Theorem uncfval 17484

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 17465	. . . 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 769	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → 𝑐 = ⟨“𝐶𝐷𝐸”⟩)
5	4	fveq1d 6672	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (𝑐‘1) = (⟨“𝐶𝐷𝐸”⟩‘1))
6		uncfval.c	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ Cat)
7		s3fv1 14254	. . . . . . . 8 ⊢ (𝐷 ∈ Cat → (⟨“𝐶𝐷𝐸”⟩‘1) = 𝐷)
8	6, 7	syl 17	. . . . . . 7 ⊢ (𝜑 → (⟨“𝐶𝐷𝐸”⟩‘1) = 𝐷)
9	8	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (⟨“𝐶𝐷𝐸”⟩‘1) = 𝐷)
10	5, 9	eqtrd 2856	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (𝑐‘1) = 𝐷)
11	4	fveq1d 6672	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (𝑐‘2) = (⟨“𝐶𝐷𝐸”⟩‘2))
12		uncfval.d	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Cat)
13		s3fv2 14255	. . . . . . . 8 ⊢ (𝐸 ∈ Cat → (⟨“𝐶𝐷𝐸”⟩‘2) = 𝐸)
14	12, 13	syl 17	. . . . . . 7 ⊢ (𝜑 → (⟨“𝐶𝐷𝐸”⟩‘2) = 𝐸)
15	14	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (⟨“𝐶𝐷𝐸”⟩‘2) = 𝐸)
16	11, 15	eqtrd 2856	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (𝑐‘2) = 𝐸)
17	10, 16	oveq12d 7174	. . . 4 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → ((𝑐‘1) evalF (𝑐‘2)) = (𝐷 evalF 𝐸))
18		simprr 771	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → 𝑓 = 𝐺)
19	4	fveq1d 6672	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (𝑐‘0) = (⟨“𝐶𝐷𝐸”⟩‘0))
20		uncfval.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
21		funcrcl 17133	. . . . . . . . . . . 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 497	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ Cat)
24		s3fv0 14253	. . . . . . . . . 10 ⊢ (𝐶 ∈ Cat → (⟨“𝐶𝐷𝐸”⟩‘0) = 𝐶)
25	23, 24	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (⟨“𝐶𝐷𝐸”⟩‘0) = 𝐶)
26	25	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (⟨“𝐶𝐷𝐸”⟩‘0) = 𝐶)
27	19, 26	eqtrd 2856	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (𝑐‘0) = 𝐶)
28	27, 10	oveq12d 7174	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → ((𝑐‘0) 1stF (𝑐‘1)) = (𝐶 1stF 𝐷))
29	18, 28	oveq12d 7174	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) = (𝐺 ∘func (𝐶 1stF 𝐷)))
30	27, 10	oveq12d 7174	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → ((𝑐‘0) 2ndF (𝑐‘1)) = (𝐶 2ndF 𝐷))
31	29, 30	oveq12d 7174	. . . 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 7174	. . 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 14243	. . . 4 ⊢ ⟨“𝐶𝐷𝐸”⟩ ∈ Word V
34		elex 3512	. . . 4 ⊢ (⟨“𝐶𝐷𝐸”⟩ ∈ Word V → ⟨“𝐶𝐷𝐸”⟩ ∈ V)
35	33, 34	mp1i 13	. . 3 ⊢ (𝜑 → ⟨“𝐶𝐷𝐸”⟩ ∈ V)
36	20	elexd 3514	. . 3 ⊢ (𝜑 → 𝐺 ∈ V)
37		ovexd 7191	. . 3 ⊢ (𝜑 → ((𝐷 evalF 𝐸) ∘func ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))) ∈ V)
38	3, 32, 35, 36, 37	ovmpod 7302	. 2 ⊢ (𝜑 → (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺) = ((𝐷 evalF 𝐸) ∘func ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))
39	1, 38	syl5eq 2868	1 ⊢ (𝜑 → 𝐹 = ((𝐷 evalF 𝐸) ∘func ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  Vcvv 3494  ‘cfv 6355  (class class class)co 7156   ∈ cmpo 7158  0cc0 10537  1c1 10538  2c2 11693  Word cword 13862  ⟨“cs3 14204  Catccat 16935   Func cfunc 17124   ∘_func ccofu 17126   FuncCat cfuc 17212   1^st_F c1stf 17419   2^nd_F c2ndf 17420   ⟨,⟩_F cprf 17421   eval_F cevlf 17459   uncurry_F cuncf 17461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-oadd 8106  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-card 9368  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-2 11701  df-n0 11899  df-z 11983  df-uz 12245  df-fz 12894  df-fzo 13035  df-hash 13692  df-word 13863  df-concat 13923  df-s1 13950  df-s2 14210  df-s3 14211  df-func 17128  df-uncf 17465

This theorem is referenced by:  uncfcl  17485  uncf1  17486  uncf2  17487