fucorid - Mathbox for Zhi Wang

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Theorem fucorid 50140

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 2763	. . . 4 ⊢ (Id‘𝐷) = (Id‘𝐷)
4		fucorid.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
5	1, 2, 3, 4	fucid 18026	. . 3 ⊢ (𝜑 → (𝐼‘𝐹) = ((Id‘𝐷) ∘ (1^st ‘𝐹)))
6	5	oveq2d 7426	. 2 ⊢ (𝜑 → (𝐴(⟨𝐺, 𝐹⟩𝑃⟨𝐻, 𝐹⟩)(𝐼‘𝐹)) = (𝐴(⟨𝐺, 𝐹⟩𝑃⟨𝐻, 𝐹⟩)((Id‘𝐷) ∘ (1^st ‘𝐹))))
7	4	func1st2nd 49854	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
8	7	funcrcl2 49857	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
9		eqid 2763	. . . . . . . 8 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
10		fucorid.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ (𝐺(𝐷 Nat 𝐸)𝐻))
11	9, 10	nat1st2nd 18006	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩(𝐷 Nat 𝐸)⟨(1^st ‘𝐻), (2^nd ‘𝐻)⟩))
12	9, 11	natrcl2 50002	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐺)(𝐷 Func 𝐸)(2^nd ‘𝐺))
13	12	funcrcl2 49857	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
14	12	funcrcl3 49858	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ Cat)
15		eqidd 2764	. . . . . 6 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = (⟨𝐶, 𝐷⟩ ∘_F 𝐸))
16	8, 13, 14, 15	fucoelvv 50098	. . . . 5 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) ∈ (V × V))
17		1st2nd2 8021	. . . . 5 ⊢ ((⟨𝐶, 𝐷⟩ ∘_F 𝐸) ∈ (V × V) → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨(1^st ‘(⟨𝐶, 𝐷⟩ ∘_F 𝐸)), (2^nd ‘(⟨𝐶, 𝐷⟩ ∘_F 𝐸))⟩)
18	16, 17	syl 18	. . . 4 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨(1^st ‘(⟨𝐶, 𝐷⟩ ∘_F 𝐸)), (2^nd ‘(⟨𝐶, 𝐷⟩ ∘_F 𝐸))⟩)
19		fucolid.p	. . . . 5 ⊢ (𝜑 → (2^nd ‘(⟨𝐶, 𝐷⟩ ∘_F 𝐸)) = 𝑃)
20	19	opeq2d 4845	. . . 4 ⊢ (𝜑 → ⟨(1^st ‘(⟨𝐶, 𝐷⟩ ∘_F 𝐸)), (2^nd ‘(⟨𝐶, 𝐷⟩ ∘_F 𝐸))⟩ = ⟨(1^st ‘(⟨𝐶, 𝐷⟩ ∘_F 𝐸)), 𝑃⟩)
21	18, 20	eqtrd 2798	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨(1^st ‘(⟨𝐶, 𝐷⟩ ∘_F 𝐸)), 𝑃⟩)
22		eqidd 2764	. . 3 ⊢ (𝜑 → ⟨𝐺, 𝐹⟩ = ⟨𝐺, 𝐹⟩)
23		eqidd 2764	. . 3 ⊢ (𝜑 → ⟨𝐻, 𝐹⟩ = ⟨𝐻, 𝐹⟩)
24		eqid 2763	. . . 4 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
25	1, 24, 3, 4	fucidcl 18020	. . 3 ⊢ (𝜑 → ((Id‘𝐷) ∘ (1^st ‘𝐹)) ∈ (𝐹(𝐶 Nat 𝐷)𝐹))
26	21, 22, 23, 25, 10	fuco22a 50128	. 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)))))
27		eqid 2763	. . . . . . . . 9 ⊢ (Base‘𝐶) = (Base‘𝐶)
28		eqid 2763	. . . . . . . . 9 ⊢ (Base‘𝐷) = (Base‘𝐷)
29	7	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
30	27, 28, 29	funcf1 17918	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1^st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
31		simpr 489	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
32	30, 31	fvco3d 6982	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((Id‘𝐷) ∘ (1^st ‘𝐹))‘𝑥) = ((Id‘𝐷)‘((1^st ‘𝐹)‘𝑥)))
33	32	fveq2d 6885	. . . . . 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))))
34		eqid 2763	. . . . . . 7 ⊢ (Id‘𝐸) = (Id‘𝐸)
35	12	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1^st ‘𝐺)(𝐷 Func 𝐸)(2^nd ‘𝐺))
36	30, 31	ffvelcdmd 7080	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
37	28, 3, 34, 35, 36	funcid 17922	. . . . . 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))))
38	33, 37	eqtrd 2798	. . . . 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))))
39	38	oveq2d 7426	. . . 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)))))
40		eqid 2763	. . . . 5 ⊢ (Base‘𝐸) = (Base‘𝐸)
41		eqid 2763	. . . . 5 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
42	14	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐸 ∈ Cat)
43	28, 40, 35	funcf1 17918	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1^st ‘𝐺):(Base‘𝐷)⟶(Base‘𝐸))
44	43, 36	ffvelcdmd 7080	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑥)) ∈ (Base‘𝐸))
45		eqid 2763	. . . . 5 ⊢ (comp‘𝐸) = (comp‘𝐸)
46	9, 11	natrcl3 50003	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐻)(𝐷 Func 𝐸)(2^nd ‘𝐻))
47	46	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1^st ‘𝐻)(𝐷 Func 𝐸)(2^nd ‘𝐻))
48	28, 40, 47	funcf1 17918	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1^st ‘𝐻):(Base‘𝐷)⟶(Base‘𝐸))
49	48, 36	ffvelcdmd 7080	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐻)‘((1^st ‘𝐹)‘𝑥)) ∈ (Base‘𝐸))
50	11	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐴 ∈ (⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩(𝐷 Nat 𝐸)⟨(1^st ‘𝐻), (2^nd ‘𝐻)⟩))
51	9, 50, 28, 41, 36	natcl 18008	. . . . 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))))
52	40, 41, 34, 42, 44, 45, 49, 51	catrid 17735	. . . 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)))
53	39, 52	eqtrd 2798	. . 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)))
54	53	mpteq2dva 5204	. 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))))
55	6, 26, 54	3eqtrd 2802	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 400 = wceq 1570 ∈ wcel 2143 Vcvv 3455 ⟨cop 4595 class class class wbr 5109 ↦ cmpt 5192 × cxp 5659 ∘ ccom 5665 ‘cfv 6536 (class class class)co 7410 1^st c1st 7980 2^nd c2nd 7981 Basecbs 17264 Hom chom 17316 compcco 17317 Catccat 17715 Idccid 17716 Func cfunc 17906 Nat cnat 17996 FuncCat cfuc 17997 ∘_F cfuco 50094

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 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 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-er 8690 df-map 8822 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-7 12303 df-8 12304 df-9 12305 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12858 df-fz 13531 df-struct 17202 df-slot 17237 df-ndx 17249 df-base 17265 df-hom 17329 df-cco 17330 df-cat 17719 df-cid 17720 df-func 17910 df-cofu 17912 df-nat 17998 df-fuc 17999 df-fuco 50095

This theorem is referenced by: fucorid2 50141 precofval 50145

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