Theorem uncfval 18322

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 18303	. . . 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 783	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → 𝑐 = ⟨“𝐶𝐷𝐸”⟩)
5	4	fveq1d 6880	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (𝑐‘1) = (⟨“𝐶𝐷𝐸”⟩‘1))
6		uncfval.c	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ Cat)
7		s3fv1 14963	. . . . . . . 8 ⊢ (𝐷 ∈ Cat → (⟨“𝐶𝐷𝐸”⟩‘1) = 𝐷)
8	6, 7	syl 18	. . . . . . 7 ⊢ (𝜑 → (⟨“𝐶𝐷𝐸”⟩‘1) = 𝐷)
9	8	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (⟨“𝐶𝐷𝐸”⟩‘1) = 𝐷)
10	5, 9	eqtrd 2795	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (𝑐‘1) = 𝐷)
11	4	fveq1d 6880	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (𝑐‘2) = (⟨“𝐶𝐷𝐸”⟩‘2))
12		uncfval.d	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Cat)
13		s3fv2 14964	. . . . . . . 8 ⊢ (𝐸 ∈ Cat → (⟨“𝐶𝐷𝐸”⟩‘2) = 𝐸)
14	12, 13	syl 18	. . . . . . 7 ⊢ (𝜑 → (⟨“𝐶𝐷𝐸”⟩‘2) = 𝐸)
15	14	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (⟨“𝐶𝐷𝐸”⟩‘2) = 𝐸)
16	11, 15	eqtrd 2795	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (𝑐‘2) = 𝐸)
17	10, 16	oveq12d 7431	. . . 4 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → ((𝑐‘1) evalF (𝑐‘2)) = (𝐷 evalF 𝐸))
18		simprr 785	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → 𝑓 = 𝐺)
19	4	fveq1d 6880	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (𝑐‘0) = (⟨“𝐶𝐷𝐸”⟩‘0))
20		uncfval.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)))
21		funcrcl 17952	. . . . . . . . . . . 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 500	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ Cat)
24		s3fv0 14962	. . . . . . . . . 10 ⊢ (𝐶 ∈ Cat → (⟨“𝐶𝐷𝐸”⟩‘0) = 𝐶)
25	23, 24	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (⟨“𝐶𝐷𝐸”⟩‘0) = 𝐶)
26	25	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (⟨“𝐶𝐷𝐸”⟩‘0) = 𝐶)
27	19, 26	eqtrd 2795	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (𝑐‘0) = 𝐶)
28	27, 10	oveq12d 7431	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → ((𝑐‘0) 1stF (𝑐‘1)) = (𝐶 1stF 𝐷))
29	18, 28	oveq12d 7431	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → (𝑓 ∘func ((𝑐‘0) 1stF (𝑐‘1))) = (𝐺 ∘func (𝐶 1stF 𝐷)))
30	27, 10	oveq12d 7431	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 = ⟨“𝐶𝐷𝐸”⟩ ∧ 𝑓 = 𝐺)) → ((𝑐‘0) 2ndF (𝑐‘1)) = (𝐶 2ndF 𝐷))
31	29, 30	oveq12d 7431	. . . 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 7431	. . 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 14952	. . . 4 ⊢ ⟨“𝐶𝐷𝐸”⟩ ∈ Word V
34		elex 3471	. . . 4 ⊢ (⟨“𝐶𝐷𝐸”⟩ ∈ Word V → ⟨“𝐶𝐷𝐸”⟩ ∈ V)
35	33, 34	mp1i 14	. . 3 ⊢ (𝜑 → ⟨“𝐶𝐷𝐸”⟩ ∈ V)
36	20	elexd 3473	. . 3 ⊢ (𝜑 → 𝐺 ∈ V)
37		ovexd 7448	. . 3 ⊢ (𝜑 → ((𝐷 evalF 𝐸) ∘func ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))) ∈ V)
38	3, 32, 35, 36, 37	ovmpod 7565	. 2 ⊢ (𝜑 → (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺) = ((𝐷 evalF 𝐸) ∘func ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))
39	1, 38	eqtrid 2807	1 ⊢ (𝜑 → 𝐹 = ((𝐷 evalF 𝐸) ∘func ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3450  ‘cfv 6533  (class class class)co 7413   ∈ cmpo 7415  0cc0 11124  1c1 11125  2c2 12319  Word cword 14578  ⟨“cs3 14913  Catccat 17752   Func cfunc 17943   ∘_func ccofu 17945   FuncCat cfuc 18034   1^st_F c1stf 18257   2^nd_F c2ndf 18258   ⟨,⟩_F cprf 18259   eval_F cevlf 18297   uncurry_F cuncf 18299

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-er 8696  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-card 9944  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-nn 12258  df-2 12327  df-n0 12529  df-z 12616  df-uz 12888  df-fz 13562  df-fzo 13710  df-hash 14395  df-word 14579  df-concat 14636  df-s1 14663  df-s2 14919  df-s3 14920  df-func 17947  df-uncf 18303

This theorem is used by:  uncfcl  18323  uncf1  18324  uncf2  18325