Theorem precofval 50025

Description: Value of the pre-composition functor as a transposed curry of the functor composition bifunctor. (Contributed by Zhi Wang, 11-Oct-2025.)

Hypotheses

Ref	Expression
precofval.q	⊢ 𝑄 = (𝐶 FuncCat 𝐷)
precofval.r	⊢ 𝑅 = (𝐷 FuncCat 𝐸)
precofval.o	⊢ (𝜑 → ⚬ = (⟨𝑄, 𝑅⟩ curryF ((⟨𝐶, 𝐷⟩ ∘F 𝐸) ∘func (𝑄 swapF 𝑅))))
precofval.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
precofval.e	⊢ (𝜑 → 𝐸 ∈ Cat)
precofval.k	⊢ (𝜑 → 𝐾 = ((1st ‘ ⚬ )‘𝐹))

Assertion

Ref	Expression
precofval	⊢ (𝜑 → 𝐾 = ⟨(𝑔 ∈ (𝐷 Func 𝐸) ↦ (𝑔 ∘func 𝐹)), (𝑔 ∈ (𝐷 Func 𝐸), ℎ ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ) ↦ (𝑥 ∈ (Base‘𝐶) ↦ (𝑎‘((1st ‘𝐹)‘𝑥)))))⟩)

Distinct variable groups:   𝐶,𝑎,𝑔,ℎ,𝑥   𝐷,𝑎,𝑔,ℎ,𝑥   𝐸,𝑎,𝑔,ℎ,𝑥   𝐹,𝑎,𝑔,ℎ,𝑥   𝑄,𝑎,𝑔,ℎ   𝑅,𝑎,𝑔,ℎ   𝜑,𝑎,𝑔,ℎ,𝑥

Allowed substitution hints:   𝑄(𝑥)   𝑅(𝑥)   𝐾(𝑥,𝑔,ℎ,𝑎)   ⚬ (𝑥,𝑔,ℎ,𝑎)

Proof of Theorem precofval

Step	Hyp	Ref	Expression
1		precofval.o	. . 3 ⊢ (𝜑 → ⚬ = (⟨𝑄, 𝑅⟩ curryF ((⟨𝐶, 𝐷⟩ ∘F 𝐸) ∘func (𝑄 swapF 𝑅))))
2		precofval.q	. . . 4 ⊢ 𝑄 = (𝐶 FuncCat 𝐷)
3	2	fucbas 18016	. . 3 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
4		precofval.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
5	4	func1st2nd 49734	. . . . 5 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
6	5	funcrcl2 49737	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
7	5	funcrcl3 49738	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
8	2, 6, 7	fuccat 18026	. . 3 ⊢ (𝜑 → 𝑄 ∈ Cat)
9		precofval.r	. . . 4 ⊢ 𝑅 = (𝐷 FuncCat 𝐸)
10		precofval.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat)
11	9, 7, 10	fuccat 18026	. . 3 ⊢ (𝜑 → 𝑅 ∈ Cat)
12	9, 2	oveq12i 7420	. . . 4 ⊢ (𝑅 ×c 𝑄) = ((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷))
13		eqid 2769	. . . 4 ⊢ (𝐶 FuncCat 𝐸) = (𝐶 FuncCat 𝐸)
14	12, 13, 6, 7, 10	fucofunca 50018	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) ∈ ((𝑅 ×c 𝑄) Func (𝐶 FuncCat 𝐸)))
15		precofval.k	. . 3 ⊢ (𝜑 → 𝐾 = ((1st ‘ ⚬ )‘𝐹))
16	9	fucbas 18016	. . 3 ⊢ (𝐷 Func 𝐸) = (Base‘𝑅)
17		eqid 2769	. . . 4 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
18	9, 17	fuchom 18017	. . 3 ⊢ (𝐷 Nat 𝐸) = (Hom ‘𝑅)
19		eqid 2769	. . 3 ⊢ (Id‘𝑄) = (Id‘𝑄)
20	1, 3, 8, 11, 14, 4, 15, 16, 18, 19	tposcurf1 49957	. 2 ⊢ (𝜑 → 𝐾 = ⟨(𝑔 ∈ (𝐷 Func 𝐸) ↦ (𝑔(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))𝐹)), (𝑔 ∈ (𝐷 Func 𝐸), ℎ ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ) ↦ (𝑎(⟨𝑔, 𝐹⟩(2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))⟨ℎ, 𝐹⟩)((Id‘𝑄)‘𝐹))))⟩)
21		eqidd 2770	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 Func 𝐸)) → (1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)) = (1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)))
22	4	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 Func 𝐸)) → 𝐹 ∈ (𝐶 Func 𝐷))
23		simpr 489	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 Func 𝐸)) → 𝑔 ∈ (𝐷 Func 𝐸))
24	21, 22, 23	fuco11b 49995	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐷 Func 𝐸)) → (𝑔(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))𝐹) = (𝑔 ∘func 𝐹))
25	24	mpteq2dva 5205	. . 3 ⊢ (𝜑 → (𝑔 ∈ (𝐷 Func 𝐸) ↦ (𝑔(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))𝐹)) = (𝑔 ∈ (𝐷 Func 𝐸) ↦ (𝑔 ∘func 𝐹)))
26		eqidd 2770	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) → (2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)) = (2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)))
27		simpr 489	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) → 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ))
28	4	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) → 𝐹 ∈ (𝐶 Func 𝐷))
29	26, 19, 2, 27, 28	fucorid 50020	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) → (𝑎(⟨𝑔, 𝐹⟩(2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))⟨ℎ, 𝐹⟩)((Id‘𝑄)‘𝐹)) = (𝑥 ∈ (Base‘𝐶) ↦ (𝑎‘((1st ‘𝐹)‘𝑥))))
30	29	mpteq2dva 5205	. . . 4 ⊢ (𝜑 → (𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ) ↦ (𝑎(⟨𝑔, 𝐹⟩(2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))⟨ℎ, 𝐹⟩)((Id‘𝑄)‘𝐹))) = (𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ) ↦ (𝑥 ∈ (Base‘𝐶) ↦ (𝑎‘((1st ‘𝐹)‘𝑥)))))
31	30	mpoeq3dv 7487	. . 3 ⊢ (𝜑 → (𝑔 ∈ (𝐷 Func 𝐸), ℎ ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ) ↦ (𝑎(⟨𝑔, 𝐹⟩(2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))⟨ℎ, 𝐹⟩)((Id‘𝑄)‘𝐹)))) = (𝑔 ∈ (𝐷 Func 𝐸), ℎ ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ) ↦ (𝑥 ∈ (Base‘𝐶) ↦ (𝑎‘((1st ‘𝐹)‘𝑥))))))
32	25, 31	opeq12d 4847	. 2 ⊢ (𝜑 → ⟨(𝑔 ∈ (𝐷 Func 𝐸) ↦ (𝑔(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))𝐹)), (𝑔 ∈ (𝐷 Func 𝐸), ℎ ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ) ↦ (𝑎(⟨𝑔, 𝐹⟩(2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))⟨ℎ, 𝐹⟩)((Id‘𝑄)‘𝐹))))⟩ = ⟨(𝑔 ∈ (𝐷 Func 𝐸) ↦ (𝑔 ∘func 𝐹)), (𝑔 ∈ (𝐷 Func 𝐸), ℎ ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ) ↦ (𝑥 ∈ (Base‘𝐶) ↦ (𝑎‘((1st ‘𝐹)‘𝑥)))))⟩)
33	20, 32	eqtrd 2804	1 ⊢ (𝜑 → 𝐾 = ⟨(𝑔 ∈ (𝐷 Func 𝐸) ↦ (𝑔 ∘func 𝐹)), (𝑔 ∈ (𝐷 Func 𝐸), ℎ ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ) ↦ (𝑥 ∈ (Base‘𝐶) ↦ (𝑎‘((1st ‘𝐹)‘𝑥)))))⟩)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ⟨cop 4597   ↦ cmpt 5193  ‘cfv 6534  (class class class)co 7408   ∈ cmpo 7410  1^st c1st 7980  2^nd c2nd 7981  Basecbs 17265  Catccat 17716  Idccid 17717   Func cfunc 17907   ∘_func ccofu 17909   Nat cnat 17997   FuncCat cfuc 17998   ×_c cxpc 18220   curry_F ccurf 18262   swap_F cswapf 49917   ∘_F cfuco 49974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-cnex 11152  ax-resscn 11153  ax-1cn 11154  ax-icn 11155  ax-addcl 11156  ax-addrcl 11157  ax-mulcl 11158  ax-mulrcl 11159  ax-mulcom 11160  ax-addass 11161  ax-mulass 11162  ax-distr 11163  ax-i2m1 11164  ax-1ne0 11165  ax-1rid 11166  ax-rnegex 11167  ax-rrecex 11168  ax-cnre 11169  ax-pre-lttri 11170  ax-pre-lttrn 11171  ax-pre-ltadd 11172  ax-pre-mulgt0 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6300  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-map 8822  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-pnf 11241  df-mnf 11242  df-xr 11243  df-ltxr 11244  df-le 11245  df-sub 11439  df-neg 11440  df-nn 12230  df-2 12299  df-3 12300  df-4 12301  df-5 12302  df-6 12303  df-7 12304  df-8 12305  df-9 12306  df-n0 12501  df-z 12588  df-dec 12708  df-uz 12859  df-fz 13532  df-struct 17203  df-slot 17238  df-ndx 17250  df-base 17266  df-hom 17330  df-cco 17331  df-cat 17720  df-cid 17721  df-func 17911  df-cofu 17913  df-nat 17999  df-fuc 18000  df-xpc 18224  df-curf 18266  df-swapf 49918  df-fuco 49975

This theorem is referenced by:  precofval2  50027