fucoid - Mathbox for Zhi Wang

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Theorem fucoid 49823

Description: Each identity morphism in the source category is mapped to the corresponding identity morphism in the target category. See also fucoid2 49824. (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 7400	. . . . 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 2736	. . . . 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 6641	. . . 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 49555	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ Cat)
7		eqid 2736	. . . . . 6 ⊢ (Base‘𝐸) = (Base‘𝐸)
8		eqid 2736	. . . . . 6 ⊢ (Id‘𝐸) = (Id‘𝐸)
9	7, 8	cidfn 17645	. . . . 5 ⊢ (𝐸 ∈ Cat → (Id‘𝐸) Fn (Base‘𝐸))
10	6, 9	syl 17	. . . 4 ⊢ (𝜑 → (Id‘𝐸) Fn (Base‘𝐸))
11		eqid 2736	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
12	11, 7, 5	funcf1 17833	. . . . 5 ⊢ (𝜑 → 𝐾:(Base‘𝐷)⟶(Base‘𝐸))
13		eqid 2736	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
14		fucoid.f	. . . . . 6 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
15	13, 11, 14	funcf1 17833	. . . . 5 ⊢ (𝜑 → 𝐹:(Base‘𝐶)⟶(Base‘𝐷))
16	12, 15	fcod 6693	. . . 4 ⊢ (𝜑 → (𝐾 ∘ 𝐹):(Base‘𝐶)⟶(Base‘𝐸))
17		fnfco 6705	. . . 4 ⊢ (((Id‘𝐸) Fn (Base‘𝐸) ∧ (𝐾 ∘ 𝐹):(Base‘𝐶)⟶(Base‘𝐸)) → ((Id‘𝐸) ∘ (𝐾 ∘ 𝐹)) Fn (Base‘𝐶))
18	10, 16, 17	syl2anc 585	. . 3 ⊢ (𝜑 → ((Id‘𝐸) ∘ (𝐾 ∘ 𝐹)) Fn (Base‘𝐶))
19		2fveq3 6845	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝐾‘(𝐹‘𝑥)) = (𝐾‘(𝐹‘𝑤)))
20	19, 19	opeq12d 4824	. . . . . . 7 ⊢ (𝑥 = 𝑤 → ⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝐹‘𝑥))⟩ = ⟨(𝐾‘(𝐹‘𝑤)), (𝐾‘(𝐹‘𝑤))⟩)
21	20, 19	oveq12d 7385	. . . . . 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 6845	. . . . . 6 ⊢ (𝑥 = 𝑤 → (((Id‘𝐸) ∘ 𝐾)‘(𝐹‘𝑥)) = (((Id‘𝐸) ∘ 𝐾)‘(𝐹‘𝑤)))
23		fveq2 6840	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
24	23, 23	oveq12d 7385	. . . . . . 7 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑥)𝐿(𝐹‘𝑥)) = ((𝐹‘𝑤)𝐿(𝐹‘𝑤)))
25		fveq2 6840	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (((Id‘𝐷) ∘ 𝐹)‘𝑥) = (((Id‘𝐷) ∘ 𝐹)‘𝑤))
26	24, 25	fveq12d 6847	. . . . . 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 7388	. . . . 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 7402	. . . . 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 6971	. . . 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 2736	. . . . . 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 7036	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → (𝐹‘𝑤) ∈ (Base‘𝐷))
35	33, 34	ffvelcdmd 7037	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → (𝐾‘(𝐹‘𝑤)) ∈ (Base‘𝐸))
36		eqid 2736	. . . . . 6 ⊢ (comp‘𝐸) = (comp‘𝐸)
37	7, 31, 8, 32, 35	catidcl 17648	. . . . . 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 17649	. . . . 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 6940	. . . . . 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 6940	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → (((Id‘𝐷) ∘ 𝐹)‘𝑤) = ((Id‘𝐷)‘(𝐹‘𝑤)))
42	41	fveq2d 6844	. . . . . . 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 2736	. . . . . . . 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 17837	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → (((𝐹‘𝑤)𝐿(𝐹‘𝑤))‘((Id‘𝐷)‘(𝐹‘𝑤))) = ((Id‘𝐸)‘(𝐾‘(𝐹‘𝑤))))
46	42, 45	eqtrd 2771	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → (((𝐹‘𝑤)𝐿(𝐹‘𝑤))‘(((Id‘𝐷) ∘ 𝐹)‘𝑤)) = ((Id‘𝐸)‘(𝐾‘(𝐹‘𝑤))))
47	39, 46	oveq12d 7385	. . . . 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 6940	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → (((Id‘𝐸) ∘ (𝐾 ∘ 𝐹))‘𝑤) = ((Id‘𝐸)‘((𝐾 ∘ 𝐹)‘𝑤)))
50	40, 28	fvco3d 6940	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → ((𝐾 ∘ 𝐹)‘𝑤) = (𝐾‘(𝐹‘𝑤)))
51	50	fveq2d 6844	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → ((Id‘𝐸)‘((𝐾 ∘ 𝐹)‘𝑤)) = ((Id‘𝐸)‘(𝐾‘(𝐹‘𝑤))))
52	49, 51	eqtrd 2771	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → (((Id‘𝐸) ∘ (𝐾 ∘ 𝐹))‘𝑤) = ((Id‘𝐸)‘(𝐾‘(𝐹‘𝑤))))
53	38, 47, 52	3eqtr4d 2781	. . . 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 2771	. . 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 6986	. 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 6844	. . . . . 6 ⊢ (𝜑 → ( 1 ‘𝑈) = ( 1 ‘⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩))
58		fucoid.t	. . . . . . 7 ⊢ 𝑇 = ((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷))
59		eqid 2736	. . . . . . . 8 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
60	5	funcrcl2 49554	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ Cat)
61	59, 60, 6	fuccat 17940	. . . . . . 7 ⊢ (𝜑 → (𝐷 FuncCat 𝐸) ∈ Cat)
62		eqid 2736	. . . . . . . 8 ⊢ (𝐶 FuncCat 𝐷) = (𝐶 FuncCat 𝐷)
63	14	funcrcl2 49554	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ Cat)
64	62, 63, 60	fuccat 17940	. . . . . . 7 ⊢ (𝜑 → (𝐶 FuncCat 𝐷) ∈ Cat)
65	59	fucbas 17930	. . . . . . 7 ⊢ (𝐷 Func 𝐸) = (Base‘(𝐷 FuncCat 𝐸))
66	62	fucbas 17930	. . . . . . 7 ⊢ (𝐶 Func 𝐷) = (Base‘(𝐶 FuncCat 𝐷))
67		eqid 2736	. . . . . . 7 ⊢ (Id‘(𝐷 FuncCat 𝐸)) = (Id‘(𝐷 FuncCat 𝐸))
68		eqid 2736	. . . . . . 7 ⊢ (Id‘(𝐶 FuncCat 𝐷)) = (Id‘(𝐶 FuncCat 𝐷))
69		fucoid.1	. . . . . . 7 ⊢ 1 = (Id‘𝑇)
70		df-br 5086	. . . . . . . 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 5086	. . . . . . . 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 18155	. . . . . 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 17941	. . . . . . . 8 ⊢ (𝜑 → ((Id‘(𝐷 FuncCat 𝐸))‘⟨𝐾, 𝐿⟩) = ((Id‘𝐸) ∘ (1^st ‘⟨𝐾, 𝐿⟩)))
76		relfunc 17829	. . . . . . . . . . . 12 ⊢ Rel (𝐷 Func 𝐸)
77	76	brrelex1i 5687	. . . . . . . . . . 11 ⊢ (𝐾(𝐷 Func 𝐸)𝐿 → 𝐾 ∈ V)
78	5, 77	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ V)
79	76	brrelex2i 5688	. . . . . . . . . . 11 ⊢ (𝐾(𝐷 Func 𝐸)𝐿 → 𝐿 ∈ V)
80	5, 79	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ V)
81		op1stg 7954	. . . . . . . . . 10 ⊢ ((𝐾 ∈ V ∧ 𝐿 ∈ V) → (1^st ‘⟨𝐾, 𝐿⟩) = 𝐾)
82	78, 80, 81	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘⟨𝐾, 𝐿⟩) = 𝐾)
83	82	coeq2d 5817	. . . . . . . 8 ⊢ (𝜑 → ((Id‘𝐸) ∘ (1^st ‘⟨𝐾, 𝐿⟩)) = ((Id‘𝐸) ∘ 𝐾))
84	75, 83	eqtrd 2771	. . . . . . 7 ⊢ (𝜑 → ((Id‘(𝐷 FuncCat 𝐸))‘⟨𝐾, 𝐿⟩) = ((Id‘𝐸) ∘ 𝐾))
85	62, 68, 43, 73	fucid 17941	. . . . . . . 8 ⊢ (𝜑 → ((Id‘(𝐶 FuncCat 𝐷))‘⟨𝐹, 𝐺⟩) = ((Id‘𝐷) ∘ (1^st ‘⟨𝐹, 𝐺⟩)))
86		relfunc 17829	. . . . . . . . . . . 12 ⊢ Rel (𝐶 Func 𝐷)
87	86	brrelex1i 5687	. . . . . . . . . . 11 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → 𝐹 ∈ V)
88	14, 87	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ V)
89	86	brrelex2i 5688	. . . . . . . . . . 11 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → 𝐺 ∈ V)
90	14, 89	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ V)
91		op1stg 7954	. . . . . . . . . 10 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (1^st ‘⟨𝐹, 𝐺⟩) = 𝐹)
92	88, 90, 91	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘⟨𝐹, 𝐺⟩) = 𝐹)
93	92	coeq2d 5817	. . . . . . . 8 ⊢ (𝜑 → ((Id‘𝐷) ∘ (1^st ‘⟨𝐹, 𝐺⟩)) = ((Id‘𝐷) ∘ 𝐹))
94	85, 93	eqtrd 2771	. . . . . . 7 ⊢ (𝜑 → ((Id‘(𝐶 FuncCat 𝐷))‘⟨𝐹, 𝐺⟩) = ((Id‘𝐷) ∘ 𝐹))
95	84, 94	opeq12d 4824	. . . . . 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 2775	. . . . 5 ⊢ (𝜑 → ( 1 ‘𝑈) = ⟨((Id‘𝐸) ∘ 𝐾), ((Id‘𝐷) ∘ 𝐹)⟩)
97	96	fveq2d 6844	. . . 4 ⊢ (𝜑 → ((𝑈𝑃𝑈)‘( 1 ‘𝑈)) = ((𝑈𝑃𝑈)‘⟨((Id‘𝐸) ∘ 𝐾), ((Id‘𝐷) ∘ 𝐹)⟩))
98		df-ov 7370	. . . 4 ⊢ (((Id‘𝐸) ∘ 𝐾)(𝑈𝑃𝑈)((Id‘𝐷) ∘ 𝐹)) = ((𝑈𝑃𝑈)‘⟨((Id‘𝐸) ∘ 𝐾), ((Id‘𝐷) ∘ 𝐹)⟩)
99	97, 98	eqtr4di 2789	. . 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 2736	. . . . . . 7 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
102	62, 101	fuchom 17931	. . . . . 6 ⊢ (𝐶 Nat 𝐷) = (Hom ‘(𝐶 FuncCat 𝐷))
103	66, 102, 68, 64, 73	catidcl 17648	. . . . 5 ⊢ (𝜑 → ((Id‘(𝐶 FuncCat 𝐷))‘⟨𝐹, 𝐺⟩) ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝐹, 𝐺⟩))
104	94, 103	eqeltrrd 2837	. . . 4 ⊢ (𝜑 → ((Id‘𝐷) ∘ 𝐹) ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝐹, 𝐺⟩))
105		eqid 2736	. . . . . . 7 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
106	59, 105	fuchom 17931	. . . . . 6 ⊢ (𝐷 Nat 𝐸) = (Hom ‘(𝐷 FuncCat 𝐸))
107	65, 106, 67, 61, 71	catidcl 17648	. . . . 5 ⊢ (𝜑 → ((Id‘(𝐷 FuncCat 𝐸))‘⟨𝐾, 𝐿⟩) ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝐾, 𝐿⟩))
108	84, 107	eqeltrrd 2837	. . . 4 ⊢ (𝜑 → ((Id‘𝐸) ∘ 𝐾) ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝐾, 𝐿⟩))
109	100, 56, 56, 104, 108	fuco22 49814	. . 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 2771	. 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 49809	. 2 ⊢ (𝜑 → (𝐼‘(𝑂‘𝑈)) = ((Id‘𝐸) ∘ (𝐾 ∘ 𝐹)))
114	55, 110, 113	3eqtr4d 2781	1 ⊢ (𝜑 → ((𝑈𝑃𝑈)‘( 1 ‘𝑈)) = (𝐼‘(𝑂‘𝑈)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 Vcvv 3429 ⟨cop 4573 class class class wbr 5085 ↦ cmpt 5166 ∘ ccom 5635 Fn wfn 6493 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 1^st c1st 7940 Basecbs 17179 Hom chom 17231 compcco 17232 Catccat 17630 Idccid 17631 Func cfunc 17821 Nat cnat 17911 FuncCat cfuc 17912 ×_c cxpc 18134 ∘_F cfuco 49791

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 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-er 8643 df-map 8775 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-fz 13462 df-struct 17117 df-slot 17152 df-ndx 17164 df-base 17180 df-hom 17244 df-cco 17245 df-cat 17634 df-cid 17635 df-func 17825 df-cofu 17827 df-nat 17913 df-fuc 17914 df-xpc 18138 df-fuco 49792

This theorem is referenced by: fucoid2 49824

W3C validator