fucorid - Mathbox for Zhi Wang

	Mathbox for Zhi Wang		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > fucorid		Structured version Visualization version GIF version

Theorem fucorid 49394

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 2731	. . . 4 ⊢ (Id‘𝐷) = (Id‘𝐷)
4		fucorid.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
5	1, 2, 3, 4	fucid 17876	. . 3 ⊢ (𝜑 → (𝐼‘𝐹) = ((Id‘𝐷) ∘ (1^st ‘𝐹)))
6	5	oveq2d 7357	. 2 ⊢ (𝜑 → (𝐴(⟨𝐺, 𝐹⟩𝑃⟨𝐻, 𝐹⟩)(𝐼‘𝐹)) = (𝐴(⟨𝐺, 𝐹⟩𝑃⟨𝐻, 𝐹⟩)((Id‘𝐷) ∘ (1^st ‘𝐹))))
7	4	func1st2nd 49108	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
8	7	funcrcl2 49111	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
9		eqid 2731	. . . . . . . 8 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
10		fucorid.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ (𝐺(𝐷 Nat 𝐸)𝐻))
11	9, 10	nat1st2nd 17856	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩(𝐷 Nat 𝐸)⟨(1^st ‘𝐻), (2^nd ‘𝐻)⟩))
12	9, 11	natrcl2 49256	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐺)(𝐷 Func 𝐸)(2^nd ‘𝐺))
13	12	funcrcl2 49111	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
14	12	funcrcl3 49112	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ Cat)
15		eqidd 2732	. . . . . 6 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = (⟨𝐶, 𝐷⟩ ∘_F 𝐸))
16	8, 13, 14, 15	fucoelvv 49352	. . . . 5 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) ∈ (V × V))
17		1st2nd2 7955	. . . . 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 17	. . . 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 4827	. . . 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 2766	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨(1^st ‘(⟨𝐶, 𝐷⟩ ∘_F 𝐸)), 𝑃⟩)
22		eqidd 2732	. . 3 ⊢ (𝜑 → ⟨𝐺, 𝐹⟩ = ⟨𝐺, 𝐹⟩)
23		eqidd 2732	. . 3 ⊢ (𝜑 → ⟨𝐻, 𝐹⟩ = ⟨𝐻, 𝐹⟩)
24		eqid 2731	. . . 4 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
25	1, 24, 3, 4	fucidcl 17870	. . 3 ⊢ (𝜑 → ((Id‘𝐷) ∘ (1^st ‘𝐹)) ∈ (𝐹(𝐶 Nat 𝐷)𝐹))
26	21, 22, 23, 25, 10	fuco22a 49382	. 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 2731	. . . . . . . . 9 ⊢ (Base‘𝐶) = (Base‘𝐶)
28		eqid 2731	. . . . . . . . 9 ⊢ (Base‘𝐷) = (Base‘𝐷)
29	7	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
30	27, 28, 29	funcf1 17768	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1^st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
31		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
32	30, 31	fvco3d 6917	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((Id‘𝐷) ∘ (1^st ‘𝐹))‘𝑥) = ((Id‘𝐷)‘((1^st ‘𝐹)‘𝑥)))
33	32	fveq2d 6821	. . . . . 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 2731	. . . . . . 7 ⊢ (Id‘𝐸) = (Id‘𝐸)
35	12	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1^st ‘𝐺)(𝐷 Func 𝐸)(2^nd ‘𝐺))
36	30, 31	ffvelcdmd 7013	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
37	28, 3, 34, 35, 36	funcid 17772	. . . . . 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 2766	. . . . 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 7357	. . . 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 2731	. . . . 5 ⊢ (Base‘𝐸) = (Base‘𝐸)
41		eqid 2731	. . . . 5 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
42	14	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐸 ∈ Cat)
43	28, 40, 35	funcf1 17768	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1^st ‘𝐺):(Base‘𝐷)⟶(Base‘𝐸))
44	43, 36	ffvelcdmd 7013	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑥)) ∈ (Base‘𝐸))
45		eqid 2731	. . . . 5 ⊢ (comp‘𝐸) = (comp‘𝐸)
46	9, 11	natrcl3 49257	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐻)(𝐷 Func 𝐸)(2^nd ‘𝐻))
47	46	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1^st ‘𝐻)(𝐷 Func 𝐸)(2^nd ‘𝐻))
48	28, 40, 47	funcf1 17768	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1^st ‘𝐻):(Base‘𝐷)⟶(Base‘𝐸))
49	48, 36	ffvelcdmd 7013	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐻)‘((1^st ‘𝐹)‘𝑥)) ∈ (Base‘𝐸))
50	11	adantr 480	. . . . . 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 17858	. . . . 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 17585	. . . 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 2766	. . 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 5179	. 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 2770	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 1541 ∈ wcel 2111 Vcvv 3436 ⟨cop 4577 class class class wbr 5086 ↦ cmpt 5167 × cxp 5609 ∘ ccom 5615 ‘cfv 6476 (class class class)co 7341 1^st c1st 7914 2^nd c2nd 7915 Basecbs 17115 Hom chom 17167 compcco 17168 Catccat 17565 Idccid 17566 Func cfunc 17756 Nat cnat 17846 FuncCat cfuc 17847 ∘_F cfuco 49348

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5212 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-tp 4576 df-op 4578 df-uni 4855 df-iun 4938 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7792 df-1st 7916 df-2nd 7917 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-er 8617 df-map 8747 df-ixp 8817 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-nn 12121 df-2 12183 df-3 12184 df-4 12185 df-5 12186 df-6 12187 df-7 12188 df-8 12189 df-9 12190 df-n0 12377 df-z 12464 df-dec 12584 df-uz 12728 df-fz 13403 df-struct 17053 df-slot 17088 df-ndx 17100 df-base 17116 df-hom 17180 df-cco 17181 df-cat 17569 df-cid 17570 df-func 17760 df-cofu 17762 df-nat 17848 df-fuc 17849 df-fuco 49349

This theorem is referenced by: fucorid2 49395 precofval 49399

W3C validator