Theorem postcofval 49369

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

Hypotheses

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

Assertion

Ref	Expression
postcofval	⊢ (𝜑 → 𝐾 = ⟨(𝑔 ∈ (𝐶 Func 𝐷) ↦ (𝐹 ∘func 𝑔)), (𝑔 ∈ (𝐶 Func 𝐷), ℎ ∈ (𝐶 Func 𝐷) ↦ (𝑎 ∈ (𝑔(𝐶 Nat 𝐷)ℎ) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((((1st ‘𝑔)‘𝑥)(2nd ‘𝐹)((1st ‘ℎ)‘𝑥))‘(𝑎‘𝑥)))))⟩)

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

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

Proof of Theorem postcofval

Step	Hyp	Ref	Expression
1		postcofval.o	. . 3 ⊢ ⚬ = (⟨𝑅, 𝑄⟩ curryF (⟨𝐶, 𝐷⟩ ∘F 𝐸))
2		postcofval.r	. . . 4 ⊢ 𝑅 = (𝐷 FuncCat 𝐸)
3	2	fucbas 17889	. . 3 ⊢ (𝐷 Func 𝐸) = (Base‘𝑅)
4		postcofval.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
5	4	func1st2nd 49081	. . . . 5 ⊢ (𝜑 → (1st ‘𝐹)(𝐷 Func 𝐸)(2nd ‘𝐹))
6	5	funcrcl2 49084	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
7	5	funcrcl3 49085	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat)
8	2, 6, 7	fuccat 17899	. . 3 ⊢ (𝜑 → 𝑅 ∈ Cat)
9		postcofval.q	. . . 4 ⊢ 𝑄 = (𝐶 FuncCat 𝐷)
10		postcofval.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
11	9, 10, 6	fuccat 17899	. . 3 ⊢ (𝜑 → 𝑄 ∈ Cat)
12	2, 9	oveq12i 7365	. . . 4 ⊢ (𝑅 ×c 𝑄) = ((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷))
13		eqid 2729	. . . 4 ⊢ (𝐶 FuncCat 𝐸) = (𝐶 FuncCat 𝐸)
14	12, 13, 10, 6, 7	fucofunca 49365	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) ∈ ((𝑅 ×c 𝑄) Func (𝐶 FuncCat 𝐸)))
15	9	fucbas 17889	. . 3 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
16		postcofval.k	. . 3 ⊢ 𝐾 = ((1st ‘ ⚬ )‘𝐹)
17		eqid 2729	. . . 4 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
18	9, 17	fuchom 17890	. . 3 ⊢ (𝐶 Nat 𝐷) = (Hom ‘𝑄)
19		eqid 2729	. . 3 ⊢ (Id‘𝑅) = (Id‘𝑅)
20	1, 3, 8, 11, 14, 15, 4, 16, 18, 19	curf1 18150	. 2 ⊢ (𝜑 → 𝐾 = ⟨(𝑔 ∈ (𝐶 Func 𝐷) ↦ (𝐹(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))𝑔)), (𝑔 ∈ (𝐶 Func 𝐷), ℎ ∈ (𝐶 Func 𝐷) ↦ (𝑎 ∈ (𝑔(𝐶 Nat 𝐷)ℎ) ↦ (((Id‘𝑅)‘𝐹)(⟨𝐹, 𝑔⟩(2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))⟨𝐹, ℎ⟩)𝑎)))⟩)
21		eqidd 2730	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐶 Func 𝐷)) → (1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)) = (1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)))
22		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐶 Func 𝐷)) → 𝑔 ∈ (𝐶 Func 𝐷))
23	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐶 Func 𝐷)) → 𝐹 ∈ (𝐷 Func 𝐸))
24	21, 22, 23	fuco11b 49342	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐶 Func 𝐷)) → (𝐹(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))𝑔) = (𝐹 ∘func 𝑔))
25	24	mpteq2dva 5188	. . 3 ⊢ (𝜑 → (𝑔 ∈ (𝐶 Func 𝐷) ↦ (𝐹(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))𝑔)) = (𝑔 ∈ (𝐶 Func 𝐷) ↦ (𝐹 ∘func 𝑔)))
26		eqidd 2730	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑔(𝐶 Nat 𝐷)ℎ)) → (2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)) = (2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)))
27		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑔(𝐶 Nat 𝐷)ℎ)) → 𝑎 ∈ (𝑔(𝐶 Nat 𝐷)ℎ))
28	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑔(𝐶 Nat 𝐷)ℎ)) → 𝐹 ∈ (𝐷 Func 𝐸))
29	26, 19, 2, 27, 28	fucolid 49366	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑔(𝐶 Nat 𝐷)ℎ)) → (((Id‘𝑅)‘𝐹)(⟨𝐹, 𝑔⟩(2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))⟨𝐹, ℎ⟩)𝑎) = (𝑥 ∈ (Base‘𝐶) ↦ ((((1st ‘𝑔)‘𝑥)(2nd ‘𝐹)((1st ‘ℎ)‘𝑥))‘(𝑎‘𝑥))))
30	29	mpteq2dva 5188	. . . 4 ⊢ (𝜑 → (𝑎 ∈ (𝑔(𝐶 Nat 𝐷)ℎ) ↦ (((Id‘𝑅)‘𝐹)(⟨𝐹, 𝑔⟩(2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))⟨𝐹, ℎ⟩)𝑎)) = (𝑎 ∈ (𝑔(𝐶 Nat 𝐷)ℎ) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((((1st ‘𝑔)‘𝑥)(2nd ‘𝐹)((1st ‘ℎ)‘𝑥))‘(𝑎‘𝑥)))))
31	30	mpoeq3dv 7432	. . 3 ⊢ (𝜑 → (𝑔 ∈ (𝐶 Func 𝐷), ℎ ∈ (𝐶 Func 𝐷) ↦ (𝑎 ∈ (𝑔(𝐶 Nat 𝐷)ℎ) ↦ (((Id‘𝑅)‘𝐹)(⟨𝐹, 𝑔⟩(2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))⟨𝐹, ℎ⟩)𝑎))) = (𝑔 ∈ (𝐶 Func 𝐷), ℎ ∈ (𝐶 Func 𝐷) ↦ (𝑎 ∈ (𝑔(𝐶 Nat 𝐷)ℎ) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((((1st ‘𝑔)‘𝑥)(2nd ‘𝐹)((1st ‘ℎ)‘𝑥))‘(𝑎‘𝑥))))))
32	25, 31	opeq12d 4835	. 2 ⊢ (𝜑 → ⟨(𝑔 ∈ (𝐶 Func 𝐷) ↦ (𝐹(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))𝑔)), (𝑔 ∈ (𝐶 Func 𝐷), ℎ ∈ (𝐶 Func 𝐷) ↦ (𝑎 ∈ (𝑔(𝐶 Nat 𝐷)ℎ) ↦ (((Id‘𝑅)‘𝐹)(⟨𝐹, 𝑔⟩(2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))⟨𝐹, ℎ⟩)𝑎)))⟩ = ⟨(𝑔 ∈ (𝐶 Func 𝐷) ↦ (𝐹 ∘func 𝑔)), (𝑔 ∈ (𝐶 Func 𝐷), ℎ ∈ (𝐶 Func 𝐷) ↦ (𝑎 ∈ (𝑔(𝐶 Nat 𝐷)ℎ) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((((1st ‘𝑔)‘𝑥)(2nd ‘𝐹)((1st ‘ℎ)‘𝑥))‘(𝑎‘𝑥)))))⟩)
33	20, 32	eqtrd 2764	1 ⊢ (𝜑 → 𝐾 = ⟨(𝑔 ∈ (𝐶 Func 𝐷) ↦ (𝐹 ∘func 𝑔)), (𝑔 ∈ (𝐶 Func 𝐷), ℎ ∈ (𝐶 Func 𝐷) ↦ (𝑎 ∈ (𝑔(𝐶 Nat 𝐷)ℎ) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((((1st ‘𝑔)‘𝑥)(2nd ‘𝐹)((1st ‘ℎ)‘𝑥))‘(𝑎‘𝑥)))))⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ⟨cop 4585   ↦ cmpt 5176  ‘cfv 6486  (class class class)co 7353   ∈ cmpo 7355  1^st c1st 7929  2^nd c2nd 7930  Basecbs 17139  Catccat 17589  Idccid 17590   Func cfunc 17780   ∘_func ccofu 17782   Nat cnat 17870   FuncCat cfuc 17871   ×_c cxpc 18093   curry_F ccurf 18135   ∘_F cfuco 49321

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 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

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 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8632  df-map 8762  df-ixp 8832  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12148  df-2 12210  df-3 12211  df-4 12212  df-5 12213  df-6 12214  df-7 12215  df-8 12216  df-9 12217  df-n0 12404  df-z 12491  df-dec 12611  df-uz 12755  df-fz 13430  df-struct 17077  df-slot 17112  df-ndx 17124  df-base 17140  df-hom 17204  df-cco 17205  df-cat 17593  df-cid 17594  df-func 17784  df-cofu 17786  df-nat 17872  df-fuc 17873  df-xpc 18097  df-curf 18139  df-fuco 49322

This theorem is referenced by: (None)