Theorem uncfval 18278

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 18259	. . . 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 782	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → 𝑐 = ⟨“𝐶𝐷𝐸”⟩)
5	4	fveq1d 6873	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (𝑐‘1) = (⟨“𝐶𝐷𝐸”⟩‘1))
6		uncfval.c	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ Cat)
7		s3fv1 14917	. . . . . . . 8 ⊢ (𝐷 ∈ Cat → (⟨“𝐶𝐷𝐸”⟩‘1) = 𝐷)
8	6, 7	syl 18	. . . . . . 7 ⊢ (𝜑 → (⟨“𝐶𝐷𝐸”⟩‘1) = 𝐷)
9	8	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (⟨“𝐶𝐷𝐸”⟩‘1) = 𝐷)
10	5, 9	eqtrd 2800	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (𝑐‘1) = 𝐷)
11	4	fveq1d 6873	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (𝑐‘2) = (⟨“𝐶𝐷𝐸”⟩‘2))
12		uncfval.d	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Cat)
13		s3fv2 14918	. . . . . . . 8 ⊢ (𝐸 ∈ Cat → (⟨“𝐶𝐷𝐸”⟩‘2) = 𝐸)
14	12, 13	syl 18	. . . . . . 7 ⊢ (𝜑 → (⟨“𝐶𝐷𝐸”⟩‘2) = 𝐸)
15	14	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (⟨“𝐶𝐷𝐸”⟩‘2) = 𝐸)
16	11, 15	eqtrd 2800	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (𝑐‘2) = 𝐸)
17	10, 16	oveq12d 7418	. . . 4 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → ((𝑐‘1) evalF (𝑐‘2)) = (𝐷 evalF 𝐸))
18		simprr 784	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → 𝑓 = 𝐺)
19	4	fveq1d 6873	. . . . . . . 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 18	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ (𝐷 FuncCat 𝐸) ∈ Cat))
23	22	simpld 499	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ Cat)
24		s3fv0 14916	. . . . . . . . . 10 ⊢ (𝐶 ∈ Cat → (⟨“𝐶𝐷𝐸”⟩‘0) = 𝐶)
25	23, 24	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (⟨“𝐶𝐷𝐸”⟩‘0) = 𝐶)
26	25	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (⟨“𝐶𝐷𝐸”⟩‘0) = 𝐶)
27	19, 26	eqtrd 2800	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (𝑐‘0) = 𝐶)
28	27, 10	oveq12d 7418	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → ((𝑐‘0) 1stF (𝑐‘1)) = (𝐶 1stF 𝐷))
29	18, 28	oveq12d 7418	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) = (𝐺 ∘func (𝐶 1stF 𝐷)))
30	27, 10	oveq12d 7418	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → ((𝑐‘0) 2ndF (𝑐‘1)) = (𝐶 2ndF 𝐷))
31	29, 30	oveq12d 7418	. . . 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 7418	. . 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 14906	. . . 4 ⊢ ⟨“𝐶𝐷𝐸”⟩ ∈ Word V
34		elex 3478	. . . 4 ⊢ (⟨“𝐶𝐷𝐸”⟩ ∈ Word V → ⟨“𝐶𝐷𝐸”⟩ ∈ V)
35	33, 34	mp1i 14	. . 3 ⊢ (𝜑 → ⟨“𝐶𝐷𝐸”⟩ ∈ V)
36	20	elexd 3480	. . 3 ⊢ (𝜑 → 𝐺 ∈ V)
37		ovexd 7435	. . 3 ⊢ (𝜑 → ((𝐷 evalF 𝐸) ∘func ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))) ∈ V)
38	3, 32, 35, 36, 37	ovmpod 7552	. 2 ⊢ (𝜑 → (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺) = ((𝐷 evalF 𝐸) ∘func ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))
39	1, 38	eqtrid 2812	1 ⊢ (𝜑 → 𝐹 = ((𝐷 evalF 𝐸) ∘func ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  Vcvv 3457  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402  0cc0 11088  1c1 11089  2c2 12283  Word cword 14538  ⟨“cs3 14867  Catccat 17708   Func cfunc 17899   ∘_func ccofu 17901   FuncCat cfuc 17990   1^st_F c1stf 18213   2^nd_F c2ndf 18214   ⟨,⟩_F cprf 18215   eval_F cevlf 18253   uncurry_F cuncf 18255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4908  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6291  df-ord 6352  df-on 6353  df-lim 6354  df-suc 6355  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-er 8682  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-card 9913  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-nn 12222  df-2 12291  df-n0 12493  df-z 12580  df-uz 12851  df-fz 13524  df-fzo 13671  df-hash 14355  df-word 14539  df-concat 14596  df-s1 14622  df-s2 14873  df-s3 14874  df-func 17903  df-uncf 18259

This theorem is referenced by:  uncfcl  18279  uncf1  18280  uncf2  18281