fucoid - Mathbox for Zhi Wang

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Theorem fucoid 48915

Description: Each identity morphism in the source category is mapped to the corresponding identity morphism in the target category. See also fucoid2 48916. (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 7471	. . . . 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 2737	. . . . 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 6719	. . . 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 48838	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ Cat)
7		eqid 2737	. . . . . 6 ⊢ (Base‘𝐸) = (Base‘𝐸)
8		eqid 2737	. . . . . 6 ⊢ (Id‘𝐸) = (Id‘𝐸)
9	7, 8	cidfn 17733	. . . . 5 ⊢ (𝐸 ∈ Cat → (Id‘𝐸) Fn (Base‘𝐸))
10	6, 9	syl 17	. . . 4 ⊢ (𝜑 → (Id‘𝐸) Fn (Base‘𝐸))
11		eqid 2737	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
12	11, 7, 5	funcf1 17926	. . . . 5 ⊢ (𝜑 → 𝐾:(Base‘𝐷)⟶(Base‘𝐸))
13		eqid 2737	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
14		fucoid.f	. . . . . 6 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
15	13, 11, 14	funcf1 17926	. . . . 5 ⊢ (𝜑 → 𝐹:(Base‘𝐶)⟶(Base‘𝐷))
16	12, 15	fcod 6769	. . . 4 ⊢ (𝜑 → (𝐾 ∘ 𝐹):(Base‘𝐶)⟶(Base‘𝐸))
17		fnfco 6781	. . . 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 6919	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝐾‘(𝐹‘𝑥)) = (𝐾‘(𝐹‘𝑤)))
20	19, 19	opeq12d 4889	. . . . . . 7 ⊢ (𝑥 = 𝑤 → ⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝐹‘𝑥))⟩ = ⟨(𝐾‘(𝐹‘𝑤)), (𝐾‘(𝐹‘𝑤))⟩)
21	20, 19	oveq12d 7456	. . . . . 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 6919	. . . . . 6 ⊢ (𝑥 = 𝑤 → (((Id‘𝐸) ∘ 𝐾)‘(𝐹‘𝑥)) = (((Id‘𝐸) ∘ 𝐾)‘(𝐹‘𝑤)))
23		fveq2 6914	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
24	23, 23	oveq12d 7456	. . . . . . 7 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑥)𝐿(𝐹‘𝑥)) = ((𝐹‘𝑤)𝐿(𝐹‘𝑤)))
25		fveq2 6914	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (((Id‘𝐷) ∘ 𝐹)‘𝑥) = (((Id‘𝐷) ∘ 𝐹)‘𝑤))
26	24, 25	fveq12d 6921	. . . . . 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 7459	. . . . 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 7473	. . . . 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 7046	. . . 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 2737	. . . . . 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 7111	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → (𝐹‘𝑤) ∈ (Base‘𝐷))
35	33, 34	ffvelcdmd 7112	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → (𝐾‘(𝐹‘𝑤)) ∈ (Base‘𝐸))
36		eqid 2737	. . . . . 6 ⊢ (comp‘𝐸) = (comp‘𝐸)
37	7, 31, 8, 32, 35	catidcl 17736	. . . . . 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 17737	. . . . 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 7016	. . . . . 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 7016	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → (((Id‘𝐷) ∘ 𝐹)‘𝑤) = ((Id‘𝐷)‘(𝐹‘𝑤)))
42	41	fveq2d 6918	. . . . . . 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 2737	. . . . . . . 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 17930	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → (((𝐹‘𝑤)𝐿(𝐹‘𝑤))‘((Id‘𝐷)‘(𝐹‘𝑤))) = ((Id‘𝐸)‘(𝐾‘(𝐹‘𝑤))))
46	42, 45	eqtrd 2777	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → (((𝐹‘𝑤)𝐿(𝐹‘𝑤))‘(((Id‘𝐷) ∘ 𝐹)‘𝑤)) = ((Id‘𝐸)‘(𝐾‘(𝐹‘𝑤))))
47	39, 46	oveq12d 7456	. . . . 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 7016	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → (((Id‘𝐸) ∘ (𝐾 ∘ 𝐹))‘𝑤) = ((Id‘𝐸)‘((𝐾 ∘ 𝐹)‘𝑤)))
50	40, 28	fvco3d 7016	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → ((𝐾 ∘ 𝐹)‘𝑤) = (𝐾‘(𝐹‘𝑤)))
51	50	fveq2d 6918	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → ((Id‘𝐸)‘((𝐾 ∘ 𝐹)‘𝑤)) = ((Id‘𝐸)‘(𝐾‘(𝐹‘𝑤))))
52	49, 51	eqtrd 2777	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (Base‘𝐶)) → (((Id‘𝐸) ∘ (𝐾 ∘ 𝐹))‘𝑤) = ((Id‘𝐸)‘(𝐾‘(𝐹‘𝑤))))
53	38, 47, 52	3eqtr4d 2787	. . . 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 2777	. . 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 7061	. 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 6918	. . . . . 6 ⊢ (𝜑 → ( 1 ‘𝑈) = ( 1 ‘⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩))
58		fucoid.t	. . . . . . 7 ⊢ 𝑇 = ((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷))
59		eqid 2737	. . . . . . . 8 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
60	5	funcrcl2 48837	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ Cat)
61	59, 60, 6	fuccat 18036	. . . . . . 7 ⊢ (𝜑 → (𝐷 FuncCat 𝐸) ∈ Cat)
62		eqid 2737	. . . . . . . 8 ⊢ (𝐶 FuncCat 𝐷) = (𝐶 FuncCat 𝐷)
63	14	funcrcl2 48837	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ Cat)
64	62, 63, 60	fuccat 18036	. . . . . . 7 ⊢ (𝜑 → (𝐶 FuncCat 𝐷) ∈ Cat)
65	59	fucbas 18025	. . . . . . 7 ⊢ (𝐷 Func 𝐸) = (Base‘(𝐷 FuncCat 𝐸))
66	62	fucbas 18025	. . . . . . 7 ⊢ (𝐶 Func 𝐷) = (Base‘(𝐶 FuncCat 𝐷))
67		eqid 2737	. . . . . . 7 ⊢ (Id‘(𝐷 FuncCat 𝐸)) = (Id‘(𝐷 FuncCat 𝐸))
68		eqid 2737	. . . . . . 7 ⊢ (Id‘(𝐶 FuncCat 𝐷)) = (Id‘(𝐶 FuncCat 𝐷))
69		fucoid.1	. . . . . . 7 ⊢ 1 = (Id‘𝑇)
70		df-br 5152	. . . . . . . 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 5152	. . . . . . . 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 18254	. . . . . 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 18037	. . . . . . . 8 ⊢ (𝜑 → ((Id‘(𝐷 FuncCat 𝐸))‘⟨𝐾, 𝐿⟩) = ((Id‘𝐸) ∘ (1^st ‘⟨𝐾, 𝐿⟩)))
76		relfunc 17922	. . . . . . . . . . . 12 ⊢ Rel (𝐷 Func 𝐸)
77	76	brrelex1i 5749	. . . . . . . . . . 11 ⊢ (𝐾(𝐷 Func 𝐸)𝐿 → 𝐾 ∈ V)
78	5, 77	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ V)
79	76	brrelex2i 5750	. . . . . . . . . . 11 ⊢ (𝐾(𝐷 Func 𝐸)𝐿 → 𝐿 ∈ V)
80	5, 79	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ V)
81		op1stg 8034	. . . . . . . . . 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 5880	. . . . . . . 8 ⊢ (𝜑 → ((Id‘𝐸) ∘ (1^st ‘⟨𝐾, 𝐿⟩)) = ((Id‘𝐸) ∘ 𝐾))
84	75, 83	eqtrd 2777	. . . . . . 7 ⊢ (𝜑 → ((Id‘(𝐷 FuncCat 𝐸))‘⟨𝐾, 𝐿⟩) = ((Id‘𝐸) ∘ 𝐾))
85	62, 68, 43, 73	fucid 18037	. . . . . . . 8 ⊢ (𝜑 → ((Id‘(𝐶 FuncCat 𝐷))‘⟨𝐹, 𝐺⟩) = ((Id‘𝐷) ∘ (1^st ‘⟨𝐹, 𝐺⟩)))
86		relfunc 17922	. . . . . . . . . . . 12 ⊢ Rel (𝐶 Func 𝐷)
87	86	brrelex1i 5749	. . . . . . . . . . 11 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → 𝐹 ∈ V)
88	14, 87	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ V)
89	86	brrelex2i 5750	. . . . . . . . . . 11 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → 𝐺 ∈ V)
90	14, 89	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ V)
91		op1stg 8034	. . . . . . . . . 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 5880	. . . . . . . 8 ⊢ (𝜑 → ((Id‘𝐷) ∘ (1^st ‘⟨𝐹, 𝐺⟩)) = ((Id‘𝐷) ∘ 𝐹))
94	85, 93	eqtrd 2777	. . . . . . 7 ⊢ (𝜑 → ((Id‘(𝐶 FuncCat 𝐷))‘⟨𝐹, 𝐺⟩) = ((Id‘𝐷) ∘ 𝐹))
95	84, 94	opeq12d 4889	. . . . . 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 2781	. . . . 5 ⊢ (𝜑 → ( 1 ‘𝑈) = ⟨((Id‘𝐸) ∘ 𝐾), ((Id‘𝐷) ∘ 𝐹)⟩)
97	96	fveq2d 6918	. . . 4 ⊢ (𝜑 → ((𝑈𝑃𝑈)‘( 1 ‘𝑈)) = ((𝑈𝑃𝑈)‘⟨((Id‘𝐸) ∘ 𝐾), ((Id‘𝐷) ∘ 𝐹)⟩))
98		df-ov 7441	. . . 4 ⊢ (((Id‘𝐸) ∘ 𝐾)(𝑈𝑃𝑈)((Id‘𝐷) ∘ 𝐹)) = ((𝑈𝑃𝑈)‘⟨((Id‘𝐸) ∘ 𝐾), ((Id‘𝐷) ∘ 𝐹)⟩)
99	97, 98	eqtr4di 2795	. . 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 2737	. . . . . . 7 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
102	62, 101	fuchom 18026	. . . . . 6 ⊢ (𝐶 Nat 𝐷) = (Hom ‘(𝐶 FuncCat 𝐷))
103	66, 102, 68, 64, 73	catidcl 17736	. . . . 5 ⊢ (𝜑 → ((Id‘(𝐶 FuncCat 𝐷))‘⟨𝐹, 𝐺⟩) ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝐹, 𝐺⟩))
104	94, 103	eqeltrrd 2842	. . . 4 ⊢ (𝜑 → ((Id‘𝐷) ∘ 𝐹) ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝐹, 𝐺⟩))
105		eqid 2737	. . . . . . 7 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
106	59, 105	fuchom 18026	. . . . . 6 ⊢ (𝐷 Nat 𝐸) = (Hom ‘(𝐷 FuncCat 𝐸))
107	65, 106, 67, 61, 71	catidcl 17736	. . . . 5 ⊢ (𝜑 → ((Id‘(𝐷 FuncCat 𝐸))‘⟨𝐾, 𝐿⟩) ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝐾, 𝐿⟩))
108	84, 107	eqeltrrd 2842	. . . 4 ⊢ (𝜑 → ((Id‘𝐸) ∘ 𝐾) ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝐾, 𝐿⟩))
109	100, 56, 56, 104, 108	fuco22 48906	. . 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 2777	. 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 48903	. 2 ⊢ (𝜑 → (𝐼‘(𝑂‘𝑈)) = ((Id‘𝐸) ∘ (𝐾 ∘ 𝐹)))
114	55, 110, 113	3eqtr4d 2787	1 ⊢ (𝜑 → ((𝑈𝑃𝑈)‘( 1 ‘𝑈)) = (𝐼‘(𝑂‘𝑈)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 Vcvv 3481 ⟨cop 4640 class class class wbr 5151 ↦ cmpt 5234 ∘ ccom 5697 Fn wfn 6564 ⟶wf 6565 ‘cfv 6569 (class class class)co 7438 1^st c1st 8020 Basecbs 17254 Hom chom 17318 compcco 17319 Catccat 17718 Idccid 17719 Func cfunc 17914 Nat cnat 18005 FuncCat cfuc 18006 ×_c cxpc 18233 ∘_F cfuco 48885

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5288 ax-sep 5305 ax-nul 5315 ax-pow 5374 ax-pr 5441 ax-un 7761 ax-cnex 11218 ax-resscn 11219 ax-1cn 11220 ax-icn 11221 ax-addcl 11222 ax-addrcl 11223 ax-mulcl 11224 ax-mulrcl 11225 ax-mulcom 11226 ax-addass 11227 ax-mulass 11228 ax-distr 11229 ax-i2m1 11230 ax-1ne0 11231 ax-1rid 11232 ax-rnegex 11233 ax-rrecex 11234 ax-cnre 11235 ax-pre-lttri 11236 ax-pre-lttrn 11237 ax-pre-ltadd 11238 ax-pre-mulgt0 11239

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3483 df-sbc 3795 df-csb 3912 df-dif 3969 df-un 3971 df-in 3973 df-ss 3983 df-pss 3986 df-nul 4343 df-if 4535 df-pw 4610 df-sn 4635 df-pr 4637 df-tp 4639 df-op 4641 df-uni 4916 df-iun 5001 df-br 5152 df-opab 5214 df-mpt 5235 df-tr 5269 df-id 5587 df-eprel 5593 df-po 5601 df-so 5602 df-fr 5645 df-we 5647 df-xp 5699 df-rel 5700 df-cnv 5701 df-co 5702 df-dm 5703 df-rn 5704 df-res 5705 df-ima 5706 df-pred 6329 df-ord 6395 df-on 6396 df-lim 6397 df-suc 6398 df-iota 6522 df-fun 6571 df-fn 6572 df-f 6573 df-f1 6574 df-fo 6575 df-f1o 6576 df-fv 6577 df-riota 7395 df-ov 7441 df-oprab 7442 df-mpo 7443 df-om 7895 df-1st 8022 df-2nd 8023 df-frecs 8314 df-wrecs 8345 df-recs 8419 df-rdg 8458 df-1o 8514 df-er 8753 df-map 8876 df-ixp 8946 df-en 8994 df-dom 8995 df-sdom 8996 df-fin 8997 df-pnf 11304 df-mnf 11305 df-xr 11306 df-ltxr 11307 df-le 11308 df-sub 11501 df-neg 11502 df-nn 12274 df-2 12336 df-3 12337 df-4 12338 df-5 12339 df-6 12340 df-7 12341 df-8 12342 df-9 12343 df-n0 12534 df-z 12621 df-dec 12741 df-uz 12886 df-fz 13554 df-struct 17190 df-slot 17225 df-ndx 17237 df-base 17255 df-hom 17331 df-cco 17332 df-cat 17722 df-cid 17723 df-func 17918 df-cofu 17920 df-nat 18007 df-fuc 18008 df-xpc 18237 df-fuco 48886

This theorem is referenced by: fucoid2 48916

W3C validator