fucorid - Mathbox for Zhi Wang

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Theorem fucorid 49333

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 2730	. . . 4 ⊢ (Id‘𝐷) = (Id‘𝐷)
4		fucorid.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
5	1, 2, 3, 4	fucid 17942	. . 3 ⊢ (𝜑 → (𝐼‘𝐹) = ((Id‘𝐷) ∘ (1^st ‘𝐹)))
6	5	oveq2d 7405	. 2 ⊢ (𝜑 → (𝐴(⟨𝐺, 𝐹⟩𝑃⟨𝐻, 𝐹⟩)(𝐼‘𝐹)) = (𝐴(⟨𝐺, 𝐹⟩𝑃⟨𝐻, 𝐹⟩)((Id‘𝐷) ∘ (1^st ‘𝐹))))
7	4	func1st2nd 49053	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
8	7	funcrcl2 49056	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
9		eqid 2730	. . . . . . . 8 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
10		fucorid.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ (𝐺(𝐷 Nat 𝐸)𝐻))
11	9, 10	nat1st2nd 17922	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩(𝐷 Nat 𝐸)⟨(1^st ‘𝐻), (2^nd ‘𝐻)⟩))
12	9, 11	natrcl2 49195	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐺)(𝐷 Func 𝐸)(2^nd ‘𝐺))
13	12	funcrcl2 49056	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
14	12	funcrcl3 49057	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ Cat)
15		eqidd 2731	. . . . . 6 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = (⟨𝐶, 𝐷⟩ ∘_F 𝐸))
16	8, 13, 14, 15	fucoelvv 49291	. . . . 5 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) ∈ (V × V))
17		1st2nd2 8009	. . . . 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 4846	. . . 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 2765	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨(1^st ‘(⟨𝐶, 𝐷⟩ ∘_F 𝐸)), 𝑃⟩)
22		eqidd 2731	. . 3 ⊢ (𝜑 → ⟨𝐺, 𝐹⟩ = ⟨𝐺, 𝐹⟩)
23		eqidd 2731	. . 3 ⊢ (𝜑 → ⟨𝐻, 𝐹⟩ = ⟨𝐻, 𝐹⟩)
24		eqid 2730	. . . 4 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
25	1, 24, 3, 4	fucidcl 17936	. . 3 ⊢ (𝜑 → ((Id‘𝐷) ∘ (1^st ‘𝐹)) ∈ (𝐹(𝐶 Nat 𝐷)𝐹))
26	21, 22, 23, 25, 10	fuco22a 49321	. 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 2730	. . . . . . . . 9 ⊢ (Base‘𝐶) = (Base‘𝐶)
28		eqid 2730	. . . . . . . . 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 17834	. . . . . . . 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 6963	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((Id‘𝐷) ∘ (1^st ‘𝐹))‘𝑥) = ((Id‘𝐷)‘((1^st ‘𝐹)‘𝑥)))
33	32	fveq2d 6864	. . . . . 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 2730	. . . . . . 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 7059	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
37	28, 3, 34, 35, 36	funcid 17838	. . . . . 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 2765	. . . . 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 7405	. . . 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 2730	. . . . 5 ⊢ (Base‘𝐸) = (Base‘𝐸)
41		eqid 2730	. . . . 5 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
42	14	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐸 ∈ Cat)
43	28, 40, 35	funcf1 17834	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1^st ‘𝐺):(Base‘𝐷)⟶(Base‘𝐸))
44	43, 36	ffvelcdmd 7059	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑥)) ∈ (Base‘𝐸))
45		eqid 2730	. . . . 5 ⊢ (comp‘𝐸) = (comp‘𝐸)
46	9, 11	natrcl3 49196	. . . . . . . 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 17834	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1^st ‘𝐻):(Base‘𝐷)⟶(Base‘𝐸))
49	48, 36	ffvelcdmd 7059	. . . . 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 17924	. . . . 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 17651	. . . 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 2765	. . 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 5202	. 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 2769	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 2109 Vcvv 3450 ⟨cop 4597 class class class wbr 5109 ↦ cmpt 5190 × cxp 5638 ∘ ccom 5644 ‘cfv 6513 (class class class)co 7389 1^st c1st 7968 2^nd c2nd 7969 Basecbs 17185 Hom chom 17237 compcco 17238 Catccat 17631 Idccid 17632 Func cfunc 17822 Nat cnat 17912 FuncCat cfuc 17913 ∘_F cfuco 49287

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 2702 ax-rep 5236 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-pss 3936 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4874 df-iun 4959 df-br 5110 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6276 df-ord 6337 df-on 6338 df-lim 6339 df-suc 6340 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-riota 7346 df-ov 7392 df-oprab 7393 df-mpo 7394 df-om 7845 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8380 df-1o 8436 df-er 8673 df-map 8803 df-ixp 8873 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-pnf 11216 df-mnf 11217 df-xr 11218 df-ltxr 11219 df-le 11220 df-sub 11413 df-neg 11414 df-nn 12188 df-2 12250 df-3 12251 df-4 12252 df-5 12253 df-6 12254 df-7 12255 df-8 12256 df-9 12257 df-n0 12449 df-z 12536 df-dec 12656 df-uz 12800 df-fz 13475 df-struct 17123 df-slot 17158 df-ndx 17170 df-base 17186 df-hom 17250 df-cco 17251 df-cat 17635 df-cid 17636 df-func 17826 df-cofu 17828 df-nat 17914 df-fuc 17915 df-fuco 49288

This theorem is referenced by: fucorid2 49334 precofval 49338

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