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Theorem unccur 35039
 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 6491 . . . . . . . . 9 (𝐹:(𝐴 × 𝐵)⟶𝐶𝐹 Fn (𝐴 × 𝐵))
21anim1i 617 . . . . . . . 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 35036 . . . . . . . . . . 11 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝑥𝐴𝑦𝐵) ∧ 𝐵 ∈ (𝑉 ∖ {∅})) → ((curry 𝐹𝑥)‘𝑦) = (𝑥𝐹𝑦))
64, 5sylanbr 585 . . . . . . . . . 10 (((𝐹 Fn (𝐴 × 𝐵) ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝐵 ∈ (𝑉 ∖ {∅})) → ((curry 𝐹𝑥)‘𝑦) = (𝑥𝐹𝑦))
76an32s 651 . . . . . . . . 9 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐵 ∈ (𝑉 ∖ {∅})) ∧ (𝑥𝐴𝑦𝐵)) → ((curry 𝐹𝑥)‘𝑦) = (𝑥𝐹𝑦))
87eqeq1d 2803 . . . . . . . 8 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐵 ∈ (𝑉 ∖ {∅})) ∧ (𝑥𝐴𝑦𝐵)) → (((curry 𝐹𝑥)‘𝑦) = 𝑧 ↔ (𝑥𝐹𝑦) = 𝑧))
9 eqcom 2808 . . . . . . . 8 ((𝑥𝐹𝑦) = 𝑧𝑧 = (𝑥𝐹𝑦))
108, 9syl6bb 290 . . . . . . 7 (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐵 ∈ (𝑉 ∖ {∅})) ∧ (𝑥𝐴𝑦𝐵)) → (((curry 𝐹𝑥)‘𝑦) = 𝑧𝑧 = (𝑥𝐹𝑦)))
113, 10sylan 583 . . . . . 6 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ (𝑥𝐴𝑦𝐵)) → (((curry 𝐹𝑥)‘𝑦) = 𝑧𝑧 = (𝑥𝐹𝑦)))
12 curf 35034 . . . . . . . . . 10 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → curry 𝐹:𝐴⟶(𝐶m 𝐵))
1312ffvelrnda 6832 . . . . . . . . 9 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ 𝑥𝐴) → (curry 𝐹𝑥) ∈ (𝐶m 𝐵))
14 elmapfn 8416 . . . . . . . . 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 6697 . . . . . . . 8 (((curry 𝐹𝑥) Fn 𝐵𝑦𝐵) → (((curry 𝐹𝑥)‘𝑦) = 𝑧𝑦(curry 𝐹𝑥)𝑧))
1715, 16sylan 583 . . . . . . 7 ((((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ 𝑥𝐴) ∧ 𝑦𝐵) → (((curry 𝐹𝑥)‘𝑦) = 𝑧𝑦(curry 𝐹𝑥)𝑧))
1817anasss 470 . . . . . 6 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ (𝑥𝐴𝑦𝐵)) → (((curry 𝐹𝑥)‘𝑦) = 𝑧𝑦(curry 𝐹𝑥)𝑧))
19 ibar 532 . . . . . . 7 ((𝑥𝐴𝑦𝐵) → (𝑧 = (𝑥𝐹𝑦) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = (𝑥𝐹𝑦))))
2019adantl 485 . . . . . 6 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ (𝑥𝐴𝑦𝐵)) → (𝑧 = (𝑥𝐹𝑦) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = (𝑥𝐹𝑦))))
2111, 18, 203bitr3d 312 . . . . 5 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ (𝑥𝐴𝑦𝐵)) → (𝑦(curry 𝐹𝑥)𝑧 ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = (𝑥𝐹𝑦))))
22 df-br 5034 . . . . . . . . . . 11 (𝑦(curry 𝐹𝑥)𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ (curry 𝐹𝑥))
23 elfvdm 6681 . . . . . . . . . . 11 (⟨𝑦, 𝑧⟩ ∈ (curry 𝐹𝑥) → 𝑥 ∈ dom curry 𝐹)
2422, 23sylbi 220 . . . . . . . . . 10 (𝑦(curry 𝐹𝑥)𝑧𝑥 ∈ dom curry 𝐹)
25 fdm 6499 . . . . . . . . . . . 12 (curry 𝐹:𝐴⟶(𝐶m 𝐵) → dom curry 𝐹 = 𝐴)
2625eleq2d 2878 . . . . . . . . . . 11 (curry 𝐹:𝐴⟶(𝐶m 𝐵) → (𝑥 ∈ dom curry 𝐹𝑥𝐴))
2726biimpa 480 . . . . . . . . . 10 ((curry 𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥 ∈ dom curry 𝐹) → 𝑥𝐴)
2824, 27sylan2 595 . . . . . . . . 9 ((curry 𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑦(curry 𝐹𝑥)𝑧) → 𝑥𝐴)
29 ffvelrn 6830 . . . . . . . . . . . . 13 ((curry 𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → (curry 𝐹𝑥) ∈ (𝐶m 𝐵))
30 elmapi 8415 . . . . . . . . . . . . 13 ((curry 𝐹𝑥) ∈ (𝐶m 𝐵) → (curry 𝐹𝑥):𝐵𝐶)
31 fdm 6499 . . . . . . . . . . . . 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 3447 . . . . . . . . . . . . 13 𝑦 ∈ V
34 vex 3447 . . . . . . . . . . . . 13 𝑧 ∈ V
3533, 34breldm 5745 . . . . . . . . . . . 12 (𝑦(curry 𝐹𝑥)𝑧𝑦 ∈ dom (curry 𝐹𝑥))
36 eleq2 2881 . . . . . . . . . . . . 13 (dom (curry 𝐹𝑥) = 𝐵 → (𝑦 ∈ dom (curry 𝐹𝑥) ↔ 𝑦𝐵))
3736biimpa 480 . . . . . . . . . . . 12 ((dom (curry 𝐹𝑥) = 𝐵𝑦 ∈ dom (curry 𝐹𝑥)) → 𝑦𝐵)
3832, 35, 37syl2an 598 . . . . . . . . . . 11 (((curry 𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) ∧ 𝑦(curry 𝐹𝑥)𝑧) → 𝑦𝐵)
3938an32s 651 . . . . . . . . . 10 (((curry 𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑦(curry 𝐹𝑥)𝑧) ∧ 𝑥𝐴) → 𝑦𝐵)
4028, 39mpdan 686 . . . . . . . . 9 ((curry 𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑦(curry 𝐹𝑥)𝑧) → 𝑦𝐵)
4128, 40jca 515 . . . . . . . 8 ((curry 𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑦(curry 𝐹𝑥)𝑧) → (𝑥𝐴𝑦𝐵))
4212, 41sylan 583 . . . . . . 7 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ 𝑦(curry 𝐹𝑥)𝑧) → (𝑥𝐴𝑦𝐵))
4342stoic1a 1774 . . . . . 6 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ ¬ (𝑥𝐴𝑦𝐵)) → ¬ 𝑦(curry 𝐹𝑥)𝑧)
44 simpl 486 . . . . . . . 8 (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = (𝑥𝐹𝑦)) → (𝑥𝐴𝑦𝐵))
4544con3i 157 . . . . . . 7 (¬ (𝑥𝐴𝑦𝐵) → ¬ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = (𝑥𝐹𝑦)))
4645adantl 485 . . . . . 6 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ ¬ (𝑥𝐴𝑦𝐵)) → ¬ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = (𝑥𝐹𝑦)))
4743, 462falsed 380 . . . . 5 (((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) ∧ ¬ (𝑥𝐴𝑦𝐵)) → (𝑦(curry 𝐹𝑥)𝑧 ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = (𝑥𝐹𝑦))))
4821, 47pm2.61dan 812 . . . 4 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → (𝑦(curry 𝐹𝑥)𝑧 ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = (𝑥𝐹𝑦))))
4948oprabbidv 7203 . . 3 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(curry 𝐹𝑥)𝑧} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = (𝑥𝐹𝑦))})
50 df-unc 7921 . . 3 uncurry curry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(curry 𝐹𝑥)𝑧}
51 df-mpo 7144 . . 3 (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = (𝑥𝐹𝑦))}
5249, 50, 513eqtr4g 2861 . 2 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → uncurry curry 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦)))
53 fnov 7265 . . . 4 (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦)))
541, 53sylib 221 . . 3 (𝐹:(𝐴 × 𝐵)⟶𝐶𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦)))
55543ad2ant1 1130 . 2 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦)))
5652, 55eqtr4d 2839 1 ((𝐹:(𝐴 × 𝐵)⟶𝐶𝐵 ∈ (𝑉 ∖ {∅}) ∧ 𝐶𝑊) → uncurry curry 𝐹 = 𝐹)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112   ∖ cdif 3881  ∅c0 4246  {csn 4528  ⟨cop 4534   class class class wbr 5033   × cxp 5521  dom cdm 5523   Fn wfn 6323  ⟶wf 6324  ‘cfv 6328  (class class class)co 7139  {coprab 7140   ∈ cmpo 7141  curry ccur 7918  uncurry cunc 7919   ↑m cmap 8393 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-cur 7920  df-unc 7921  df-map 8395 This theorem is referenced by: (None)
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