Theorem uncf 37966

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 7022	. . . . . 6 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐶 ↑m 𝐵))
2		elmapi 8786	. . . . . 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 7025	. . . 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 3182	. 2 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥)‘𝑦) ∈ 𝐶)
7		df-unc 8208	. . . . 5 ⊢ uncurry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹‘𝑥)𝑧}
8		df-br 5073	. . . . . . . . . . 11 ⊢ (𝑦(𝐹‘𝑥)𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ (𝐹‘𝑥))
9		elfvdm 6861	. . . . . . . . . . 11 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐹‘𝑥) → 𝑥 ∈ dom 𝐹)
10	8, 9	sylbi 218	. . . . . . . . . 10 ⊢ (𝑦(𝐹‘𝑥)𝑧 → 𝑥 ∈ dom 𝐹)
11		fdm 6664	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → dom 𝐹 = 𝐴)
12	11	eleq2d 2825	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴))
13	10, 12	imbitrid 245	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → (𝑦(𝐹‘𝑥)𝑧 → 𝑥 ∈ 𝐴))
14	13	pm4.71rd 567	. . . . . . . 8 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → (𝑦(𝐹‘𝑥)𝑧 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦(𝐹‘𝑥)𝑧)))
15		elmapfun 8803	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑥) ∈ (𝐶 ↑m 𝐵) → Fun (𝐹‘𝑥))
16		funbrfv2b 6884	. . . . . . . . . . 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 6664	. . . . . . . . . . . . 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 2825	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ dom (𝐹‘𝑥) ↔ 𝑦 ∈ 𝐵))
21		eqcom 2746	. . . . . . . . . . . 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 638	. . . . . . . . . 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 280	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑦(𝐹‘𝑥)𝑧 ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦))))
25	24	pm5.32da 584	. . . . . . . 8 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → ((𝑥 ∈ 𝐴 ∧ 𝑦(𝐹‘𝑥)𝑧) ↔ (𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦)))))
26	14, 25	bitrd 280	. . . . . . 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 290	. . . . . 6 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → (𝑦(𝐹‘𝑥)𝑧 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦))))
29	28	oprabbidv 7422	. . . . 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 2786	. . . 4 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → uncurry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦))})
31	30	feq1d 6637	. . 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 7361	. . . . 5 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ((𝐹‘𝑥)‘𝑦)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦))}
33	32	eqcomi 2748	. . . 4 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦))} = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ((𝐹‘𝑥)‘𝑦))
34	33	fmpo 8010	. . 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 290	. 2 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → (uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥)‘𝑦) ∈ 𝐶))
36	6, 35	mpbird 258	1 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053  ⟨cop 4561   class class class wbr 5072   × cxp 5616  dom cdm 5618  Fun wfun 6479  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356  {coprab 7357   ∈ cmpo 7358  uncurry cunc 8206   ↑_m cmap 8763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-unc 8208  df-map 8765

This theorem is referenced by:  curunc  37969  matunitlindflem2  37984