∧ 𝑦(curry 𝐹𝑥)𝑧) → 𝑦𝐵)
3938an32s 631 . . . . . . . . . 10 (((curry 𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝑦(curry 𝐹𝑥)𝑧) ∧ 𝑥𝐴) → 𝑦𝐵)
4028, 39mpdan 667 . . . . . . . . 9 ((curry 𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝑦(curry 𝐹𝑥)𝑧) → 𝑦𝐵)
4128, 40jca 501 . . . . . . . 8 ((curry 𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝑦(curry 𝐹𝑥)𝑧) → (𝑥𝐴𝑦𝐵))
4212, 41sylan 569 . . . . . . 7 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ 𝑦(curry 𝐹𝑥)𝑧) → (𝑥𝐴𝑦𝐵))
4342stoic1a 1845 . . . . . 6 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ ¬ (𝑥𝐴𝑦𝐵)) → ¬ 𝑦(curry 𝐹𝑥)𝑧)
44 simpl 468 . . . . . . . 8 (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = (𝑥𝐹𝑦)) → (𝑥𝐴𝑦𝐵))
4544con3i 151 . . . . . . 7 (¬ (𝑥𝐴𝑦𝐵) → ¬ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = (𝑥𝐹𝑦)))
4645adantl 467 . . . . . 6 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ ¬ (𝑥𝐴𝑦𝐵)) → ¬ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = (𝑥𝐹𝑦)))
4743, 462falsed 365 . . . . 5 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ ¬ (𝑥𝐴𝑦𝐵)) → (𝑦(curry 𝐹𝑥)𝑧 ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = (𝑥𝐹𝑦))))
4821, 47pm2.61dan 814 . . . 4 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → (𝑦(curry 𝐹𝑥)𝑧 ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = (𝑥𝐹𝑦))))
4948oprabbidv 6854 . . 3 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(curry 𝐹𝑥)𝑧} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = (𝑥𝐹𝑦))})
50 df-unc 7544 . . 3 uncurry curry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(curry 𝐹𝑥)𝑧}
51 df-mpt2 6796 . . 3 (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = (𝑥𝐹𝑦))}
5249, 50, 513eqtr4g 2830 . 2 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → uncurry curry 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦)))
53 fnov 6913 . . . 4 (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦)))
541, 53sylib 208 . . 3 (𝐹:(𝐴 × 𝐵)⟶𝐶𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦)))
55543ad2ant1 1127 . 2 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦)))
5652, 55eqtr4d 2808 1 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → uncurry curry 𝐹 = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  cdif 3720  c0 4063  {csn 4316  cop 4322   class class class wbr 4786   × cxp 5247  dom cdm 5249   Fn wfn 6024  wf 6025  cfv 6029  (class class class)co 6791  {coprab 6792  cmpt2 6793  curry ccur 7541  uncurry cunc 7542  𝑚 cmap 8007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-1st 7313  df-2nd 7314  df-cur 7543  df-unc 7544  df-map 8009
This theorem is referenced by: (None)
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