Theorem uncf 35756

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		ffvelrn 6959	. . . . . 6 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐶 ↑m 𝐵))
2		elmapi 8637	. . . . . 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	ffvelrnda 6961	. . . 4 ⊢ (((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → ((𝐹‘𝑥)‘𝑦) ∈ 𝐶)
5	4	anasss 467	. . 3 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ((𝐹‘𝑥)‘𝑦) ∈ 𝐶)
6	5	ralrimivva 3123	. 2 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥)‘𝑦) ∈ 𝐶)
7		df-unc 8084	. . . . 5 ⊢ uncurry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹‘𝑥)𝑧}
8		df-br 5075	. . . . . . . . . . 11 ⊢ (𝑦(𝐹‘𝑥)𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ (𝐹‘𝑥))
9		elfvdm 6806	. . . . . . . . . . 11 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐹‘𝑥) → 𝑥 ∈ dom 𝐹)
10	8, 9	sylbi 216	. . . . . . . . . 10 ⊢ (𝑦(𝐹‘𝑥)𝑧 → 𝑥 ∈ dom 𝐹)
11		fdm 6609	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → dom 𝐹 = 𝐴)
12	11	eleq2d 2824	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴))
13	10, 12	syl5ib 243	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → (𝑦(𝐹‘𝑥)𝑧 → 𝑥 ∈ 𝐴))
14	13	pm4.71rd 563	. . . . . . . 8 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → (𝑦(𝐹‘𝑥)𝑧 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦(𝐹‘𝑥)𝑧)))
15		elmapfun 8654	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑥) ∈ (𝐶 ↑m 𝐵) → Fun (𝐹‘𝑥))
16		funbrfv2b 6827	. . . . . . . . . . 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 6609	. . . . . . . . . . . . 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 2824	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ dom (𝐹‘𝑥) ↔ 𝑦 ∈ 𝐵))
21		eqcom 2745	. . . . . . . . . . . 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 631	. . . . . . . . . 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 278	. . . . . . . . 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 278	. . . . . . 7 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → (𝑦(𝐹‘𝑥)𝑧 ↔ (𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦)))))
27		anass 469	. . . . . . 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 7341	. . . . 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 2790	. . . 4 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → uncurry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦))})
31	30	feq1d 6585	. . 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 7280	. . . . 5 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ((𝐹‘𝑥)‘𝑦)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦))}
33	32	eqcomi 2747	. . . 4 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦))} = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ((𝐹‘𝑥)‘𝑦))
34	33	fmpo 7908	. . 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 256	1 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ⟨cop 4567   class class class wbr 5074   × cxp 5587  dom cdm 5589  Fun wfun 6427  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  {coprab 7276   ∈ cmpo 7277  uncurry cunc 8082   ↑_m cmap 8615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-unc 8084  df-map 8617

This theorem is referenced by:  curunc  35759  matunitlindflem2  35774