Theorem postcofval 49353

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 17925	. . 3 ⊢ (𝐷 Func 𝐸) = (Base‘𝑅)
4		postcofval.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
5	4	func1st2nd 49065	. . . . 5 ⊢ (𝜑 → (1st ‘𝐹)(𝐷 Func 𝐸)(2nd ‘𝐹))
6	5	funcrcl2 49068	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
7	5	funcrcl3 49069	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat)
8	2, 6, 7	fuccat 17935	. . 3 ⊢ (𝜑 → 𝑅 ∈ Cat)
9		postcofval.q	. . . 4 ⊢ 𝑄 = (𝐶 FuncCat 𝐷)
10		postcofval.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
11	9, 10, 6	fuccat 17935	. . 3 ⊢ (𝜑 → 𝑄 ∈ Cat)
12	2, 9	oveq12i 7399	. . . 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 49349	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) ∈ ((𝑅 ×c 𝑄) Func (𝐶 FuncCat 𝐸)))
15	9	fucbas 17925	. . 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 17926	. . 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 18186	. 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 49326	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐶 Func 𝐷)) → (𝐹(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))𝑔) = (𝐹 ∘func 𝑔))
25	24	mpteq2dva 5200	. . 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 49350	. . . . 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 5200	. . . 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 7468	. . 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 4845	. 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 4595   ↦ cmpt 5188  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389  1^st c1st 7966  2^nd c2nd 7967  Basecbs 17179  Catccat 17625  Idccid 17626   Func cfunc 17816   ∘_func ccofu 17818   Nat cnat 17906   FuncCat cfuc 17907   ×_c cxpc 18129   curry_F ccurf 18171   ∘_F cfuco 49305

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 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

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 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-map 8801  df-ixp 8871  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-uz 12794  df-fz 13469  df-struct 17117  df-slot 17152  df-ndx 17164  df-base 17180  df-hom 17244  df-cco 17245  df-cat 17629  df-cid 17630  df-func 17820  df-cofu 17822  df-nat 17908  df-fuc 17909  df-xpc 18133  df-curf 18175  df-fuco 49306

This theorem is referenced by: (None)