fucorid - Mathbox for Zhi Wang

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Theorem fucorid 49030

Description: Pre-composing a natural transformation with the identity natural transformation of a functor is pre-composing it with the object part of the functor, in maps-to notation. (Contributed by Zhi Wang, 11-Oct-2025.)

**Hypotheses**
Ref	Expression
fucolid.p	⊢ (𝜑 → (2^nd ‘(⟨𝐶, 𝐷⟩ ∘_F 𝐸)) = 𝑃)
fucolid.i	⊢ 𝐼 = (Id‘𝑄)
fucorid.q	⊢ 𝑄 = (𝐶 FuncCat 𝐷)
fucorid.a	⊢ (𝜑 → 𝐴 ∈ (𝐺(𝐷 Nat 𝐸)𝐻))
fucorid.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))

**Assertion**
Ref	Expression
fucorid	⊢ (𝜑 → (𝐴(⟨𝐺, 𝐹⟩𝑃⟨𝐻, 𝐹⟩)(𝐼‘𝐹)) = (𝑥 ∈ (Base‘𝐶) ↦ (𝐴‘((1^st ‘𝐹)‘𝑥))))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐶 𝑥,𝐷 𝑥,𝐸 𝑥,𝐹 𝑥,𝐺 𝑥,𝐻 𝜑,𝑥 Allowed substitution hints: 𝑃(𝑥) 𝑄(𝑥) 𝐼(𝑥)

**Proof of Theorem fucorid**
Step	Hyp	Ref	Expression
1		fucorid.q	. . . 4 ⊢ 𝑄 = (𝐶 FuncCat 𝐷)
2		fucolid.i	. . . 4 ⊢ 𝐼 = (Id‘𝑄)
3		eqid 2736	. . . 4 ⊢ (Id‘𝐷) = (Id‘𝐷)
4		fucorid.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
5	1, 2, 3, 4	fucid 18015	. . 3 ⊢ (𝜑 → (𝐼‘𝐹) = ((Id‘𝐷) ∘ (1^st ‘𝐹)))
6	5	oveq2d 7445	. 2 ⊢ (𝜑 → (𝐴(⟨𝐺, 𝐹⟩𝑃⟨𝐻, 𝐹⟩)(𝐼‘𝐹)) = (𝐴(⟨𝐺, 𝐹⟩𝑃⟨𝐻, 𝐹⟩)((Id‘𝐷) ∘ (1^st ‘𝐹))))
7		eqid 2736	. . . . . . . 8 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
8	1, 7, 3, 4	fucidcl 18009	. . . . . . . . 9 ⊢ (𝜑 → ((Id‘𝐷) ∘ (1^st ‘𝐹)) ∈ (𝐹(𝐶 Nat 𝐷)𝐹))
9	7, 8	nat1st2nd 17995	. . . . . . . 8 ⊢ (𝜑 → ((Id‘𝐷) ∘ (1^st ‘𝐹)) ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩(𝐶 Nat 𝐷)⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩))
10	7, 9	natrcl2 48923	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
11	10	funcrcl2 48885	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
12		eqid 2736	. . . . . . . 8 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
13		fucorid.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ (𝐺(𝐷 Nat 𝐸)𝐻))
14	12, 13	nat1st2nd 17995	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩(𝐷 Nat 𝐸)⟨(1^st ‘𝐻), (2^nd ‘𝐻)⟩))
15	12, 14	natrcl2 48923	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐺)(𝐷 Func 𝐸)(2^nd ‘𝐺))
16	15	funcrcl2 48885	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
17	15	funcrcl3 48886	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ Cat)
18		eqidd 2737	. . . . . 6 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = (⟨𝐶, 𝐷⟩ ∘_F 𝐸))
19	11, 16, 17, 18	fucoelvv 48988	. . . . 5 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) ∈ (V × V))
20		1st2nd2 8049	. . . . 5 ⊢ ((⟨𝐶, 𝐷⟩ ∘_F 𝐸) ∈ (V × V) → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨(1^st ‘(⟨𝐶, 𝐷⟩ ∘_F 𝐸)), (2^nd ‘(⟨𝐶, 𝐷⟩ ∘_F 𝐸))⟩)
21	19, 20	syl 17	. . . 4 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨(1^st ‘(⟨𝐶, 𝐷⟩ ∘_F 𝐸)), (2^nd ‘(⟨𝐶, 𝐷⟩ ∘_F 𝐸))⟩)
22		fucolid.p	. . . . 5 ⊢ (𝜑 → (2^nd ‘(⟨𝐶, 𝐷⟩ ∘_F 𝐸)) = 𝑃)
23	22	opeq2d 4878	. . . 4 ⊢ (𝜑 → ⟨(1^st ‘(⟨𝐶, 𝐷⟩ ∘_F 𝐸)), (2^nd ‘(⟨𝐶, 𝐷⟩ ∘_F 𝐸))⟩ = ⟨(1^st ‘(⟨𝐶, 𝐷⟩ ∘_F 𝐸)), 𝑃⟩)
24	21, 23	eqtrd 2776	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨(1^st ‘(⟨𝐶, 𝐷⟩ ∘_F 𝐸)), 𝑃⟩)
25		eqidd 2737	. . 3 ⊢ (𝜑 → ⟨𝐺, 𝐹⟩ = ⟨𝐺, 𝐹⟩)
26		eqidd 2737	. . 3 ⊢ (𝜑 → ⟨𝐻, 𝐹⟩ = ⟨𝐻, 𝐹⟩)
27	24, 25, 26, 8, 13	fuco22a 49018	. 2 ⊢ (𝜑 → (𝐴(⟨𝐺, 𝐹⟩𝑃⟨𝐻, 𝐹⟩)((Id‘𝐷) ∘ (1^st ‘𝐹))) = (𝑥 ∈ (Base‘𝐶) ↦ ((𝐴‘((1^st ‘𝐹)‘𝑥))(⟨((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑥)), ((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝐻)‘((1^st ‘𝐹)‘𝑥)))((((1^st ‘𝐹)‘𝑥)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑥))‘(((Id‘𝐷) ∘ (1^st ‘𝐹))‘𝑥)))))
28		eqid 2736	. . . . . . . . 9 ⊢ (Base‘𝐶) = (Base‘𝐶)
29		eqid 2736	. . . . . . . . 9 ⊢ (Base‘𝐷) = (Base‘𝐷)
30	10	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
31	28, 29, 30	funcf1 17907	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1^st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
32		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
33	31, 32	fvco3d 7007	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((Id‘𝐷) ∘ (1^st ‘𝐹))‘𝑥) = ((Id‘𝐷)‘((1^st ‘𝐹)‘𝑥)))
34	33	fveq2d 6908	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((((1^st ‘𝐹)‘𝑥)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑥))‘(((Id‘𝐷) ∘ (1^st ‘𝐹))‘𝑥)) = ((((1^st ‘𝐹)‘𝑥)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑥))‘((Id‘𝐷)‘((1^st ‘𝐹)‘𝑥))))
35		eqid 2736	. . . . . . 7 ⊢ (Id‘𝐸) = (Id‘𝐸)
36	15	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1^st ‘𝐺)(𝐷 Func 𝐸)(2^nd ‘𝐺))
37	31, 32	ffvelcdmd 7103	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
38	29, 3, 35, 36, 37	funcid 17911	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((((1^st ‘𝐹)‘𝑥)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑥))‘((Id‘𝐷)‘((1^st ‘𝐹)‘𝑥))) = ((Id‘𝐸)‘((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑥))))
39	34, 38	eqtrd 2776	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((((1^st ‘𝐹)‘𝑥)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑥))‘(((Id‘𝐷) ∘ (1^st ‘𝐹))‘𝑥)) = ((Id‘𝐸)‘((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑥))))
40	39	oveq2d 7445	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝐴‘((1^st ‘𝐹)‘𝑥))(⟨((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑥)), ((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝐻)‘((1^st ‘𝐹)‘𝑥)))((((1^st ‘𝐹)‘𝑥)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑥))‘(((Id‘𝐷) ∘ (1^st ‘𝐹))‘𝑥))) = ((𝐴‘((1^st ‘𝐹)‘𝑥))(⟨((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑥)), ((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝐻)‘((1^st ‘𝐹)‘𝑥)))((Id‘𝐸)‘((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑥)))))
41		eqid 2736	. . . . 5 ⊢ (Base‘𝐸) = (Base‘𝐸)
42		eqid 2736	. . . . 5 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
43	17	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐸 ∈ Cat)
44	29, 41, 36	funcf1 17907	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1^st ‘𝐺):(Base‘𝐷)⟶(Base‘𝐸))
45	44, 37	ffvelcdmd 7103	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑥)) ∈ (Base‘𝐸))
46		eqid 2736	. . . . 5 ⊢ (comp‘𝐸) = (comp‘𝐸)
47	12, 14	natrcl3 48924	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐻)(𝐷 Func 𝐸)(2^nd ‘𝐻))
48	47	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1^st ‘𝐻)(𝐷 Func 𝐸)(2^nd ‘𝐻))
49	29, 41, 48	funcf1 17907	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1^st ‘𝐻):(Base‘𝐷)⟶(Base‘𝐸))
50	49, 37	ffvelcdmd 7103	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐻)‘((1^st ‘𝐹)‘𝑥)) ∈ (Base‘𝐸))
51	14	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐴 ∈ (⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩(𝐷 Nat 𝐸)⟨(1^st ‘𝐻), (2^nd ‘𝐻)⟩))
52	12, 51, 29, 42, 37	natcl 17997	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝐴‘((1^st ‘𝐹)‘𝑥)) ∈ (((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑥))(Hom ‘𝐸)((1^st ‘𝐻)‘((1^st ‘𝐹)‘𝑥))))
53	41, 42, 35, 43, 45, 46, 50, 52	catrid 17723	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝐴‘((1^st ‘𝐹)‘𝑥))(⟨((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑥)), ((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝐻)‘((1^st ‘𝐹)‘𝑥)))((Id‘𝐸)‘((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑥)))) = (𝐴‘((1^st ‘𝐹)‘𝑥)))
54	40, 53	eqtrd 2776	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝐴‘((1^st ‘𝐹)‘𝑥))(⟨((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑥)), ((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝐻)‘((1^st ‘𝐹)‘𝑥)))((((1^st ‘𝐹)‘𝑥)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑥))‘(((Id‘𝐷) ∘ (1^st ‘𝐹))‘𝑥))) = (𝐴‘((1^st ‘𝐹)‘𝑥)))
55	54	mpteq2dva 5240	. 2 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ ((𝐴‘((1^st ‘𝐹)‘𝑥))(⟨((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑥)), ((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝐻)‘((1^st ‘𝐹)‘𝑥)))((((1^st ‘𝐹)‘𝑥)(2^nd ‘𝐺)((1^st ‘𝐹)‘𝑥))‘(((Id‘𝐷) ∘ (1^st ‘𝐹))‘𝑥)))) = (𝑥 ∈ (Base‘𝐶) ↦ (𝐴‘((1^st ‘𝐹)‘𝑥))))
56	6, 27, 55	3eqtrd 2780	1 ⊢ (𝜑 → (𝐴(⟨𝐺, 𝐹⟩𝑃⟨𝐻, 𝐹⟩)(𝐼‘𝐹)) = (𝑥 ∈ (Base‘𝐶) ↦ (𝐴‘((1^st ‘𝐹)‘𝑥))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3479 ⟨cop 4630 class class class wbr 5141 ↦ cmpt 5223 × cxp 5681 ∘ ccom 5687 ‘cfv 6559 (class class class)co 7429 1^st c1st 8008 2^nd c2nd 8009 Basecbs 17243 Hom chom 17304 compcco 17305 Catccat 17703 Idccid 17704 Func cfunc 17895 Nat cnat 17985 FuncCat cfuc 17986 ∘_F cfuco 48984

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-cnex 11207 ax-resscn 11208 ax-1cn 11209 ax-icn 11210 ax-addcl 11211 ax-addrcl 11212 ax-mulcl 11213 ax-mulrcl 11214 ax-mulcom 11215 ax-addass 11216 ax-mulass 11217 ax-distr 11218 ax-i2m1 11219 ax-1ne0 11220 ax-1rid 11221 ax-rnegex 11222 ax-rrecex 11223 ax-cnre 11224 ax-pre-lttri 11225 ax-pre-lttrn 11226 ax-pre-ltadd 11227 ax-pre-mulgt0 11228

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4906 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-riota 7386 df-ov 7432 df-oprab 7433 df-mpo 7434 df-om 7884 df-1st 8010 df-2nd 8011 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-1o 8502 df-er 8741 df-map 8864 df-ixp 8934 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-pnf 11293 df-mnf 11294 df-xr 11295 df-ltxr 11296 df-le 11297 df-sub 11490 df-neg 11491 df-nn 12263 df-2 12325 df-3 12326 df-4 12327 df-5 12328 df-6 12329 df-7 12330 df-8 12331 df-9 12332 df-n0 12523 df-z 12610 df-dec 12730 df-uz 12875 df-fz 13544 df-struct 17180 df-slot 17215 df-ndx 17227 df-base 17244 df-hom 17317 df-cco 17318 df-cat 17707 df-cid 17708 df-func 17899 df-cofu 17901 df-nat 17987 df-fuc 17988 df-fuco 48985

This theorem is referenced by: fucorid2 49031 precofval 49035

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