Theorem postcofval 49109

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 17980	. . 3 ⊢ (𝐷 Func 𝐸) = (Base‘𝑅)
4		relfunc 17879	. . . . . 6 ⊢ Rel (𝐷 Func 𝐸)
5		postcofval.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
6		1st2ndbr 8049	. . . . . 6 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐹 ∈ (𝐷 Func 𝐸)) → (1st ‘𝐹)(𝐷 Func 𝐸)(2nd ‘𝐹))
7	4, 5, 6	sylancr 587	. . . . 5 ⊢ (𝜑 → (1st ‘𝐹)(𝐷 Func 𝐸)(2nd ‘𝐹))
8	7	funcrcl2 48937	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
9	7	funcrcl3 48938	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat)
10	2, 8, 9	fuccat 17990	. . 3 ⊢ (𝜑 → 𝑅 ∈ Cat)
11		postcofval.q	. . . 4 ⊢ 𝑄 = (𝐶 FuncCat 𝐷)
12		postcofval.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
13	11, 12, 8	fuccat 17990	. . 3 ⊢ (𝜑 → 𝑄 ∈ Cat)
14	2, 11	oveq12i 7425	. . . 4 ⊢ (𝑅 ×c 𝑄) = ((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷))
15		eqid 2734	. . . 4 ⊢ (𝐶 FuncCat 𝐸) = (𝐶 FuncCat 𝐸)
16	14, 15, 12, 8, 9	fucofunca 49105	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) ∈ ((𝑅 ×c 𝑄) Func (𝐶 FuncCat 𝐸)))
17	11	fucbas 17980	. . 3 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
18		postcofval.k	. . 3 ⊢ 𝐾 = ((1st ‘ ⚬ )‘𝐹)
19		eqid 2734	. . . 4 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
20	11, 19	fuchom 17981	. . 3 ⊢ (𝐶 Nat 𝐷) = (Hom ‘𝑄)
21		eqid 2734	. . 3 ⊢ (Id‘𝑅) = (Id‘𝑅)
22	1, 3, 10, 13, 16, 17, 5, 18, 20, 21	curf1 18241	. 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)))⟩)
23		eqidd 2735	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐶 Func 𝐷)) → (1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)) = (1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)))
24		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐶 Func 𝐷)) → 𝑔 ∈ (𝐶 Func 𝐷))
25	5	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐶 Func 𝐷)) → 𝐹 ∈ (𝐷 Func 𝐸))
26	23, 24, 25	fuco11b 49082	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝐶 Func 𝐷)) → (𝐹(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))𝑔) = (𝐹 ∘func 𝑔))
27	26	mpteq2dva 5222	. . 3 ⊢ (𝜑 → (𝑔 ∈ (𝐶 Func 𝐷) ↦ (𝐹(1st ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸))𝑔)) = (𝑔 ∈ (𝐶 Func 𝐷) ↦ (𝐹 ∘func 𝑔)))
28		eqidd 2735	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑔(𝐶 Nat 𝐷)ℎ)) → (2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)) = (2nd ‘(⟨𝐶, 𝐷⟩ ∘F 𝐸)))
29		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑔(𝐶 Nat 𝐷)ℎ)) → 𝑎 ∈ (𝑔(𝐶 Nat 𝐷)ℎ))
30	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑔(𝐶 Nat 𝐷)ℎ)) → 𝐹 ∈ (𝐷 Func 𝐸))
31	28, 21, 2, 29, 30	fucolid 49106	. . . . 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))))
32	31	mpteq2dva 5222	. . . 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)))))
33	32	mpoeq3dv 7494	. . 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))))))
34	27, 33	opeq12d 4861	. 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)))))⟩)
35	22, 34	eqtrd 2769	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 1539   ∈ wcel 2107  ⟨cop 4612   class class class wbr 5123   ↦ cmpt 5205  Rel wrel 5670  ‘cfv 6541  (class class class)co 7413   ∈ cmpo 7415  1^st c1st 7994  2^nd c2nd 7995  Basecbs 17230  Catccat 17679  Idccid 17680   Func cfunc 17871   ∘_func ccofu 17873   Nat cnat 17961   FuncCat cfuc 17962   ×_c cxpc 18184   curry_F ccurf 18226   ∘_F cfuco 49061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737  ax-cnex 11193  ax-resscn 11194  ax-1cn 11195  ax-icn 11196  ax-addcl 11197  ax-addrcl 11198  ax-mulcl 11199  ax-mulrcl 11200  ax-mulcom 11201  ax-addass 11202  ax-mulass 11203  ax-distr 11204  ax-i2m1 11205  ax-1ne0 11206  ax-1rid 11207  ax-rnegex 11208  ax-rrecex 11209  ax-cnre 11210  ax-pre-lttri 11211  ax-pre-lttrn 11212  ax-pre-ltadd 11213  ax-pre-mulgt0 11214

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-tr 5240  df-id 5558  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6301  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7870  df-1st 7996  df-2nd 7997  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-1o 8488  df-er 8727  df-map 8850  df-ixp 8920  df-en 8968  df-dom 8969  df-sdom 8970  df-fin 8971  df-pnf 11279  df-mnf 11280  df-xr 11281  df-ltxr 11282  df-le 11283  df-sub 11476  df-neg 11477  df-nn 12249  df-2 12311  df-3 12312  df-4 12313  df-5 12314  df-6 12315  df-7 12316  df-8 12317  df-9 12318  df-n0 12510  df-z 12597  df-dec 12717  df-uz 12861  df-fz 13530  df-struct 17167  df-slot 17202  df-ndx 17214  df-base 17231  df-hom 17298  df-cco 17299  df-cat 17683  df-cid 17684  df-func 17875  df-cofu 17877  df-nat 17963  df-fuc 17964  df-xpc 18188  df-curf 18230  df-fuco 49062

This theorem is referenced by: (None)