fucoid - Mathbox for Zhi Wang

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Theorem fucoid 50563

Description: Each identity morphism in the source category is mapped to the corresponding identity morphism in the target category. See also fucoid2 50564. (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 7453	. . . . 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 2760	. . . . 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 6682	. . . 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 50287	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ Cat)
7		eqid 2760	. . . . . 6 ⊢ (Base‘𝐸) = (Base‘𝐸)
8		eqid 2760	. . . . . 6 ⊢ (Id‘𝐸) = (Id‘𝐸)
9	7, 8	cidfn 17858	. . . . 5 ⊢ (𝐸 ∈ Cat → (Id‘𝐸) Fn (Base‘𝐸))
10	6, 9	syl 18	. . . 4 ⊢ (𝜑 → (Id‘𝐸) Fn (Base‘𝐸))
11		eqid 2760	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
12	11, 7, 5	funcf1 18046	. . . . 5 ⊢ (𝜑 → 𝐾:(Base‘𝐷)⟶(Base‘𝐸))
13		eqid 2760	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
14		fucoid.f	. . . . . 6 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
15	13, 11, 14	funcf1 18046	. . . . 5 ⊢ (𝜑 → 𝐹:(Base‘𝐶)⟶(Base‘𝐷))
16	12, 15	fcod 6735	. . . 4 ⊢ (𝜑 → (𝐾 ∘ 𝐹):(Base‘𝐶)⟶(Base‘𝐸))
17		fnfco 6747	. . . 4 ⊢ (((Id‘𝐸) Fn (Base‘𝐸) ∧ (𝐾 ∘ 𝐹):(Base‘𝐶)⟶(Base‘𝐸)) → ((Id‘𝐸) ∘ (𝐾 ∘ 𝐹)) Fn (Base‘𝐶))
18	10, 16, 17	syl2anc 596	. . 3 ⊢ (𝜑 → ((Id‘𝐸) ∘ (𝐾 ∘ 𝐹)) Fn (Base‘𝐶))
19		2fveq3 6890	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝐾‘(𝐹‘𝑥)) = (𝐾‘(𝐹‘𝑤)))
20	19, 19	opeq12d 4847	. . . . . . 7 ⊢ (𝑥 = 𝑤 → ⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝐹‘𝑥))⟩ = ⟨(𝐾‘(𝐹‘𝑤)), (𝐾‘(𝐹‘𝑤))⟩)
21	20, 19	oveq12d 7438	. . . . . 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 6890	. . . . . 6 ⊢ (𝑥 = 𝑤 → (((Id‘𝐸) ∘ 𝐾)‘(𝐹‘𝑥)) = (((Id‘𝐸) ∘ 𝐾)‘(𝐹‘𝑤)))
23		fveq2 6885	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
24	23, 23	oveq12d 7438	. . . . . . 7 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑥)𝐿(𝐹‘𝑥)) = ((𝐹‘𝑤)𝐿(𝐹‘𝑤)))
25		fveq2 6885	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (((Id‘𝐷) ∘ 𝐹)‘𝑥) = (((Id‘𝐷) ∘ 𝐹)‘𝑤))
26	24, 25	fveq12d 6892	. . . . . 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 7441	. . . . 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 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → 𝑤 ∈ (Base‘𝐶))
29		ovexd 7455	. . . . 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 7017	. . . 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 2760	. . . . . 6 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
32	6	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → 𝐸 ∈ Cat)
33	12	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → 𝐾:(Base‘𝐷)⟶(Base‘𝐸))
34	15	ffvelcdmda 7084	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → (𝐹‘𝑤) ∈ (Base‘𝐷))
35	33, 34	ffvelcdmd 7085	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → (𝐾‘(𝐹‘𝑤)) ∈ (Base‘𝐸))
36		eqid 2760	. . . . . 6 ⊢ (comp‘𝐸) = (comp‘𝐸)
37	7, 31, 8, 32, 35	catidcl 17861	. . . . . 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 17862	. . . . 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 6986	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → (((Id‘𝐸) ∘ 𝐾)‘(𝐹‘𝑤)) = ((Id‘𝐸)‘(𝐾‘(𝐹‘𝑤))))
40	15	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → 𝐹:(Base‘𝐶)⟶(Base‘𝐷))
41	40, 28	fvco3d 6986	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → (((Id‘𝐷) ∘ 𝐹)‘𝑤) = ((Id‘𝐷)‘(𝐹‘𝑤)))
42	41	fveq2d 6889	. . . . . . 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 2760	. . . . . . . 8 ⊢ (Id‘𝐷) = (Id‘𝐷)
44	5	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → 𝐾(𝐷 Func 𝐸)𝐿)
45	11, 43, 8, 44, 34	funcid 18050	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → (((𝐹‘𝑤)𝐿(𝐹‘𝑤))‘((Id‘𝐷)‘(𝐹‘𝑤))) = ((Id‘𝐸)‘(𝐾‘(𝐹‘𝑤))))
46	42, 45	eqtrd 2795	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → (((𝐹‘𝑤)𝐿(𝐹‘𝑤))‘(((Id‘𝐷) ∘ 𝐹)‘𝑤)) = ((Id‘𝐸)‘(𝐾‘(𝐹‘𝑤))))
47	39, 46	oveq12d 7438	. . . . 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 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → (𝐾 ∘ 𝐹):(Base‘𝐶)⟶(Base‘𝐸))
49	48, 28	fvco3d 6986	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → (((Id‘𝐸) ∘ (𝐾 ∘ 𝐹))‘𝑤) = ((Id‘𝐸)‘((𝐾 ∘ 𝐹)‘𝑤)))
50	40, 28	fvco3d 6986	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → ((𝐾 ∘ 𝐹)‘𝑤) = (𝐾‘(𝐹‘𝑤)))
51	50	fveq2d 6889	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → ((Id‘𝐸)‘((𝐾 ∘ 𝐹)‘𝑤)) = ((Id‘𝐸)‘(𝐾‘(𝐹‘𝑤))))
52	49, 51	eqtrd 2795	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → (((Id‘𝐸) ∘ (𝐾 ∘ 𝐹))‘𝑤) = ((Id‘𝐸)‘(𝐾‘(𝐹‘𝑤))))
53	38, 47, 52	3eqtr4d 2805	. . . 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 2795	. . 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 7032	. 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 6889	. . . . . 6 ⊢ (𝜑 → ( 1 ‘𝑈) = ( 1 ‘⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩))
58		fucoid.t	. . . . . . 7 ⊢ 𝑇 = ((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷))
59		eqid 2760	. . . . . . . 8 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
60	5	funcrcl2 50286	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ Cat)
61	59, 60, 6	fuccat 18153	. . . . . . 7 ⊢ (𝜑 → (𝐷 FuncCat 𝐸) ∈ Cat)
62		eqid 2760	. . . . . . . 8 ⊢ (𝐶 FuncCat 𝐷) = (𝐶 FuncCat 𝐷)
63	14	funcrcl2 50286	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ Cat)
64	62, 63, 60	fuccat 18153	. . . . . . 7 ⊢ (𝜑 → (𝐶 FuncCat 𝐷) ∈ Cat)
65	59	fucbas 18143	. . . . . . 7 ⊢ (𝐷 Func 𝐸) = (Base‘(𝐷 FuncCat 𝐸))
66	62	fucbas 18143	. . . . . . 7 ⊢ (𝐶 Func 𝐷) = (Base‘(𝐶 FuncCat 𝐷))
67		eqid 2760	. . . . . . 7 ⊢ (Id‘(𝐷 FuncCat 𝐸)) = (Id‘(𝐷 FuncCat 𝐸))
68		eqid 2760	. . . . . . 7 ⊢ (Id‘(𝐶 FuncCat 𝐷)) = (Id‘(𝐶 FuncCat 𝐷))
69		fucoid.1	. . . . . . 7 ⊢ 1 = (Id‘𝑇)
70		df-br 5103	. . . . . . . 8 ⊢ (𝐾(𝐷 Func 𝐸)𝐿 ↔ ⟨𝐾, 𝐿⟩ ∈ (𝐷 Func 𝐸))
71	5, 70	sylib 221	. . . . . . 7 ⊢ (𝜑 → ⟨𝐾, 𝐿⟩ ∈ (𝐷 Func 𝐸))
72		df-br 5103	. . . . . . . 8 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
73	14, 72	sylib 221	. . . . . . 7 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
74	58, 61, 64, 65, 66, 67, 68, 69, 71, 73	xpcid 18368	. . . . . 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 18154	. . . . . . . 8 ⊢ (𝜑 → ((Id‘(𝐷 FuncCat 𝐸))‘⟨𝐾, 𝐿⟩) = ((Id‘𝐸) ∘ (1^st ‘⟨𝐾, 𝐿⟩)))
76		relfunc 18042	. . . . . . . . . . . 12 ⊢ Rel (𝐷 Func 𝐸)
77	76	brrelex1i 5706	. . . . . . . . . . 11 ⊢ (𝐾(𝐷 Func 𝐸)𝐿 → 𝐾 ∈ V)
78	5, 77	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ V)
79	76	brrelex2i 5707	. . . . . . . . . . 11 ⊢ (𝐾(𝐷 Func 𝐸)𝐿 → 𝐿 ∈ V)
80	5, 79	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ V)
81		op1stg 8013	. . . . . . . . . 10 ⊢ ((𝐾 ∈ V ∧ 𝐿 ∈ V) → (1^st ‘⟨𝐾, 𝐿⟩) = 𝐾)
82	78, 80, 81	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘⟨𝐾, 𝐿⟩) = 𝐾)
83	82	coeq2d 5839	. . . . . . . 8 ⊢ (𝜑 → ((Id‘𝐸) ∘ (1^st ‘⟨𝐾, 𝐿⟩)) = ((Id‘𝐸) ∘ 𝐾))
84	75, 83	eqtrd 2795	. . . . . . 7 ⊢ (𝜑 → ((Id‘(𝐷 FuncCat 𝐸))‘⟨𝐾, 𝐿⟩) = ((Id‘𝐸) ∘ 𝐾))
85	62, 68, 43, 73	fucid 18154	. . . . . . . 8 ⊢ (𝜑 → ((Id‘(𝐶 FuncCat 𝐷))‘⟨𝐹, 𝐺⟩) = ((Id‘𝐷) ∘ (1^st ‘⟨𝐹, 𝐺⟩)))
86		relfunc 18042	. . . . . . . . . . . 12 ⊢ Rel (𝐶 Func 𝐷)
87	86	brrelex1i 5706	. . . . . . . . . . 11 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → 𝐹 ∈ V)
88	14, 87	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ V)
89	86	brrelex2i 5707	. . . . . . . . . . 11 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → 𝐺 ∈ V)
90	14, 89	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ V)
91		op1stg 8013	. . . . . . . . . 10 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (1^st ‘⟨𝐹, 𝐺⟩) = 𝐹)
92	88, 90, 91	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘⟨𝐹, 𝐺⟩) = 𝐹)
93	92	coeq2d 5839	. . . . . . . 8 ⊢ (𝜑 → ((Id‘𝐷) ∘ (1^st ‘⟨𝐹, 𝐺⟩)) = ((Id‘𝐷) ∘ 𝐹))
94	85, 93	eqtrd 2795	. . . . . . 7 ⊢ (𝜑 → ((Id‘(𝐶 FuncCat 𝐷))‘⟨𝐹, 𝐺⟩) = ((Id‘𝐷) ∘ 𝐹))
95	84, 94	opeq12d 4847	. . . . . 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 2799	. . . . 5 ⊢ (𝜑 → ( 1 ‘𝑈) = ⟨((Id‘𝐸) ∘ 𝐾), ((Id‘𝐷) ∘ 𝐹)⟩)
97	96	fveq2d 6889	. . . 4 ⊢ (𝜑 → ((𝑈𝑃𝑈)‘( 1 ‘𝑈)) = ((𝑈𝑃𝑈)‘⟨((Id‘𝐸) ∘ 𝐾), ((Id‘𝐷) ∘ 𝐹)⟩))
98		df-ov 7423	. . . 4 ⊢ (((Id‘𝐸) ∘ 𝐾)(𝑈𝑃𝑈)((Id‘𝐷) ∘ 𝐹)) = ((𝑈𝑃𝑈)‘⟨((Id‘𝐸) ∘ 𝐾), ((Id‘𝐷) ∘ 𝐹)⟩)
99	97, 98	eqtr4di 2813	. . 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 2760	. . . . . . 7 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
102	62, 101	fuchom 18144	. . . . . 6 ⊢ (𝐶 Nat 𝐷) = (Hom ‘(𝐶 FuncCat 𝐷))
103	66, 102, 68, 64, 73	catidcl 17861	. . . . 5 ⊢ (𝜑 → ((Id‘(𝐶 FuncCat 𝐷))‘⟨𝐹, 𝐺⟩) ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝐹, 𝐺⟩))
104	94, 103	eqeltrrd 2861	. . . 4 ⊢ (𝜑 → ((Id‘𝐷) ∘ 𝐹) ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝐹, 𝐺⟩))
105		eqid 2760	. . . . . . 7 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
106	59, 105	fuchom 18144	. . . . . 6 ⊢ (𝐷 Nat 𝐸) = (Hom ‘(𝐷 FuncCat 𝐸))
107	65, 106, 67, 61, 71	catidcl 17861	. . . . 5 ⊢ (𝜑 → ((Id‘(𝐷 FuncCat 𝐸))‘⟨𝐾, 𝐿⟩) ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝐾, 𝐿⟩))
108	84, 107	eqeltrrd 2861	. . . 4 ⊢ (𝜑 → ((Id‘𝐸) ∘ 𝐾) ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝐾, 𝐿⟩))
109	100, 56, 56, 104, 108	fuco22 50554	. . 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 2795	. 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 50549	. 2 ⊢ (𝜑 → (𝐼‘(𝑂‘𝑈)) = ((Id‘𝐸) ∘ (𝐾 ∘ 𝐹)))
114	55, 110, 113	3eqtr4d 2805	1 ⊢ (𝜑 → ((𝑈𝑃𝑈)‘( 1 ‘𝑈)) = (𝐼‘(𝑂‘𝑈)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2144 Vcvv 3450 ⟨cop 4834 class class class wbr 5102 ↦ cmpt 5185 ∘ ccom 5654 Fn wfn 6533 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 1^st c1st 7999 Basecbs 17392 Hom chom 17444 compcco 17445 Catccat 17843 Idccid 17844 Func cfunc 18034 Nat cnat 18124 FuncCat cfuc 18125 ×_c cxpc 18347 ∘_F cfuco 50531

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11259 ax-resscn 11260 ax-1cn 11261 ax-icn 11262 ax-addcl 11263 ax-addrcl 11264 ax-mulcl 11265 ax-mulrcl 11266 ax-mulcom 11267 ax-addass 11268 ax-mulass 11269 ax-distr 11270 ax-i2m1 11271 ax-1ne0 11272 ax-1rid 11273 ax-rnegex 11274 ax-rrecex 11275 ax-cnre 11276 ax-pre-lttri 11277 ax-pre-lttrn 11278 ax-pre-ltadd 11279 ax-pre-mulgt0 11280

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8302 df-wrecs 8333 df-recs 8382 df-rdg 8421 df-1o 8479 df-er 8720 df-map 8852 df-ixp 8929 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-pnf 11348 df-mnf 11349 df-xr 11350 df-ltxr 11351 df-le 11352 df-sub 11546 df-neg 11547 df-nn 12339 df-2 12408 df-3 12409 df-4 12410 df-5 12411 df-6 12412 df-7 12413 df-8 12414 df-9 12415 df-n0 12610 df-z 12697 df-dec 12818 df-uz 12969 df-fz 13643 df-struct 17330 df-slot 17365 df-ndx 17377 df-base 17393 df-hom 17457 df-cco 17458 df-cat 17847 df-cid 17848 df-func 18038 df-cofu 18040 df-nat 18126 df-fuc 18127 df-xpc 18351 df-fuco 50532

This theorem is used by: fucoid2 50564

W3C validator