fucorid - Mathbox for Zhi Wang

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Theorem fucorid 49849

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 2737	. . . 4 ⊢ (Id‘𝐷) = (Id‘𝐷)
4		fucorid.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
5	1, 2, 3, 4	fucid 17932	. . 3 ⊢ (𝜑 → (𝐼‘𝐹) = ((Id‘𝐷) ∘ (1^st ‘𝐹)))
6	5	oveq2d 7376	. 2 ⊢ (𝜑 → (𝐴(⟨𝐺, 𝐹⟩𝑃⟨𝐻, 𝐹⟩)(𝐼‘𝐹)) = (𝐴(⟨𝐺, 𝐹⟩𝑃⟨𝐻, 𝐹⟩)((Id‘𝐷) ∘ (1^st ‘𝐹))))
7	4	func1st2nd 49563	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
8	7	funcrcl2 49566	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
9		eqid 2737	. . . . . . . 8 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
10		fucorid.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ (𝐺(𝐷 Nat 𝐸)𝐻))
11	9, 10	nat1st2nd 17912	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩(𝐷 Nat 𝐸)⟨(1^st ‘𝐻), (2^nd ‘𝐻)⟩))
12	9, 11	natrcl2 49711	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐺)(𝐷 Func 𝐸)(2^nd ‘𝐺))
13	12	funcrcl2 49566	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
14	12	funcrcl3 49567	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ Cat)
15		eqidd 2738	. . . . . 6 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = (⟨𝐶, 𝐷⟩ ∘_F 𝐸))
16	8, 13, 14, 15	fucoelvv 49807	. . . . 5 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) ∈ (V × V))
17		1st2nd2 7974	. . . . 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 4824	. . . 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 2772	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨(1^st ‘(⟨𝐶, 𝐷⟩ ∘_F 𝐸)), 𝑃⟩)
22		eqidd 2738	. . 3 ⊢ (𝜑 → ⟨𝐺, 𝐹⟩ = ⟨𝐺, 𝐹⟩)
23		eqidd 2738	. . 3 ⊢ (𝜑 → ⟨𝐻, 𝐹⟩ = ⟨𝐻, 𝐹⟩)
24		eqid 2737	. . . 4 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
25	1, 24, 3, 4	fucidcl 17926	. . 3 ⊢ (𝜑 → ((Id‘𝐷) ∘ (1^st ‘𝐹)) ∈ (𝐹(𝐶 Nat 𝐷)𝐹))
26	21, 22, 23, 25, 10	fuco22a 49837	. 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 2737	. . . . . . . . 9 ⊢ (Base‘𝐶) = (Base‘𝐶)
28		eqid 2737	. . . . . . . . 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 17824	. . . . . . . 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 6934	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((Id‘𝐷) ∘ (1^st ‘𝐹))‘𝑥) = ((Id‘𝐷)‘((1^st ‘𝐹)‘𝑥)))
33	32	fveq2d 6838	. . . . . 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 2737	. . . . . . 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 7031	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
37	28, 3, 34, 35, 36	funcid 17828	. . . . . 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 2772	. . . . 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 7376	. . . 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 2737	. . . . 5 ⊢ (Base‘𝐸) = (Base‘𝐸)
41		eqid 2737	. . . . 5 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
42	14	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐸 ∈ Cat)
43	28, 40, 35	funcf1 17824	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1^st ‘𝐺):(Base‘𝐷)⟶(Base‘𝐸))
44	43, 36	ffvelcdmd 7031	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐺)‘((1^st ‘𝐹)‘𝑥)) ∈ (Base‘𝐸))
45		eqid 2737	. . . . 5 ⊢ (comp‘𝐸) = (comp‘𝐸)
46	9, 11	natrcl3 49712	. . . . . . . 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 17824	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1^st ‘𝐻):(Base‘𝐷)⟶(Base‘𝐸))
49	48, 36	ffvelcdmd 7031	. . . . 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 17914	. . . . 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 17641	. . . 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 2772	. . 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 2776	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 1542 ∈ wcel 2114 Vcvv 3430 ⟨cop 4574 class class class wbr 5086 ↦ cmpt 5167 × cxp 5622 ∘ ccom 5628 ‘cfv 6492 (class class class)co 7360 1^st c1st 7933 2^nd c2nd 7934 Basecbs 17170 Hom chom 17222 compcco 17223 Catccat 17621 Idccid 17622 Func cfunc 17812 Nat cnat 17902 FuncCat cfuc 17903 ∘_F cfuco 49803

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-er 8636 df-map 8768 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-fz 13453 df-struct 17108 df-slot 17143 df-ndx 17155 df-base 17171 df-hom 17235 df-cco 17236 df-cat 17625 df-cid 17626 df-func 17816 df-cofu 17818 df-nat 17904 df-fuc 17905 df-fuco 49804

This theorem is referenced by: fucorid2 49850 precofval 49854

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