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Theorem uncf 35316
 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 6840 . . . . . 6 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → (𝐹𝑥) ∈ (𝐶m 𝐵))
2 elmapi 8438 . . . . . 6 ((𝐹𝑥) ∈ (𝐶m 𝐵) → (𝐹𝑥):𝐵𝐶)
31, 2syl 17 . . . . 5 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → (𝐹𝑥):𝐵𝐶)
43ffvelrnda 6842 . . . 4 (((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) ∧ 𝑦𝐵) → ((𝐹𝑥)‘𝑦) ∈ 𝐶)
54anasss 470 . . 3 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ (𝑥𝐴𝑦𝐵)) → ((𝐹𝑥)‘𝑦) ∈ 𝐶)
65ralrimivva 3120 . 2 (𝐹:𝐴⟶(𝐶m 𝐵) → ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥)‘𝑦) ∈ 𝐶)
7 df-unc 7944 . . . . 5 uncurry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹𝑥)𝑧}
8 df-br 5033 . . . . . . . . . . 11 (𝑦(𝐹𝑥)𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ (𝐹𝑥))
9 elfvdm 6690 . . . . . . . . . . 11 (⟨𝑦, 𝑧⟩ ∈ (𝐹𝑥) → 𝑥 ∈ dom 𝐹)
108, 9sylbi 220 . . . . . . . . . 10 (𝑦(𝐹𝑥)𝑧𝑥 ∈ dom 𝐹)
11 fdm 6506 . . . . . . . . . . 11 (𝐹:𝐴⟶(𝐶m 𝐵) → dom 𝐹 = 𝐴)
1211eleq2d 2837 . . . . . . . . . 10 (𝐹:𝐴⟶(𝐶m 𝐵) → (𝑥 ∈ dom 𝐹𝑥𝐴))
1310, 12syl5ib 247 . . . . . . . . 9 (𝐹:𝐴⟶(𝐶m 𝐵) → (𝑦(𝐹𝑥)𝑧𝑥𝐴))
1413pm4.71rd 566 . . . . . . . 8 (𝐹:𝐴⟶(𝐶m 𝐵) → (𝑦(𝐹𝑥)𝑧 ↔ (𝑥𝐴𝑦(𝐹𝑥)𝑧)))
15 elmapfun 8448 . . . . . . . . . . 11 ((𝐹𝑥) ∈ (𝐶m 𝐵) → Fun (𝐹𝑥))
16 funbrfv2b 6711 . . . . . . . . . . 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 6506 . . . . . . . . . . . . 13 ((𝐹𝑥):𝐵𝐶 → dom (𝐹𝑥) = 𝐵)
191, 2, 183syl 18 . . . . . . . . . . . 12 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → dom (𝐹𝑥) = 𝐵)
2019eleq2d 2837 . . . . . . . . . . 11 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → (𝑦 ∈ dom (𝐹𝑥) ↔ 𝑦𝐵))
21 eqcom 2765 . . . . . . . . . . . 12 (((𝐹𝑥)‘𝑦) = 𝑧𝑧 = ((𝐹𝑥)‘𝑦))
2221a1i 11 . . . . . . . . . . 11 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → (((𝐹𝑥)‘𝑦) = 𝑧𝑧 = ((𝐹𝑥)‘𝑦)))
2320, 22anbi12d 633 . . . . . . . . . 10 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → ((𝑦 ∈ dom (𝐹𝑥) ∧ ((𝐹𝑥)‘𝑦) = 𝑧) ↔ (𝑦𝐵𝑧 = ((𝐹𝑥)‘𝑦))))
2417, 23bitrd 282 . . . . . . . . 9 ((𝐹:𝐴⟶(𝐶m 𝐵) ∧ 𝑥𝐴) → (𝑦(𝐹𝑥)𝑧 ↔ (𝑦𝐵𝑧 = ((𝐹𝑥)‘𝑦))))
2524pm5.32da 582 . . . . . . . 8 (𝐹:𝐴⟶(𝐶m 𝐵) → ((𝑥𝐴𝑦(𝐹𝑥)𝑧) ↔ (𝑥𝐴 ∧ (𝑦𝐵𝑧 = ((𝐹𝑥)‘𝑦)))))
2614, 25bitrd 282 . . . . . . 7 (𝐹:𝐴⟶(𝐶m 𝐵) → (𝑦(𝐹𝑥)𝑧 ↔ (𝑥𝐴 ∧ (𝑦𝐵𝑧 = ((𝐹𝑥)‘𝑦)))))
27 anass 472 . . . . . . 7 (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = ((𝐹𝑥)‘𝑦)) ↔ (𝑥𝐴 ∧ (𝑦𝐵𝑧 = ((𝐹𝑥)‘𝑦))))
2826, 27bitr4di 292 . . . . . 6 (𝐹:𝐴⟶(𝐶m 𝐵) → (𝑦(𝐹𝑥)𝑧 ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = ((𝐹𝑥)‘𝑦))))
2928oprabbidv 7214 . . . . 5 (𝐹:𝐴⟶(𝐶m 𝐵) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹𝑥)𝑧} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = ((𝐹𝑥)‘𝑦))})
307, 29syl5eq 2805 . . . 4 (𝐹:𝐴⟶(𝐶m 𝐵) → uncurry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = ((𝐹𝑥)‘𝑦))})
3130feq1d 6483 . . 3 (𝐹:𝐴⟶(𝐶m 𝐵) → (uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = ((𝐹𝑥)‘𝑦))}:(𝐴 × 𝐵)⟶𝐶))
32 df-mpo 7155 . . . . 5 (𝑥𝐴, 𝑦𝐵 ↦ ((𝐹𝑥)‘𝑦)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = ((𝐹𝑥)‘𝑦))}
3332eqcomi 2767 . . . 4 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = ((𝐹𝑥)‘𝑦))} = (𝑥𝐴, 𝑦𝐵 ↦ ((𝐹𝑥)‘𝑦))
3433fmpo 7770 . . 3 (∀𝑥𝐴𝑦𝐵 ((𝐹𝑥)‘𝑦) ∈ 𝐶 ↔ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = ((𝐹𝑥)‘𝑦))}:(𝐴 × 𝐵)⟶𝐶)
3531, 34bitr4di 292 . 2 (𝐹:𝐴⟶(𝐶m 𝐵) → (uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ ∀𝑥𝐴𝑦𝐵 ((𝐹𝑥)‘𝑦) ∈ 𝐶))
366, 35mpbird 260 1 (𝐹:𝐴⟶(𝐶m 𝐵) → uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ⟨cop 4528   class class class wbr 5032   × cxp 5522  dom cdm 5524  Fun wfun 6329  ⟶wf 6331  ‘cfv 6335  (class class class)co 7150  {coprab 7151   ∈ cmpo 7152  uncurry cunc 7942   ↑m cmap 8416 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7693  df-2nd 7694  df-unc 7944  df-map 8418 This theorem is referenced by:  curunc  35319  matunitlindflem2  35334
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