Theorem uncf 38250

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 7076	. . . . . 6 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ (𝐶 ↑m 𝐵))
2		elmapi 8842	. . . . . 6 ⊢ ((𝐹‘𝑥) ∈ (𝐶 ↑m 𝐵) → (𝐹‘𝑥):𝐵⟶𝐶)
3	1, 2	syl 18	. . . . 5 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥):𝐵⟶𝐶)
4	3	ffvelcdmda 7079	. . . 4 ⊢ (((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → ((𝐹‘𝑥)‘𝑦) ∈ 𝐶)
5	4	anasss 471	. . 3 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ((𝐹‘𝑥)‘𝑦) ∈ 𝐶)
6	5	ralrimivva 3208	. 2 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥)‘𝑦) ∈ 𝐶)
7		df-unc 8260	. . . . 5 ⊢ uncurry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐹‘𝑥)𝑧}
8		df-br 5110	. . . . . . . . . . 11 ⊢ (𝑦(𝐹‘𝑥)𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ (𝐹‘𝑥))
9		elfvdm 6915	. . . . . . . . . . 11 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐹‘𝑥) → 𝑥 ∈ dom 𝐹)
10	8, 9	sylbi 220	. . . . . . . . . 10 ⊢ (𝑦(𝐹‘𝑥)𝑧 → 𝑥 ∈ dom 𝐹)
11		fdm 6715	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → dom 𝐹 = 𝐴)
12	11	eleq2d 2849	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴))
13	10, 12	imbitrid 247	. . . . . . . . 9 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → (𝑦(𝐹‘𝑥)𝑧 → 𝑥 ∈ 𝐴))
14	13	pm4.71rd 571	. . . . . . . 8 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → (𝑦(𝐹‘𝑥)𝑧 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦(𝐹‘𝑥)𝑧)))
15		elmapfun 8859	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑥) ∈ (𝐶 ↑m 𝐵) → Fun (𝐹‘𝑥))
16		funbrfv2b 6938	. . . . . . . . . . 11 ⊢ (Fun (𝐹‘𝑥) → (𝑦(𝐹‘𝑥)𝑧 ↔ (𝑦 ∈ dom (𝐹‘𝑥) ∧ ((𝐹‘𝑥)‘𝑦) = 𝑧)))
17	1, 15, 16	3syl 19	. . . . . . . . . 10 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑦(𝐹‘𝑥)𝑧 ↔ (𝑦 ∈ dom (𝐹‘𝑥) ∧ ((𝐹‘𝑥)‘𝑦) = 𝑧)))
18		fdm 6715	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑥):𝐵⟶𝐶 → dom (𝐹‘𝑥) = 𝐵)
19	1, 2, 18	3syl 19	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → dom (𝐹‘𝑥) = 𝐵)
20	19	eleq2d 2849	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ dom (𝐹‘𝑥) ↔ 𝑦 ∈ 𝐵))
21		eqcom 2770	. . . . . . . . . . . 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 643	. . . . . . . . . 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 282	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶(𝐶 ↑m 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑦(𝐹‘𝑥)𝑧 ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦))))
25	24	pm5.32da 589	. . . . . . . 8 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → ((𝑥 ∈ 𝐴 ∧ 𝑦(𝐹‘𝑥)𝑧) ↔ (𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦)))))
26	14, 25	bitrd 282	. . . . . . 7 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → (𝑦(𝐹‘𝑥)𝑧 ↔ (𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦)))))
27		anass 473	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦))))
28	26, 27	bitr4di 292	. . . . . 6 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → (𝑦(𝐹‘𝑥)𝑧 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦))))
29	28	oprabbidv 7476	. . . . 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 2810	. . . 4 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → uncurry 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦))})
31	30	feq1d 6687	. . 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 7415	. . . . 5 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ((𝐹‘𝑥)‘𝑦)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦))}
33	32	eqcomi 2772	. . . 4 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = ((𝐹‘𝑥)‘𝑦))} = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ((𝐹‘𝑥)‘𝑦))
34	33	fmpo 8061	. . 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 292	. 2 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → (uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥)‘𝑦) ∈ 𝐶))
36	6, 35	mpbird 260	1 ⊢ (𝐹:𝐴⟶(𝐶 ↑m 𝐵) → uncurry 𝐹:(𝐴 × 𝐵)⟶𝐶)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ⟨cop 4595   class class class wbr 5109   × cxp 5659  dom cdm 5661  Fun wfun 6530  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  {coprab 7411   ∈ cmpo 7412  uncurry cunc 8258   ↑_m cmap 8820

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-unc 8260  df-map 8822

This theorem is referenced by:  curunc  38253  matunitlindflem2  38268