fucoid - Mathbox for Zhi Wang

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Theorem fucoid 49379

Description: Each identity morphism in the source category is mapped to the corresponding identity morphism in the target category. See also fucoid2 49380. (Contributed by Zhi Wang, 30-Sep-2025.)

**Hypotheses**
Ref	Expression
fucoid.o	⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨𝑂, 𝑃⟩)
fucoid.t	⊢ 𝑇 = ((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷))
fucoid.1	⊢ 1 = (Id‘𝑇)
fucoid.q	⊢ 𝑄 = (𝐶 FuncCat 𝐸)
fucoid.i	⊢ 𝐼 = (Id‘𝑄)
fucoid.f	⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
fucoid.k	⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)
fucoid.u	⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)

**Assertion**
Ref	Expression
fucoid	⊢ (𝜑 → ((𝑈𝑃𝑈)‘( 1 ‘𝑈)) = (𝐼‘(𝑂‘𝑈)))

Proof of Theorem fucoid Dummy variables 𝑤 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 7379	. . . . 5 ⊢ ((((Id‘𝐸) ∘ 𝐾)‘(𝐹‘𝑥))(⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝐹‘𝑥))⟩(comp‘𝐸)(𝐾‘(𝐹‘𝑥)))(((𝐹‘𝑥)𝐿(𝐹‘𝑥))‘(((Id‘𝐷) ∘ 𝐹)‘𝑥))) ∈ V
2		eqid 2731	. . . . 5 ⊢ (𝑥 ∈ (Base‘𝐶) ↦ ((((Id‘𝐸) ∘ 𝐾)‘(𝐹‘𝑥))(⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝐹‘𝑥))⟩(comp‘𝐸)(𝐾‘(𝐹‘𝑥)))(((𝐹‘𝑥)𝐿(𝐹‘𝑥))‘(((Id‘𝐷) ∘ 𝐹)‘𝑥)))) = (𝑥 ∈ (Base‘𝐶) ↦ ((((Id‘𝐸) ∘ 𝐾)‘(𝐹‘𝑥))(⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝐹‘𝑥))⟩(comp‘𝐸)(𝐾‘(𝐹‘𝑥)))(((𝐹‘𝑥)𝐿(𝐹‘𝑥))‘(((Id‘𝐷) ∘ 𝐹)‘𝑥))))
3	1, 2	fnmpti 6624	. . . 4 ⊢ (𝑥 ∈ (Base‘𝐶) ↦ ((((Id‘𝐸) ∘ 𝐾)‘(𝐹‘𝑥))(⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝐹‘𝑥))⟩(comp‘𝐸)(𝐾‘(𝐹‘𝑥)))(((𝐹‘𝑥)𝐿(𝐹‘𝑥))‘(((Id‘𝐷) ∘ 𝐹)‘𝑥)))) Fn (Base‘𝐶)
4	3	a1i 11	. . 3 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ ((((Id‘𝐸) ∘ 𝐾)‘(𝐹‘𝑥))(⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝐹‘𝑥))⟩(comp‘𝐸)(𝐾‘(𝐹‘𝑥)))(((𝐹‘𝑥)𝐿(𝐹‘𝑥))‘(((Id‘𝐷) ∘ 𝐹)‘𝑥)))) Fn (Base‘𝐶))
5		fucoid.k	. . . . . 6 ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)
6	5	funcrcl3 49111	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ Cat)
7		eqid 2731	. . . . . 6 ⊢ (Base‘𝐸) = (Base‘𝐸)
8		eqid 2731	. . . . . 6 ⊢ (Id‘𝐸) = (Id‘𝐸)
9	7, 8	cidfn 17582	. . . . 5 ⊢ (𝐸 ∈ Cat → (Id‘𝐸) Fn (Base‘𝐸))
10	6, 9	syl 17	. . . 4 ⊢ (𝜑 → (Id‘𝐸) Fn (Base‘𝐸))
11		eqid 2731	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
12	11, 7, 5	funcf1 17770	. . . . 5 ⊢ (𝜑 → 𝐾:(Base‘𝐷)⟶(Base‘𝐸))
13		eqid 2731	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
14		fucoid.f	. . . . . 6 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
15	13, 11, 14	funcf1 17770	. . . . 5 ⊢ (𝜑 → 𝐹:(Base‘𝐶)⟶(Base‘𝐷))
16	12, 15	fcod 6676	. . . 4 ⊢ (𝜑 → (𝐾 ∘ 𝐹):(Base‘𝐶)⟶(Base‘𝐸))
17		fnfco 6688	. . . 4 ⊢ (((Id‘𝐸) Fn (Base‘𝐸) ∧ (𝐾 ∘ 𝐹):(Base‘𝐶)⟶(Base‘𝐸)) → ((Id‘𝐸) ∘ (𝐾 ∘ 𝐹)) Fn (Base‘𝐶))
18	10, 16, 17	syl2anc 584	. . 3 ⊢ (𝜑 → ((Id‘𝐸) ∘ (𝐾 ∘ 𝐹)) Fn (Base‘𝐶))
19		2fveq3 6827	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝐾‘(𝐹‘𝑥)) = (𝐾‘(𝐹‘𝑤)))
20	19, 19	opeq12d 4833	. . . . . . 7 ⊢ (𝑥 = 𝑤 → ⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝐹‘𝑥))⟩ = ⟨(𝐾‘(𝐹‘𝑤)), (𝐾‘(𝐹‘𝑤))⟩)
21	20, 19	oveq12d 7364	. . . . . 6 ⊢ (𝑥 = 𝑤 → (⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝐹‘𝑥))⟩(comp‘𝐸)(𝐾‘(𝐹‘𝑥))) = (⟨(𝐾‘(𝐹‘𝑤)), (𝐾‘(𝐹‘𝑤))⟩(comp‘𝐸)(𝐾‘(𝐹‘𝑤))))
22		2fveq3 6827	. . . . . 6 ⊢ (𝑥 = 𝑤 → (((Id‘𝐸) ∘ 𝐾)‘(𝐹‘𝑥)) = (((Id‘𝐸) ∘ 𝐾)‘(𝐹‘𝑤)))
23		fveq2 6822	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
24	23, 23	oveq12d 7364	. . . . . . 7 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑥)𝐿(𝐹‘𝑥)) = ((𝐹‘𝑤)𝐿(𝐹‘𝑤)))
25		fveq2 6822	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (((Id‘𝐷) ∘ 𝐹)‘𝑥) = (((Id‘𝐷) ∘ 𝐹)‘𝑤))
26	24, 25	fveq12d 6829	. . . . . 6 ⊢ (𝑥 = 𝑤 → (((𝐹‘𝑥)𝐿(𝐹‘𝑥))‘(((Id‘𝐷) ∘ 𝐹)‘𝑥)) = (((𝐹‘𝑤)𝐿(𝐹‘𝑤))‘(((Id‘𝐷) ∘ 𝐹)‘𝑤)))
27	21, 22, 26	oveq123d 7367	. . . . 5 ⊢ (𝑥 = 𝑤 → ((((Id‘𝐸) ∘ 𝐾)‘(𝐹‘𝑥))(⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝐹‘𝑥))⟩(comp‘𝐸)(𝐾‘(𝐹‘𝑥)))(((𝐹‘𝑥)𝐿(𝐹‘𝑥))‘(((Id‘𝐷) ∘ 𝐹)‘𝑥))) = ((((Id‘𝐸) ∘ 𝐾)‘(𝐹‘𝑤))(⟨(𝐾‘(𝐹‘𝑤)), (𝐾‘(𝐹‘𝑤))⟩(comp‘𝐸)(𝐾‘(𝐹‘𝑤)))(((𝐹‘𝑤)𝐿(𝐹‘𝑤))‘(((Id‘𝐷) ∘ 𝐹)‘𝑤))))
28		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → 𝑤 ∈ (Base‘𝐶))
29		ovexd 7381	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → ((((Id‘𝐸) ∘ 𝐾)‘(𝐹‘𝑤))(⟨(𝐾‘(𝐹‘𝑤)), (𝐾‘(𝐹‘𝑤))⟩(comp‘𝐸)(𝐾‘(𝐹‘𝑤)))(((𝐹‘𝑤)𝐿(𝐹‘𝑤))‘(((Id‘𝐷) ∘ 𝐹)‘𝑤))) ∈ V)
30	2, 27, 28, 29	fvmptd3 6952	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → ((𝑥 ∈ (Base‘𝐶) ↦ ((((Id‘𝐸) ∘ 𝐾)‘(𝐹‘𝑥))(⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝐹‘𝑥))⟩(comp‘𝐸)(𝐾‘(𝐹‘𝑥)))(((𝐹‘𝑥)𝐿(𝐹‘𝑥))‘(((Id‘𝐷) ∘ 𝐹)‘𝑥))))‘𝑤) = ((((Id‘𝐸) ∘ 𝐾)‘(𝐹‘𝑤))(⟨(𝐾‘(𝐹‘𝑤)), (𝐾‘(𝐹‘𝑤))⟩(comp‘𝐸)(𝐾‘(𝐹‘𝑤)))(((𝐹‘𝑤)𝐿(𝐹‘𝑤))‘(((Id‘𝐷) ∘ 𝐹)‘𝑤))))
31		eqid 2731	. . . . . 6 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
32	6	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → 𝐸 ∈ Cat)
33	12	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → 𝐾:(Base‘𝐷)⟶(Base‘𝐸))
34	15	ffvelcdmda 7017	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → (𝐹‘𝑤) ∈ (Base‘𝐷))
35	33, 34	ffvelcdmd 7018	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → (𝐾‘(𝐹‘𝑤)) ∈ (Base‘𝐸))
36		eqid 2731	. . . . . 6 ⊢ (comp‘𝐸) = (comp‘𝐸)
37	7, 31, 8, 32, 35	catidcl 17585	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → ((Id‘𝐸)‘(𝐾‘(𝐹‘𝑤))) ∈ ((𝐾‘(𝐹‘𝑤))(Hom ‘𝐸)(𝐾‘(𝐹‘𝑤))))
38	7, 31, 8, 32, 35, 36, 35, 37	catlid 17586	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → (((Id‘𝐸)‘(𝐾‘(𝐹‘𝑤)))(⟨(𝐾‘(𝐹‘𝑤)), (𝐾‘(𝐹‘𝑤))⟩(comp‘𝐸)(𝐾‘(𝐹‘𝑤)))((Id‘𝐸)‘(𝐾‘(𝐹‘𝑤)))) = ((Id‘𝐸)‘(𝐾‘(𝐹‘𝑤))))
39	33, 34	fvco3d 6922	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → (((Id‘𝐸) ∘ 𝐾)‘(𝐹‘𝑤)) = ((Id‘𝐸)‘(𝐾‘(𝐹‘𝑤))))
40	15	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → 𝐹:(Base‘𝐶)⟶(Base‘𝐷))
41	40, 28	fvco3d 6922	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → (((Id‘𝐷) ∘ 𝐹)‘𝑤) = ((Id‘𝐷)‘(𝐹‘𝑤)))
42	41	fveq2d 6826	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → (((𝐹‘𝑤)𝐿(𝐹‘𝑤))‘(((Id‘𝐷) ∘ 𝐹)‘𝑤)) = (((𝐹‘𝑤)𝐿(𝐹‘𝑤))‘((Id‘𝐷)‘(𝐹‘𝑤))))
43		eqid 2731	. . . . . . . 8 ⊢ (Id‘𝐷) = (Id‘𝐷)
44	5	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → 𝐾(𝐷 Func 𝐸)𝐿)
45	11, 43, 8, 44, 34	funcid 17774	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → (((𝐹‘𝑤)𝐿(𝐹‘𝑤))‘((Id‘𝐷)‘(𝐹‘𝑤))) = ((Id‘𝐸)‘(𝐾‘(𝐹‘𝑤))))
46	42, 45	eqtrd 2766	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → (((𝐹‘𝑤)𝐿(𝐹‘𝑤))‘(((Id‘𝐷) ∘ 𝐹)‘𝑤)) = ((Id‘𝐸)‘(𝐾‘(𝐹‘𝑤))))
47	39, 46	oveq12d 7364	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → ((((Id‘𝐸) ∘ 𝐾)‘(𝐹‘𝑤))(⟨(𝐾‘(𝐹‘𝑤)), (𝐾‘(𝐹‘𝑤))⟩(comp‘𝐸)(𝐾‘(𝐹‘𝑤)))(((𝐹‘𝑤)𝐿(𝐹‘𝑤))‘(((Id‘𝐷) ∘ 𝐹)‘𝑤))) = (((Id‘𝐸)‘(𝐾‘(𝐹‘𝑤)))(⟨(𝐾‘(𝐹‘𝑤)), (𝐾‘(𝐹‘𝑤))⟩(comp‘𝐸)(𝐾‘(𝐹‘𝑤)))((Id‘𝐸)‘(𝐾‘(𝐹‘𝑤)))))
48	16	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → (𝐾 ∘ 𝐹):(Base‘𝐶)⟶(Base‘𝐸))
49	48, 28	fvco3d 6922	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → (((Id‘𝐸) ∘ (𝐾 ∘ 𝐹))‘𝑤) = ((Id‘𝐸)‘((𝐾 ∘ 𝐹)‘𝑤)))
50	40, 28	fvco3d 6922	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → ((𝐾 ∘ 𝐹)‘𝑤) = (𝐾‘(𝐹‘𝑤)))
51	50	fveq2d 6826	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → ((Id‘𝐸)‘((𝐾 ∘ 𝐹)‘𝑤)) = ((Id‘𝐸)‘(𝐾‘(𝐹‘𝑤))))
52	49, 51	eqtrd 2766	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → (((Id‘𝐸) ∘ (𝐾 ∘ 𝐹))‘𝑤) = ((Id‘𝐸)‘(𝐾‘(𝐹‘𝑤))))
53	38, 47, 52	3eqtr4d 2776	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → ((((Id‘𝐸) ∘ 𝐾)‘(𝐹‘𝑤))(⟨(𝐾‘(𝐹‘𝑤)), (𝐾‘(𝐹‘𝑤))⟩(comp‘𝐸)(𝐾‘(𝐹‘𝑤)))(((𝐹‘𝑤)𝐿(𝐹‘𝑤))‘(((Id‘𝐷) ∘ 𝐹)‘𝑤))) = (((Id‘𝐸) ∘ (𝐾 ∘ 𝐹))‘𝑤))
54	30, 53	eqtrd 2766	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → ((𝑥 ∈ (Base‘𝐶) ↦ ((((Id‘𝐸) ∘ 𝐾)‘(𝐹‘𝑥))(⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝐹‘𝑥))⟩(comp‘𝐸)(𝐾‘(𝐹‘𝑥)))(((𝐹‘𝑥)𝐿(𝐹‘𝑥))‘(((Id‘𝐷) ∘ 𝐹)‘𝑥))))‘𝑤) = (((Id‘𝐸) ∘ (𝐾 ∘ 𝐹))‘𝑤))
55	4, 18, 54	eqfnfvd 6967	. 2 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ ((((Id‘𝐸) ∘ 𝐾)‘(𝐹‘𝑥))(⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝐹‘𝑥))⟩(comp‘𝐸)(𝐾‘(𝐹‘𝑥)))(((𝐹‘𝑥)𝐿(𝐹‘𝑥))‘(((Id‘𝐷) ∘ 𝐹)‘𝑥)))) = ((Id‘𝐸) ∘ (𝐾 ∘ 𝐹)))
56		fucoid.u	. . . . . . 7 ⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)
57	56	fveq2d 6826	. . . . . 6 ⊢ (𝜑 → ( 1 ‘𝑈) = ( 1 ‘⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩))
58		fucoid.t	. . . . . . 7 ⊢ 𝑇 = ((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷))
59		eqid 2731	. . . . . . . 8 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
60	5	funcrcl2 49110	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ Cat)
61	59, 60, 6	fuccat 17877	. . . . . . 7 ⊢ (𝜑 → (𝐷 FuncCat 𝐸) ∈ Cat)
62		eqid 2731	. . . . . . . 8 ⊢ (𝐶 FuncCat 𝐷) = (𝐶 FuncCat 𝐷)
63	14	funcrcl2 49110	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ Cat)
64	62, 63, 60	fuccat 17877	. . . . . . 7 ⊢ (𝜑 → (𝐶 FuncCat 𝐷) ∈ Cat)
65	59	fucbas 17867	. . . . . . 7 ⊢ (𝐷 Func 𝐸) = (Base‘(𝐷 FuncCat 𝐸))
66	62	fucbas 17867	. . . . . . 7 ⊢ (𝐶 Func 𝐷) = (Base‘(𝐶 FuncCat 𝐷))
67		eqid 2731	. . . . . . 7 ⊢ (Id‘(𝐷 FuncCat 𝐸)) = (Id‘(𝐷 FuncCat 𝐸))
68		eqid 2731	. . . . . . 7 ⊢ (Id‘(𝐶 FuncCat 𝐷)) = (Id‘(𝐶 FuncCat 𝐷))
69		fucoid.1	. . . . . . 7 ⊢ 1 = (Id‘𝑇)
70		df-br 5092	. . . . . . . 8 ⊢ (𝐾(𝐷 Func 𝐸)𝐿 ↔ ⟨𝐾, 𝐿⟩ ∈ (𝐷 Func 𝐸))
71	5, 70	sylib 218	. . . . . . 7 ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (𝐷 Func 𝐸))
72		df-br 5092	. . . . . . . 8 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
73	14, 72	sylib 218	. . . . . . 7 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
74	58, 61, 64, 65, 66, 67, 68, 69, 71, 73	xpcid 18092	. . . . . 6 ⊢ (𝜑 → ( 1 ‘⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩) = ⟨((Id‘(𝐷 FuncCat 𝐸))‘⟨𝐾, 𝐿⟩), ((Id‘(𝐶 FuncCat 𝐷))‘⟨𝐹, 𝐺⟩)⟩)
75	59, 67, 8, 71	fucid 17878	. . . . . . . 8 ⊢ (𝜑 → ((Id‘(𝐷 FuncCat 𝐸))‘⟨𝐾, 𝐿⟩) = ((Id‘𝐸) ∘ (1^st ‘⟨𝐾, 𝐿⟩)))
76		relfunc 17766	. . . . . . . . . . . 12 ⊢ Rel (𝐷 Func 𝐸)
77	76	brrelex1i 5672	. . . . . . . . . . 11 ⊢ (𝐾(𝐷 Func 𝐸)𝐿 → 𝐾 ∈ V)
78	5, 77	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ V)
79	76	brrelex2i 5673	. . . . . . . . . . 11 ⊢ (𝐾(𝐷 Func 𝐸)𝐿 → 𝐿 ∈ V)
80	5, 79	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ V)
81		op1stg 7933	. . . . . . . . . 10 ⊢ ((𝐾 ∈ V ∧ 𝐿 ∈ V) → (1^st ‘⟨𝐾, 𝐿⟩) = 𝐾)
82	78, 80, 81	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘⟨𝐾, 𝐿⟩) = 𝐾)
83	82	coeq2d 5802	. . . . . . . 8 ⊢ (𝜑 → ((Id‘𝐸) ∘ (1^st ‘⟨𝐾, 𝐿⟩)) = ((Id‘𝐸) ∘ 𝐾))
84	75, 83	eqtrd 2766	. . . . . . 7 ⊢ (𝜑 → ((Id‘(𝐷 FuncCat 𝐸))‘⟨𝐾, 𝐿⟩) = ((Id‘𝐸) ∘ 𝐾))
85	62, 68, 43, 73	fucid 17878	. . . . . . . 8 ⊢ (𝜑 → ((Id‘(𝐶 FuncCat 𝐷))‘⟨𝐹, 𝐺⟩) = ((Id‘𝐷) ∘ (1^st ‘⟨𝐹, 𝐺⟩)))
86		relfunc 17766	. . . . . . . . . . . 12 ⊢ Rel (𝐶 Func 𝐷)
87	86	brrelex1i 5672	. . . . . . . . . . 11 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → 𝐹 ∈ V)
88	14, 87	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ V)
89	86	brrelex2i 5673	. . . . . . . . . . 11 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → 𝐺 ∈ V)
90	14, 89	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ V)
91		op1stg 7933	. . . . . . . . . 10 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (1^st ‘⟨𝐹, 𝐺⟩) = 𝐹)
92	88, 90, 91	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘⟨𝐹, 𝐺⟩) = 𝐹)
93	92	coeq2d 5802	. . . . . . . 8 ⊢ (𝜑 → ((Id‘𝐷) ∘ (1^st ‘⟨𝐹, 𝐺⟩)) = ((Id‘𝐷) ∘ 𝐹))
94	85, 93	eqtrd 2766	. . . . . . 7 ⊢ (𝜑 → ((Id‘(𝐶 FuncCat 𝐷))‘⟨𝐹, 𝐺⟩) = ((Id‘𝐷) ∘ 𝐹))
95	84, 94	opeq12d 4833	. . . . . 6 ⊢ (𝜑 → ⟨((Id‘(𝐷 FuncCat 𝐸))‘⟨𝐾, 𝐿⟩), ((Id‘(𝐶 FuncCat 𝐷))‘⟨𝐹, 𝐺⟩)⟩ = ⟨((Id‘𝐸) ∘ 𝐾), ((Id‘𝐷) ∘ 𝐹)⟩)
96	57, 74, 95	3eqtrd 2770	. . . . 5 ⊢ (𝜑 → ( 1 ‘𝑈) = ⟨((Id‘𝐸) ∘ 𝐾), ((Id‘𝐷) ∘ 𝐹)⟩)
97	96	fveq2d 6826	. . . 4 ⊢ (𝜑 → ((𝑈𝑃𝑈)‘( 1 ‘𝑈)) = ((𝑈𝑃𝑈)‘⟨((Id‘𝐸) ∘ 𝐾), ((Id‘𝐷) ∘ 𝐹)⟩))
98		df-ov 7349	. . . 4 ⊢ (((Id‘𝐸) ∘ 𝐾)(𝑈𝑃𝑈)((Id‘𝐷) ∘ 𝐹)) = ((𝑈𝑃𝑈)‘⟨((Id‘𝐸) ∘ 𝐾), ((Id‘𝐷) ∘ 𝐹)⟩)
99	97, 98	eqtr4di 2784	. . 3 ⊢ (𝜑 → ((𝑈𝑃𝑈)‘( 1 ‘𝑈)) = (((Id‘𝐸) ∘ 𝐾)(𝑈𝑃𝑈)((Id‘𝐷) ∘ 𝐹)))
100		fucoid.o	. . . 4 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨𝑂, 𝑃⟩)
101		eqid 2731	. . . . . . 7 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
102	62, 101	fuchom 17868	. . . . . 6 ⊢ (𝐶 Nat 𝐷) = (Hom ‘(𝐶 FuncCat 𝐷))
103	66, 102, 68, 64, 73	catidcl 17585	. . . . 5 ⊢ (𝜑 → ((Id‘(𝐶 FuncCat 𝐷))‘⟨𝐹, 𝐺⟩) ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝐹, 𝐺⟩))
104	94, 103	eqeltrrd 2832	. . . 4 ⊢ (𝜑 → ((Id‘𝐷) ∘ 𝐹) ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝐹, 𝐺⟩))
105		eqid 2731	. . . . . . 7 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
106	59, 105	fuchom 17868	. . . . . 6 ⊢ (𝐷 Nat 𝐸) = (Hom ‘(𝐷 FuncCat 𝐸))
107	65, 106, 67, 61, 71	catidcl 17585	. . . . 5 ⊢ (𝜑 → ((Id‘(𝐷 FuncCat 𝐸))‘⟨𝐾, 𝐿⟩) ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝐾, 𝐿⟩))
108	84, 107	eqeltrrd 2832	. . . 4 ⊢ (𝜑 → ((Id‘𝐸) ∘ 𝐾) ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝐾, 𝐿⟩))
109	100, 56, 56, 104, 108	fuco22 49370	. . 3 ⊢ (𝜑 → (((Id‘𝐸) ∘ 𝐾)(𝑈𝑃𝑈)((Id‘𝐷) ∘ 𝐹)) = (𝑥 ∈ (Base‘𝐶) ↦ ((((Id‘𝐸) ∘ 𝐾)‘(𝐹‘𝑥))(⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝐹‘𝑥))⟩(comp‘𝐸)(𝐾‘(𝐹‘𝑥)))(((𝐹‘𝑥)𝐿(𝐹‘𝑥))‘(((Id‘𝐷) ∘ 𝐹)‘𝑥)))))
110	99, 109	eqtrd 2766	. 2 ⊢ (𝜑 → ((𝑈𝑃𝑈)‘( 1 ‘𝑈)) = (𝑥 ∈ (Base‘𝐶) ↦ ((((Id‘𝐸) ∘ 𝐾)‘(𝐹‘𝑥))(⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝐹‘𝑥))⟩(comp‘𝐸)(𝐾‘(𝐹‘𝑥)))(((𝐹‘𝑥)𝐿(𝐹‘𝑥))‘(((Id‘𝐷) ∘ 𝐹)‘𝑥)))))
111		fucoid.q	. . 3 ⊢ 𝑄 = (𝐶 FuncCat 𝐸)
112		fucoid.i	. . 3 ⊢ 𝐼 = (Id‘𝑄)
113	100, 14, 5, 56, 111, 112, 8	fuco11id 49365	. 2 ⊢ (𝜑 → (𝐼‘(𝑂‘𝑈)) = ((Id‘𝐸) ∘ (𝐾 ∘ 𝐹)))
114	55, 110, 113	3eqtr4d 2776	1 ⊢ (𝜑 → ((𝑈𝑃𝑈)‘( 1 ‘𝑈)) = (𝐼‘(𝑂‘𝑈)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 Vcvv 3436 ⟨cop 4582 class class class wbr 5091 ↦ cmpt 5172 ∘ ccom 5620 Fn wfn 6476 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 1^st c1st 7919 Basecbs 17117 Hom chom 17169 compcco 17170 Catccat 17567 Idccid 17568 Func cfunc 17758 Nat cnat 17848 FuncCat cfuc 17849 ×_c cxpc 18071 ∘_F cfuco 49347

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 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11059 ax-resscn 11060 ax-1cn 11061 ax-icn 11062 ax-addcl 11063 ax-addrcl 11064 ax-mulcl 11065 ax-mulrcl 11066 ax-mulcom 11067 ax-addass 11068 ax-mulass 11069 ax-distr 11070 ax-i2m1 11071 ax-1ne0 11072 ax-1rid 11073 ax-rnegex 11074 ax-rrecex 11075 ax-cnre 11076 ax-pre-lttri 11077 ax-pre-lttrn 11078 ax-pre-ltadd 11079 ax-pre-mulgt0 11080

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 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-iun 4943 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-er 8622 df-map 8752 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-pnf 11145 df-mnf 11146 df-xr 11147 df-ltxr 11148 df-le 11149 df-sub 11343 df-neg 11344 df-nn 12123 df-2 12185 df-3 12186 df-4 12187 df-5 12188 df-6 12189 df-7 12190 df-8 12191 df-9 12192 df-n0 12379 df-z 12466 df-dec 12586 df-uz 12730 df-fz 13405 df-struct 17055 df-slot 17090 df-ndx 17102 df-base 17118 df-hom 17182 df-cco 17183 df-cat 17571 df-cid 17572 df-func 17762 df-cofu 17764 df-nat 17850 df-fuc 17851 df-xpc 18075 df-fuco 49348

This theorem is referenced by: fucoid2 49380

W3C validator