Theorem uncfval 18279

Description: Value of the uncurry functor, which is the reverse of the curry functor, taking 𝐺:𝐶⟶(𝐷⟶𝐸) to uncurry_F (𝐺):𝐶 × 𝐷⟶𝐸. (Contributed by Mario Carneiro, 13-Jan-2017.)

Hypotheses

Ref	Expression
uncfval.g	⊢ 𝐹 = (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)
uncfval.c	⊢ (𝜑 → 𝐷 ∈ Cat)
uncfval.d	⊢ (𝜑 → 𝐸 ∈ Cat)
uncfval.f	⊢ (𝜑 → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))

Assertion

Ref	Expression
uncfval	⊢ (𝜑 → 𝐹 = ((𝐷 evalF 𝐸) ∘func ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))

Proof of Theorem uncfval

Dummy variables 𝑓 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uncfval.g	. 2 ⊢ 𝐹 = (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺)
2		df-uncf 18260	. . . 4 ⊢ uncurryF = (𝑐 ∈ V, 𝑓 ∈ V ↦ (((𝑐‘1) evalF (𝑐‘2)) ∘func ((𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))))
3	2	a1i 11	. . 3 ⊢ (𝜑 → uncurryF = (𝑐 ∈ V, 𝑓 ∈ V ↦ (((𝑐‘1) evalF (𝑐‘2)) ∘func ((𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1))))))
4		simprl 771	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → 𝑐 = ⟨“𝐶𝐷𝐸”⟩)
5	4	fveq1d 6908	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (𝑐‘1) = (⟨“𝐶𝐷𝐸”⟩‘1))
6		uncfval.c	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ Cat)
7		s3fv1 14931	. . . . . . . 8 ⊢ (𝐷 ∈ Cat → (⟨“𝐶𝐷𝐸”⟩‘1) = 𝐷)
8	6, 7	syl 17	. . . . . . 7 ⊢ (𝜑 → (⟨“𝐶𝐷𝐸”⟩‘1) = 𝐷)
9	8	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (⟨“𝐶𝐷𝐸”⟩‘1) = 𝐷)
10	5, 9	eqtrd 2777	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (𝑐‘1) = 𝐷)
11	4	fveq1d 6908	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (𝑐‘2) = (⟨“𝐶𝐷𝐸”⟩‘2))
12		uncfval.d	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Cat)
13		s3fv2 14932	. . . . . . . 8 ⊢ (𝐸 ∈ Cat → (⟨“𝐶𝐷𝐸”⟩‘2) = 𝐸)
14	12, 13	syl 17	. . . . . . 7 ⊢ (𝜑 → (⟨“𝐶𝐷𝐸”⟩‘2) = 𝐸)
15	14	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (⟨“𝐶𝐷𝐸”⟩‘2) = 𝐸)
16	11, 15	eqtrd 2777	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (𝑐‘2) = 𝐸)
17	10, 16	oveq12d 7449	. . . 4 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → ((𝑐‘1) evalF (𝑐‘2)) = (𝐷 evalF 𝐸))
18		simprr 773	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → 𝑓 = 𝐺)
19	4	fveq1d 6908	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (𝑐‘0) = (⟨“𝐶𝐷𝐸”⟩‘0))
20		uncfval.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
21		funcrcl 17908	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)) → (𝐶 ∈ Cat ∧ (𝐷 FuncCat 𝐸) ∈ Cat))
22	20, 21	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ (𝐷 FuncCat 𝐸) ∈ Cat))
23	22	simpld 494	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ Cat)
24		s3fv0 14930	. . . . . . . . . 10 ⊢ (𝐶 ∈ Cat → (⟨“𝐶𝐷𝐸”⟩‘0) = 𝐶)
25	23, 24	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (⟨“𝐶𝐷𝐸”⟩‘0) = 𝐶)
26	25	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (⟨“𝐶𝐷𝐸”⟩‘0) = 𝐶)
27	19, 26	eqtrd 2777	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (𝑐‘0) = 𝐶)
28	27, 10	oveq12d 7449	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → ((𝑐‘0) 1stF (𝑐‘1)) = (𝐶 1stF 𝐷))
29	18, 28	oveq12d 7449	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) = (𝐺 ∘func (𝐶 1stF 𝐷)))
30	27, 10	oveq12d 7449	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → ((𝑐‘0) 2ndF (𝑐‘1)) = (𝐶 2ndF 𝐷))
31	29, 30	oveq12d 7449	. . . 4 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → ((𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1))) = ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))
32	17, 31	oveq12d 7449	. . 3 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (((𝑐‘1) evalF (𝑐‘2)) ∘func ((𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) ⟨,⟩F ((𝑐‘0) 2ndF (𝑐‘1)))) = ((𝐷 evalF 𝐸) ∘func ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))
33		s3cli 14920	. . . 4 ⊢ ⟨“𝐶𝐷𝐸”⟩ ∈ Word V
34		elex 3501	. . . 4 ⊢ (⟨“𝐶𝐷𝐸”⟩ ∈ Word V → ⟨“𝐶𝐷𝐸”⟩ ∈ V)
35	33, 34	mp1i 13	. . 3 ⊢ (𝜑 → ⟨“𝐶𝐷𝐸”⟩ ∈ V)
36	20	elexd 3504	. . 3 ⊢ (𝜑 → 𝐺 ∈ V)
37		ovexd 7466	. . 3 ⊢ (𝜑 → ((𝐷 evalF 𝐸) ∘func ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))) ∈ V)
38	3, 32, 35, 36, 37	ovmpod 7585	. 2 ⊢ (𝜑 → (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺) = ((𝐷 evalF 𝐸) ∘func ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))
39	1, 38	eqtrid 2789	1 ⊢ (𝜑 → 𝐹 = ((𝐷 evalF 𝐸) ∘func ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3480  ‘cfv 6561  (class class class)co 7431   ∈ cmpo 7433  0cc0 11155  1c1 11156  2c2 12321  Word cword 14552  ⟨“cs3 14881  Catccat 17707   Func cfunc 17899   ∘_func ccofu 17901   FuncCat cfuc 17990   1^st_F c1stf 18214   2^nd_F c2ndf 18215   ⟨,⟩_F cprf 18216   eval_F cevlf 18254   uncurry_F cuncf 18256

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-hash 14370  df-word 14553  df-concat 14609  df-s1 14634  df-s2 14887  df-s3 14888  df-func 17903  df-uncf 18260

This theorem is referenced by:  uncfcl  18280  uncf1  18281  uncf2  18282