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Theorem uncf 33017
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 6313 . . . . . 6 ((𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝑥𝐴) → (𝐹𝑥) ∈ (𝐶𝑚 𝐵))
2 elmapi 7823 . . . . . 6 ((𝐹𝑥) ∈ (𝐶𝑚 𝐵) → (𝐹𝑥):𝐵𝐶)
31, 2syl 17 . . . . 5 ((𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝑥𝐴) → (𝐹𝑥):𝐵𝐶)
43ffvelrnda 6315 . . . 4 (((𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝑥𝐴) ∧ 𝑦𝐵) → ((𝐹𝑥)‘𝑦) ∈ 𝐶)
54anasss 678 . . 3 ((𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ (𝑥𝐴𝑦𝐵)) → ((𝐹𝑥)‘𝑦) ∈ 𝐶)
65ralrimivva 2965 . 2 (𝐹:𝐴⟶(𝐶𝑚 𝐵) → ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥)‘𝑦) ∈ 𝐶)
7 df-unc 7339 . . . . 5 uncurry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹𝑥)𝑧}
8 df-br 4614 . . . . . . . . . . 11 (𝑦(𝐹𝑥)𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ (𝐹𝑥))
9 elfvdm 6177 . . . . . . . . . . 11 (⟨𝑦, 𝑧⟩ ∈ (𝐹𝑥) → 𝑥 ∈ dom 𝐹)
108, 9sylbi 207 . . . . . . . . . 10 (𝑦(𝐹𝑥)𝑧𝑥 ∈ dom 𝐹)
11 fdm 6008 . . . . . . . . . . 11 (𝐹:𝐴⟶(𝐶𝑚 𝐵) → dom 𝐹 = 𝐴)
1211eleq2d 2684 . . . . . . . . . 10 (𝐹:𝐴⟶(𝐶𝑚 𝐵) → (𝑥 ∈ dom 𝐹𝑥𝐴))
1310, 12syl5ib 234 . . . . . . . . 9 (𝐹:𝐴⟶(𝐶𝑚 𝐵) → (𝑦(𝐹𝑥)𝑧𝑥𝐴))
1413pm4.71rd 666 . . . . . . . 8 (𝐹:𝐴⟶(𝐶𝑚 𝐵) → (𝑦(𝐹𝑥)𝑧 ↔ (𝑥𝐴𝑦(𝐹𝑥)𝑧)))
15 elmapfun 7825 . . . . . . . . . . 11 ((𝐹𝑥) ∈ (𝐶𝑚 𝐵) → Fun (𝐹𝑥))
16 funbrfv2b 6197 . . . . . . . . . . 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 6008 . . . . . . . . . . . . 13 ((𝐹𝑥):𝐵𝐶 → dom (𝐹𝑥) = 𝐵)
191, 2, 183syl 18 . . . . . . . . . . . 12 ((𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝑥𝐴) → dom (𝐹𝑥) = 𝐵)
2019eleq2d 2684 . . . . . . . . . . 11 ((𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝑥𝐴) → (𝑦 ∈ dom (𝐹𝑥) ↔ 𝑦𝐵))
21 eqcom 2628 . . . . . . . . . . . 12 (((𝐹𝑥)‘𝑦) = 𝑧𝑧 = ((𝐹𝑥)‘𝑦))
2221a1i 11 . . . . . . . . . . 11 ((𝐹:𝐴⟶(𝐶𝑚 𝐵) ∧ 𝑥𝐴) → (((𝐹𝑥)‘𝑦) = 𝑧𝑧 = ((𝐹𝑥)‘𝑦)))
2320, 22anbi12d 746 . . . . . . . . . 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 672 . . . . . . . 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 680 . . . . . . 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 6662 . . . . 5 (𝐹:𝐴⟶(𝐶𝑚 𝐵) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹𝑥)𝑧} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = ((𝐹𝑥)‘𝑦))})
307, 29syl5eq 2667 . . . 4 (𝐹:𝐴⟶(𝐶𝑚 𝐵) → uncurry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = ((𝐹𝑥)‘𝑦))})
3130feq1d 5987 . . 3 (𝐹:𝐴⟶(𝐶𝑚 𝐵) → (uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = ((𝐹𝑥)‘𝑦))}:(𝐴 × 𝐵)⟶𝐶))
32 df-mpt2 6609 . . . . 5 (𝑥𝐴, 𝑦𝐵 ↦ ((𝐹𝑥)‘𝑦)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = ((𝐹𝑥)‘𝑦))}
3332eqcomi 2630 . . . 4 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = ((𝐹𝑥)‘𝑦))} = (𝑥𝐴, 𝑦𝐵 ↦ ((𝐹𝑥)‘𝑦))
3433fmpt2 7182 . . 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 384   = wceq 1480  wcel 1987  wral 2907  cop 4154   class class class wbr 4613   × cxp 5072  dom cdm 5074  Fun wfun 5841  wf 5843  cfv 5847  (class class class)co 6604  {coprab 6605  cmpt2 6606  uncurry cunc 7337  𝑚 cmap 7802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-unc 7339  df-map 7804
This theorem is referenced by:  curunc  33020  matunitlindflem2  33035
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