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Theorem uncf 33518
Description: Functional property of uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021.)
Assertion
Ref Expression
uncf (𝐹:𝐴⟶(𝐶𝑚 𝐵) → uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶)

Proof of Theorem uncf
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ffvelrn 6397 . . . . . 6 ((𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝑥𝐴) → (𝐹𝑥) ∈ (𝐶𝑚 𝐵))
2 elmapi 7921 . . . . . 6 ((𝐹𝑥) ∈ (𝐶𝑚 𝐵) → (𝐹𝑥):𝐵𝐶)
31, 2syl 17 . . . . 5 ((𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝑥𝐴) → (𝐹𝑥):𝐵𝐶)
43ffvelrnda 6399 . . . 4 (((𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝑥𝐴) ∧ 𝑦𝐵) → ((𝐹𝑥)‘𝑦) ∈ 𝐶)
54anasss 680 . . 3 ((𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ (𝑥𝐴𝑦𝐵)) → ((𝐹𝑥)‘𝑦) ∈ 𝐶)
65ralrimivva 3000 . 2 (𝐹:𝐴⟶(𝐶𝑚 𝐵) → ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥)‘𝑦) ∈ 𝐶)
7 df-unc 7439 . . . . 5 uncurry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹𝑥)𝑧}
8 df-br 4686 . . . . . . . . . . 11 (𝑦(𝐹𝑥)𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ (𝐹𝑥))
9 elfvdm 6258 . . . . . . . . . . 11 (⟨𝑦, 𝑧⟩ ∈ (𝐹𝑥) → 𝑥 ∈ dom 𝐹)
108, 9sylbi 207 . . . . . . . . . 10 (𝑦(𝐹𝑥)𝑧𝑥 ∈ dom 𝐹)
11 fdm 6089 . . . . . . . . . . 11 (𝐹:𝐴⟶(𝐶𝑚 𝐵) → dom 𝐹 = 𝐴)
1211eleq2d 2716 . . . . . . . . . 10 (𝐹:𝐴⟶(𝐶𝑚 𝐵) → (𝑥 ∈ dom 𝐹𝑥𝐴))
1310, 12syl5ib 234 . . . . . . . . 9 (𝐹:𝐴⟶(𝐶𝑚 𝐵) → (𝑦(𝐹𝑥)𝑧𝑥𝐴))
1413pm4.71rd 668 . . . . . . . 8 (𝐹:𝐴⟶(𝐶𝑚 𝐵) → (𝑦(𝐹𝑥)𝑧 ↔ (𝑥𝐴𝑦(𝐹𝑥)𝑧)))
15 elmapfun 7923 . . . . . . . . . . 11 ((𝐹𝑥) ∈ (𝐶𝑚 𝐵) → Fun (𝐹𝑥))
16 funbrfv2b 6279 . . . . . . . . . . 11 (Fun (𝐹𝑥) → (𝑦(𝐹𝑥)𝑧 ↔ (𝑦 ∈ dom (𝐹𝑥) ∧ ((𝐹𝑥)‘𝑦) = 𝑧)))
171, 15, 163syl 18 . . . . . . . . . 10 ((𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝑥𝐴) → (𝑦(𝐹𝑥)𝑧 ↔ (𝑦 ∈ dom (𝐹𝑥) ∧ ((𝐹𝑥)‘𝑦) = 𝑧)))
18 fdm 6089 . . . . . . . . . . . . 13 ((𝐹𝑥):𝐵𝐶 → dom (𝐹𝑥) = 𝐵)
191, 2, 183syl 18 . . . . . . . . . . . 12 ((𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝑥𝐴) → dom (𝐹𝑥) = 𝐵)
2019eleq2d 2716 . . . . . . . . . . 11 ((𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝑥𝐴) → (𝑦 ∈ dom (𝐹𝑥) ↔ 𝑦𝐵))
21 eqcom 2658 . . . . . . . . . . . 12 (((𝐹𝑥)‘𝑦) = 𝑧𝑧 = ((𝐹𝑥)‘𝑦))
2221a1i 11 . . . . . . . . . . 11 ((𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝑥𝐴) → (((𝐹𝑥)‘𝑦) = 𝑧𝑧 = ((𝐹𝑥)‘𝑦)))
2320, 22anbi12d 747 . . . . . . . . . 10 ((𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝑥𝐴) → ((𝑦 ∈ dom (𝐹𝑥) ∧ ((𝐹𝑥)‘𝑦) = 𝑧) ↔ (𝑦𝐵𝑧 = ((𝐹𝑥)‘𝑦))))
2417, 23bitrd 268 . . . . . . . . 9 ((𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝑥𝐴) → (𝑦(𝐹𝑥)𝑧 ↔ (𝑦𝐵𝑧 = ((𝐹𝑥)‘𝑦))))
2524pm5.32da 674 . . . . . . . 8 (𝐹:𝐴⟶(𝐶𝑚 𝐵) → ((𝑥𝐴𝑦(𝐹𝑥)𝑧) ↔ (𝑥𝐴 ∧ (𝑦𝐵𝑧 = ((𝐹𝑥)‘𝑦)))))
2614, 25bitrd 268 . . . . . . 7 (𝐹:𝐴⟶(𝐶𝑚 𝐵) → (𝑦(𝐹𝑥)𝑧 ↔ (𝑥𝐴 ∧ (𝑦𝐵𝑧 = ((𝐹𝑥)‘𝑦)))))
27 anass 682 . . . . . . 7 (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = ((𝐹𝑥)‘𝑦)) ↔ (𝑥𝐴 ∧ (𝑦𝐵𝑧 = ((𝐹𝑥)‘𝑦))))
2826, 27syl6bbr 278 . . . . . 6 (𝐹:𝐴⟶(𝐶𝑚 𝐵) → (𝑦(𝐹𝑥)𝑧 ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = ((𝐹𝑥)‘𝑦))))
2928oprabbidv 6751 . . . . 5 (𝐹:𝐴⟶(𝐶𝑚 𝐵) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹𝑥)𝑧} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = ((𝐹𝑥)‘𝑦))})
307, 29syl5eq 2697 . . . 4 (𝐹:𝐴⟶(𝐶𝑚 𝐵) → uncurry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = ((𝐹𝑥)‘𝑦))})
3130feq1d 6068 . . 3 (𝐹:𝐴⟶(𝐶𝑚 𝐵) → (uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = ((𝐹𝑥)‘𝑦))}:(𝐴 × 𝐵)⟶𝐶))
32 df-mpt2 6695 . . . . 5 (𝑥𝐴, 𝑦𝐵 ↦ ((𝐹𝑥)‘𝑦)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = ((𝐹𝑥)‘𝑦))}
3332eqcomi 2660 . . . 4 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = ((𝐹𝑥)‘𝑦))} = (𝑥𝐴, 𝑦𝐵 ↦ ((𝐹𝑥)‘𝑦))
3433fmpt2 7282 . . 3 (∀𝑥𝐴𝑦𝐵 ((𝐹𝑥)‘𝑦) ∈ 𝐶 ↔ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = ((𝐹𝑥)‘𝑦))}:(𝐴 × 𝐵)⟶𝐶)
3531, 34syl6bbr 278 . 2 (𝐹:𝐴⟶(𝐶𝑚 𝐵) → (uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥)‘𝑦) ∈ 𝐶))
366, 35mpbird 247 1 (𝐹:𝐴⟶(𝐶𝑚 𝐵) → uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  cop 4216   class class class wbr 4685   × cxp 5141  dom cdm 5143  Fun wfun 5920  wf 5922  cfv 5926  (class class class)co 6690  {coprab 6691  cmpt2 6692  uncurry cunc 7437  𝑚 cmap 7899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-unc 7439  df-map 7901
This theorem is referenced by:  curunc  33521  matunitlindflem2  33536
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