Theorem uncfval 18291

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 18272	. . . 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 6909	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (𝑐‘1) = (⟨“𝐶𝐷𝐸”⟩‘1))
6		uncfval.c	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ Cat)
7		s3fv1 14928	. . . . . . . 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 2775	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (𝑐‘1) = 𝐷)
11	4	fveq1d 6909	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (𝑐‘2) = (⟨“𝐶𝐷𝐸”⟩‘2))
12		uncfval.d	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Cat)
13		s3fv2 14929	. . . . . . . 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 2775	. . . . 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 6909	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (𝑐‘0) = (⟨“𝐶𝐷𝐸”⟩‘0))
20		uncfval.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
21		funcrcl 17914	. . . . . . . . . . . 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 14927	. . . . . . . . . 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 2775	. . . . . . 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 14917	. . . 4 ⊢ ⟨“𝐶𝐷𝐸”⟩ ∈ Word V
34		elex 3499	. . . 4 ⊢ (⟨“𝐶𝐷𝐸”⟩ ∈ Word V → ⟨“𝐶𝐷𝐸”⟩ ∈ V)
35	33, 34	mp1i 13	. . 3 ⊢ (𝜑 → ⟨“𝐶𝐷𝐸”⟩ ∈ V)
36	20	elexd 3502	. . 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 2787	1 ⊢ (𝜑 → 𝐹 = ((𝐷 evalF 𝐸) ∘func ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  Vcvv 3478  ‘cfv 6563  (class class class)co 7431   ∈ cmpo 7433  0cc0 11153  1c1 11154  2c2 12319  Word cword 14549  ⟨“cs3 14878  Catccat 17709   Func cfunc 17905   ∘_func ccofu 17907   FuncCat cfuc 17997   1^st_F c1stf 18225   2^nd_F c2ndf 18226   ⟨,⟩_F cprf 18227   eval_F cevlf 18266   uncurry_F cuncf 18268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692  df-hash 14367  df-word 14550  df-concat 14606  df-s1 14631  df-s2 14884  df-s3 14885  df-func 17909  df-uncf 18272

This theorem is referenced by:  uncfcl  18292  uncf1  18293  uncf2  18294