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Theorem unccur 37304
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 6728 . . . . . . . . 9 (𝐹:(𝐴 × 𝐵)⟶𝐶𝐹 Fn (𝐴 × 𝐵))
21anim1i 613 . . . . . . . 8 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅})) → (𝐹 Fn (𝐴 × 𝐵) ∧ 𝐵 ∈ (𝑉 ∖ {∅})))
323adant3 1129 . . . . . . 7 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → (𝐹 Fn (𝐴 × 𝐵) ∧ 𝐵 ∈ (𝑉 ∖ {∅})))
4 3anass 1092 . . . . . . . . . . 11 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝑥𝐴𝑦𝐵) ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ (𝑥𝐴𝑦𝐵)))
5 curfv 37301 . . . . . . . . . . 11 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝑥𝐴𝑦𝐵) ∧ 𝐵 ∈ (𝑉 ∖ {∅})) → ((curry 𝐹𝑥)‘𝑦) = (𝑥𝐹𝑦))
64, 5sylanbr 580 . . . . . . . . . 10 (((𝐹 Fn (𝐴 × 𝐵) ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝐵 ∈ (𝑉 ∖ {∅})) → ((curry 𝐹𝑥)‘𝑦) = (𝑥𝐹𝑦))
76an32s 650 . . . . . . . . 9 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐵 ∈ (𝑉 ∖ {∅})) ∧ (𝑥𝐴𝑦𝐵)) → ((curry 𝐹𝑥)‘𝑦) = (𝑥𝐹𝑦))
87eqeq1d 2728 . . . . . . . 8 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐵 ∈ (𝑉 ∖ {∅})) ∧ (𝑥𝐴𝑦𝐵)) → (((curry 𝐹𝑥)‘𝑦) = 𝑧 ↔ (𝑥𝐹𝑦) = 𝑧))
9 eqcom 2733 . . . . . . . 8 ((𝑥𝐹𝑦) = 𝑧𝑧 = (𝑥𝐹𝑦))
108, 9bitrdi 286 . . . . . . 7 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐵 ∈ (𝑉 ∖ {∅})) ∧ (𝑥𝐴𝑦𝐵)) → (((curry 𝐹𝑥)‘𝑦) = 𝑧𝑧 = (𝑥𝐹𝑦)))
113, 10sylan 578 . . . . . 6 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ (𝑥𝐴𝑦𝐵)) → (((curry 𝐹𝑥)‘𝑦) = 𝑧𝑧 = (𝑥𝐹𝑦)))
12 curf 37299 . . . . . . . . . 10 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → curry 𝐹:𝐴⟶(𝐶m 𝐵))
1312ffvelcdmda 7098 . . . . . . . . 9 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ 𝑥𝐴) → (curry 𝐹𝑥) ∈ (𝐶m 𝐵))
14 elmapfn 8894 . . . . . . . . 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 6954 . . . . . . . 8 (((curry 𝐹𝑥) Fn 𝐵𝑦𝐵) → (((curry 𝐹𝑥)‘𝑦) = 𝑧𝑦(curry 𝐹𝑥)𝑧))
1715, 16sylan 578 . . . . . . 7 ((((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ 𝑥𝐴) ∧ 𝑦𝐵) → (((curry 𝐹𝑥)‘𝑦) = 𝑧𝑦(curry 𝐹𝑥)𝑧))
1817anasss 465 . . . . . 6 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ (𝑥𝐴𝑦𝐵)) → (((curry 𝐹𝑥)‘𝑦) = 𝑧𝑦(curry 𝐹𝑥)𝑧))
19 ibar 527 . . . . . . 7 ((𝑥𝐴𝑦𝐵) → (𝑧 = (𝑥𝐹𝑦) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = (𝑥𝐹𝑦))))
2019adantl 480 . . . . . 6 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ (𝑥𝐴𝑦𝐵)) → (𝑧 = (𝑥𝐹𝑦) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = (𝑥𝐹𝑦))))
2111, 18, 203bitr3d 308 . . . . 5 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ (𝑥𝐴𝑦𝐵)) → (𝑦(curry 𝐹𝑥)𝑧 ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = (𝑥𝐹𝑦))))
22 df-br 5154 . . . . . . . . . . 11 (𝑦(curry 𝐹𝑥)𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ (curry 𝐹𝑥))
23 elfvdm 6938 . . . . . . . . . . 11 (⟨𝑦, 𝑧⟩ ∈ (curry 𝐹𝑥) → 𝑥 ∈ dom curry 𝐹)
2422, 23sylbi 216 . . . . . . . . . 10 (𝑦(curry 𝐹𝑥)𝑧𝑥 ∈ dom curry 𝐹)
25 fdm 6737 . . . . . . . . . . . 12 (curry 𝐹:𝐴⟶(𝐶m 𝐵) → dom curry 𝐹 = 𝐴)
2625eleq2d 2812 . . . . . . . . . . 11 (curry 𝐹:𝐴⟶(𝐶m 𝐵) → (𝑥 ∈ dom curry 𝐹𝑥𝐴))
2726biimpa 475 . . . . . . . . . 10 ((curry 𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥 ∈ dom curry 𝐹) → 𝑥𝐴)
2824, 27sylan2 591 . . . . . . . . 9 ((curry 𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑦(curry 𝐹𝑥)𝑧) → 𝑥𝐴)
29 ffvelcdm 7095 . . . . . . . . . . . . 13 ((curry 𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → (curry 𝐹𝑥) ∈ (𝐶m 𝐵))
30 elmapi 8878 . . . . . . . . . . . . 13 ((curry 𝐹𝑥) ∈ (𝐶m 𝐵) → (curry 𝐹𝑥):𝐵𝐶)
31 fdm 6737 . . . . . . . . . . . . 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 3466 . . . . . . . . . . . . 13 𝑦 ∈ V
34 vex 3466 . . . . . . . . . . . . 13 𝑧 ∈ V
3533, 34breldm 5915 . . . . . . . . . . . 12 (𝑦(curry 𝐹𝑥)𝑧𝑦 ∈ dom (curry 𝐹𝑥))
36 eleq2 2815 . . . . . . . . . . . . 13 (dom (curry 𝐹𝑥) = 𝐵 → (𝑦 ∈ dom (curry 𝐹𝑥) ↔ 𝑦𝐵))
3736biimpa 475 . . . . . . . . . . . 12 ((dom (curry 𝐹𝑥) = 𝐵𝑦 ∈ dom (curry 𝐹𝑥)) → 𝑦𝐵)
3832, 35, 37syl2an 594 . . . . . . . . . . 11 (((curry 𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) ∧ 𝑦(curry 𝐹𝑥)𝑧) → 𝑦𝐵)
3938an32s 650 . . . . . . . . . 10 (((curry 𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑦(curry 𝐹𝑥)𝑧) ∧ 𝑥𝐴) → 𝑦𝐵)
4028, 39mpdan 685 . . . . . . . . 9 ((curry 𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑦(curry 𝐹𝑥)𝑧) → 𝑦𝐵)
4128, 40jca 510 . . . . . . . 8 ((curry 𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑦(curry 𝐹𝑥)𝑧) → (𝑥𝐴𝑦𝐵))
4212, 41sylan 578 . . . . . . 7 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ 𝑦(curry 𝐹𝑥)𝑧) → (𝑥𝐴𝑦𝐵))
4342stoic1a 1767 . . . . . 6 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ ¬ (𝑥𝐴𝑦𝐵)) → ¬ 𝑦(curry 𝐹𝑥)𝑧)
44 simpl 481 . . . . . . . 8 (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = (𝑥𝐹𝑦)) → (𝑥𝐴𝑦𝐵))
4544con3i 154 . . . . . . 7 (¬ (𝑥𝐴𝑦𝐵) → ¬ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = (𝑥𝐹𝑦)))
4645adantl 480 . . . . . 6 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ ¬ (𝑥𝐴𝑦𝐵)) → ¬ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = (𝑥𝐹𝑦)))
4743, 462falsed 375 . . . . 5 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ ¬ (𝑥𝐴𝑦𝐵)) → (𝑦(curry 𝐹𝑥)𝑧 ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = (𝑥𝐹𝑦))))
4821, 47pm2.61dan 811 . . . 4 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → (𝑦(curry 𝐹𝑥)𝑧 ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = (𝑥𝐹𝑦))))
4948oprabbidv 7491 . . 3 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(curry 𝐹𝑥)𝑧} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = (𝑥𝐹𝑦))})
50 df-unc 8283 . . 3 uncurry curry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(curry 𝐹𝑥)𝑧}
51 df-mpo 7429 . . 3 (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = (𝑥𝐹𝑦))}
5249, 50, 513eqtr4g 2791 . 2 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → uncurry curry 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦)))
53 fnov 7557 . . . 4 (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦)))
541, 53sylib 217 . . 3 (𝐹:(𝐴 × 𝐵)⟶𝐶𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦)))
55543ad2ant1 1130 . 2 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦)))
5652, 55eqtr4d 2769 1 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → uncurry curry 𝐹 = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  w3a 1084   = wceq 1534  wcel 2099  cdif 3944  c0 4325  {csn 4633  cop 4639   class class class wbr 5153   × cxp 5680  dom cdm 5682   Fn wfn 6549  wf 6550  cfv 6554  (class class class)co 7424  {coprab 7425  cmpo 7426  curry ccur 8280  uncurry cunc 8281  m cmap 8855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8003  df-2nd 8004  df-cur 8282  df-unc 8283  df-map 8857
This theorem is referenced by: (None)
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