Theorem precofval2 49331

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
precofval2	⊢ (𝜑 → 𝐾 = ⟨(𝑔 ∈ (𝐷 Func 𝐸) ↦ (𝑔 ∘func 𝐹)), (𝑔 ∈ (𝐷 Func 𝐸), ℎ ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ) ↦ (𝑎 ∘ (1st ‘𝐹))))⟩)

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

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

Proof of Theorem precofval2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		precofval.q	. . 3 ⊢ 𝑄 = (𝐶 FuncCat 𝐷)
2		precofval.r	. . 3 ⊢ 𝑅 = (𝐷 FuncCat 𝐸)
3		precofval.o	. . 3 ⊢ (𝜑 → ⚬ = (⟨𝑄, 𝑅⟩ curryF ((⟨𝐶, 𝐷⟩ ∘F 𝐸) ∘func (𝑄 swapF 𝑅))))
4		precofval.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
5		precofval.e	. . 3 ⊢ (𝜑 → 𝐸 ∈ Cat)
6		precofval.k	. . 3 ⊢ (𝜑 → 𝐾 = ((1st ‘ ⚬ )‘𝐹))
7	1, 2, 3, 4, 5, 6	precofval 49329	. 2 ⊢ (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)))))⟩)
8		eqid 2729	. . . . . . . 8 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
9		id 22	. . . . . . . . 9 ⊢ (𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ) → 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ))
10	8, 9	nat1st2nd 17892	. . . . . . . 8 ⊢ (𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ) → 𝑎 ∈ (⟨(1st ‘𝑔), (2nd ‘𝑔)⟩(𝐷 Nat 𝐸)⟨(1st ‘ℎ), (2nd ‘ℎ)⟩))
11		eqid 2729	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
12	8, 10, 11	natfn 17895	. . . . . . 7 ⊢ (𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ) → 𝑎 Fn (Base‘𝐷))
13		dffn2 6672	. . . . . . 7 ⊢ (𝑎 Fn (Base‘𝐷) ↔ 𝑎:(Base‘𝐷)⟶V)
14	12, 13	sylib 218	. . . . . 6 ⊢ (𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ) → 𝑎:(Base‘𝐷)⟶V)
15		eqid 2729	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
16	4	func1st2nd 49038	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
17	15, 11, 16	funcf1 17804	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
18		fcompt 7087	. . . . . 6 ⊢ ((𝑎:(Base‘𝐷)⟶V ∧ (1st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷)) → (𝑎 ∘ (1st ‘𝐹)) = (𝑥 ∈ (Base‘𝐶) ↦ (𝑎‘((1st ‘𝐹)‘𝑥))))
19	14, 17, 18	syl2anr 597	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ)) → (𝑎 ∘ (1st ‘𝐹)) = (𝑥 ∈ (Base‘𝐶) ↦ (𝑎‘((1st ‘𝐹)‘𝑥))))
20	19	mpteq2dva 5195	. . . 4 ⊢ (𝜑 → (𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ) ↦ (𝑎 ∘ (1st ‘𝐹))) = (𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ) ↦ (𝑥 ∈ (Base‘𝐶) ↦ (𝑎‘((1st ‘𝐹)‘𝑥)))))
21	20	mpoeq3dv 7448	. . 3 ⊢ (𝜑 → (𝑔 ∈ (𝐷 Func 𝐸), ℎ ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ) ↦ (𝑎 ∘ (1st ‘𝐹)))) = (𝑔 ∈ (𝐷 Func 𝐸), ℎ ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ) ↦ (𝑥 ∈ (Base‘𝐶) ↦ (𝑎‘((1st ‘𝐹)‘𝑥))))))
22	21	opeq2d 4840	. 2 ⊢ (𝜑 → ⟨(𝑔 ∈ (𝐷 Func 𝐸) ↦ (𝑔 ∘func 𝐹)), (𝑔 ∈ (𝐷 Func 𝐸), ℎ ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ) ↦ (𝑎 ∘ (1st ‘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)))))⟩)
23	7, 22	eqtr4d 2767	1 ⊢ (𝜑 → 𝐾 = ⟨(𝑔 ∈ (𝐷 Func 𝐸) ↦ (𝑔 ∘func 𝐹)), (𝑔 ∈ (𝐷 Func 𝐸), ℎ ∈ (𝐷 Func 𝐸) ↦ (𝑎 ∈ (𝑔(𝐷 Nat 𝐸)ℎ) ↦ (𝑎 ∘ (1st ‘𝐹))))⟩)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3444  ⟨cop 4591   ↦ cmpt 5183   ∘ ccom 5635   Fn wfn 6494  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369   ∈ cmpo 7371  1^st c1st 7945  2^nd c2nd 7946  Basecbs 17155  Catccat 17601   Func cfunc 17792   ∘_func ccofu 17794   Nat cnat 17882   FuncCat cfuc 17883   curry_F ccurf 18147   swap_F cswapf 49221   ∘_F cfuco 49278

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-er 8648  df-map 8778  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12419  df-z 12506  df-dec 12626  df-uz 12770  df-fz 13445  df-struct 17093  df-slot 17128  df-ndx 17140  df-base 17156  df-hom 17220  df-cco 17221  df-cat 17605  df-cid 17606  df-func 17796  df-cofu 17798  df-nat 17884  df-fuc 17885  df-xpc 18109  df-curf 18151  df-swapf 49222  df-fuco 49279

This theorem is referenced by:  precofval3  49333