Theorem uncf 37934

Description: Functional property of uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021.)

Assertion

Ref	Expression
uncf	⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶)

Proof of Theorem uncf

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ffvelcdm 7027	. . . . . 6 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐶 ↑m 𝐵))
2		elmapi 8789	. . . . . 6 ⊢ ((𝐹‘𝑥) ∈ (𝐶 ↑m 𝐵) → (𝐹‘𝑥):𝐵⟶𝐶)
3	1, 2	syl 17	. . . . 5 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥):𝐵⟶𝐶)
4	3	ffvelcdmda 7030	. . . 4 ⊢ (((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → ((𝐹‘𝑥)‘𝑦) ∈ 𝐶)
5	4	anasss 466	. . 3 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ((𝐹‘𝑥)‘𝑦) ∈ 𝐶)
6	5	ralrimivva 3181	. 2 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥)‘𝑦) ∈ 𝐶)
7		df-unc 8211	. . . . 5 ⊢ uncurry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹‘𝑥)𝑧}
8		df-br 5087	. . . . . . . . . . 11 ⊢ (𝑦(𝐹‘𝑥)𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ (𝐹‘𝑥))
9		elfvdm 6868	. . . . . . . . . . 11 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐹‘𝑥) → 𝑥 ∈ dom 𝐹)
10	8, 9	sylbi 217	. . . . . . . . . 10 ⊢ (𝑦(𝐹‘𝑥)𝑧 → 𝑥 ∈ dom 𝐹)
11		fdm 6671	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → dom 𝐹 = 𝐴)
12	11	eleq2d 2823	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴))
13	10, 12	imbitrid 244	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → (𝑦(𝐹‘𝑥)𝑧 → 𝑥 ∈ 𝐴))
14	13	pm4.71rd 562	. . . . . . . 8 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → (𝑦(𝐹‘𝑥)𝑧 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦(𝐹‘𝑥)𝑧)))
15		elmapfun 8806	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑥) ∈ (𝐶 ↑m 𝐵) → Fun (𝐹‘𝑥))
16		funbrfv2b 6891	. . . . . . . . . . 11 ⊢ (Fun (𝐹‘𝑥) → (𝑦(𝐹‘𝑥)𝑧 ↔ (𝑦 ∈ dom (𝐹‘𝑥) ∧ ((𝐹‘𝑥)‘𝑦) = 𝑧)))
17	1, 15, 16	3syl 18	. . . . . . . . . 10 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑦(𝐹‘𝑥)𝑧 ↔ (𝑦 ∈ dom (𝐹‘𝑥) ∧ ((𝐹‘𝑥)‘𝑦) = 𝑧)))
18		fdm 6671	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑥):𝐵⟶𝐶 → dom (𝐹‘𝑥) = 𝐵)
19	1, 2, 18	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → dom (𝐹‘𝑥) = 𝐵)
20	19	eleq2d 2823	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ dom (𝐹‘𝑥) ↔ 𝑦 ∈ 𝐵))
21		eqcom 2744	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥)‘𝑦) = 𝑧 ↔ 𝑧 = ((𝐹‘𝑥)‘𝑦))
22	21	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → (((𝐹‘𝑥)‘𝑦) = 𝑧 ↔ 𝑧 = ((𝐹‘𝑥)‘𝑦)))
23	20, 22	anbi12d 633	. . . . . . . . . 10 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝑦 ∈ dom (𝐹‘𝑥) ∧ ((𝐹‘𝑥)‘𝑦) = 𝑧) ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦))))
24	17, 23	bitrd 279	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑦(𝐹‘𝑥)𝑧 ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦))))
25	24	pm5.32da 579	. . . . . . . 8 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → ((𝑥 ∈ 𝐴 ∧ 𝑦(𝐹‘𝑥)𝑧) ↔ (𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦)))))
26	14, 25	bitrd 279	. . . . . . 7 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → (𝑦(𝐹‘𝑥)𝑧 ↔ (𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦)))))
27		anass 468	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦))))
28	26, 27	bitr4di 289	. . . . . 6 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → (𝑦(𝐹‘𝑥)𝑧 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦))))
29	28	oprabbidv 7426	. . . . 5 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹‘𝑥)𝑧} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦))})
30	7, 29	eqtrid 2784	. . . 4 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → uncurry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦))})
31	30	feq1d 6644	. . 3 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → (uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦))}:(𝐴 × 𝐵)⟶𝐶))
32		df-mpo 7365	. . . . 5 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ((𝐹‘𝑥)‘𝑦)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦))}
33	32	eqcomi 2746	. . . 4 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦))} = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ((𝐹‘𝑥)‘𝑦))
34	33	fmpo 8014	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥)‘𝑦) ∈ 𝐶 ↔ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦))}:(𝐴 × 𝐵)⟶𝐶)
35	31, 34	bitr4di 289	. 2 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → (uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥)‘𝑦) ∈ 𝐶))
36	6, 35	mpbird 257	1 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ⟨cop 4574   class class class wbr 5086   × cxp 5622  dom cdm 5624  Fun wfun 6486  ⟶wf 6488  ‘cfv 6492  (class class class)co 7360  {coprab 7361   ∈ cmpo 7362  uncurry cunc 8209   ↑_m cmap 8766

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-unc 8211  df-map 8768

This theorem is referenced by:  curunc  37937  matunitlindflem2  37952