fucof21 - Mathbox for Zhi Wang

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Theorem fucof21 50102

Description: The morphism part of the functor composition bifunctor maps a hom-set of the product category into a set of natural transformations. (Contributed by Zhi Wang, 30-Sep-2025.)

**Hypotheses**
Ref	Expression
fucof21.o	⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨𝑂, 𝑃⟩)
fucof21.t	⊢ 𝑇 = ((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷))
fucof21.j	⊢ 𝐽 = (Hom ‘𝑇)
fucof21.w	⊢ (𝜑 → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
fucof21.u	⊢ (𝜑 → 𝑈 ∈ 𝑊)
fucof21.v	⊢ (𝜑 → 𝑉 ∈ 𝑊)

**Assertion**
Ref	Expression
fucof21	⊢ (𝜑 → (𝑈𝑃𝑉):(𝑈𝐽𝑉)⟶((𝑂‘𝑈)(𝐶 Nat 𝐸)(𝑂‘𝑉)))

Proof of Theorem fucof21 Dummy variables 𝑎 𝑏 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fucof21.o	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨𝑂, 𝑃⟩)
2		fucof21.w	. . . . 5 ⊢ (𝜑 → 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
3		fucof21.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑊)
4		relfunc 17920	. . . . 5 ⊢ Rel (𝐷 Func 𝐸)
5		relfunc 17920	. . . . 5 ⊢ Rel (𝐶 Func 𝐷)
6	2, 3, 4, 5	fuco2eld3 50070	. . . 4 ⊢ (𝜑 → ((1^st ‘(1^st ‘𝑈))(𝐷 Func 𝐸)(2^nd ‘(1^st ‘𝑈)) ∧ (1^st ‘(2^nd ‘𝑈))(𝐶 Func 𝐷)(2^nd ‘(2^nd ‘𝑈))))
7	6	simprd 500	. . 3 ⊢ (𝜑 → (1^st ‘(2^nd ‘𝑈))(𝐶 Func 𝐷)(2^nd ‘(2^nd ‘𝑈)))
8	6	simpld 499	. . 3 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑈))(𝐷 Func 𝐸)(2^nd ‘(1^st ‘𝑈)))
9	2, 3, 4, 5	fuco2eld2 50069	. . 3 ⊢ (𝜑 → 𝑈 = ⟨⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩, ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩⟩)
10		fucof21.v	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ 𝑊)
11	2, 10, 4, 5	fuco2eld3 50070	. . . 4 ⊢ (𝜑 → ((1^st ‘(1^st ‘𝑉))(𝐷 Func 𝐸)(2^nd ‘(1^st ‘𝑉)) ∧ (1^st ‘(2^nd ‘𝑉))(𝐶 Func 𝐷)(2^nd ‘(2^nd ‘𝑉))))
12	11	simprd 500	. . 3 ⊢ (𝜑 → (1^st ‘(2^nd ‘𝑉))(𝐶 Func 𝐷)(2^nd ‘(2^nd ‘𝑉)))
13	11	simpld 499	. . 3 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑉))(𝐷 Func 𝐸)(2^nd ‘(1^st ‘𝑉)))
14	2, 10, 4, 5	fuco2eld2 50069	. . 3 ⊢ (𝜑 → 𝑉 = ⟨⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩, ⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩⟩)
15	1, 7, 8, 9, 12, 13, 14	fuco21 50091	. 2 ⊢ (𝜑 → (𝑈𝑃𝑉) = (𝑏 ∈ (⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩(𝐷 Nat 𝐸)⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩), 𝑎 ∈ (⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩(𝐶 Nat 𝐷)⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((𝑏‘((1^st ‘(2^nd ‘𝑉))‘𝑥))(⟨((1^st ‘(1^st ‘𝑈))‘((1^st ‘(2^nd ‘𝑈))‘𝑥)), ((1^st ‘(1^st ‘𝑈))‘((1^st ‘(2^nd ‘𝑉))‘𝑥))⟩(comp‘𝐸)((1^st ‘(1^st ‘𝑉))‘((1^st ‘(2^nd ‘𝑉))‘𝑥)))((((1^st ‘(2^nd ‘𝑈))‘𝑥)(2^nd ‘(1^st ‘𝑈))((1^st ‘(2^nd ‘𝑉))‘𝑥))‘(𝑎‘𝑥))))))
16	1	adantr 485	. . . 4 ⊢ ((𝜑 ∧ (𝑏 ∈ (⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩(𝐷 Nat 𝐸)⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩) ∧ 𝑎 ∈ (⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩(𝐶 Nat 𝐷)⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩))) → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨𝑂, 𝑃⟩)
17	9	adantr 485	. . . 4 ⊢ ((𝜑 ∧ (𝑏 ∈ (⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩(𝐷 Nat 𝐸)⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩) ∧ 𝑎 ∈ (⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩(𝐶 Nat 𝐷)⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩))) → 𝑈 = ⟨⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩, ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩⟩)
18	14	adantr 485	. . . 4 ⊢ ((𝜑 ∧ (𝑏 ∈ (⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩(𝐷 Nat 𝐸)⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩) ∧ 𝑎 ∈ (⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩(𝐶 Nat 𝐷)⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩))) → 𝑉 = ⟨⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩, ⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩⟩)
19		simprr 784	. . . 4 ⊢ ((𝜑 ∧ (𝑏 ∈ (⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩(𝐷 Nat 𝐸)⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩) ∧ 𝑎 ∈ (⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩(𝐶 Nat 𝐷)⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩))) → 𝑎 ∈ (⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩(𝐶 Nat 𝐷)⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩))
20		simprl 782	. . . 4 ⊢ ((𝜑 ∧ (𝑏 ∈ (⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩(𝐷 Nat 𝐸)⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩) ∧ 𝑎 ∈ (⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩(𝐶 Nat 𝐷)⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩))) → 𝑏 ∈ (⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩(𝐷 Nat 𝐸)⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩))
21	16, 17, 18, 19, 20	fuco22 50094	. . 3 ⊢ ((𝜑 ∧ (𝑏 ∈ (⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩(𝐷 Nat 𝐸)⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩) ∧ 𝑎 ∈ (⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩(𝐶 Nat 𝐷)⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩))) → (𝑏(𝑈𝑃𝑉)𝑎) = (𝑥 ∈ (Base‘𝐶) ↦ ((𝑏‘((1^st ‘(2^nd ‘𝑉))‘𝑥))(⟨((1^st ‘(1^st ‘𝑈))‘((1^st ‘(2^nd ‘𝑈))‘𝑥)), ((1^st ‘(1^st ‘𝑈))‘((1^st ‘(2^nd ‘𝑉))‘𝑥))⟩(comp‘𝐸)((1^st ‘(1^st ‘𝑉))‘((1^st ‘(2^nd ‘𝑉))‘𝑥)))((((1^st ‘(2^nd ‘𝑈))‘𝑥)(2^nd ‘(1^st ‘𝑈))((1^st ‘(2^nd ‘𝑉))‘𝑥))‘(𝑎‘𝑥)))))
22	16, 19, 20, 17, 18	fuco22nat 50101	. . 3 ⊢ ((𝜑 ∧ (𝑏 ∈ (⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩(𝐷 Nat 𝐸)⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩) ∧ 𝑎 ∈ (⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩(𝐶 Nat 𝐷)⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩))) → (𝑏(𝑈𝑃𝑉)𝑎) ∈ ((𝑂‘𝑈)(𝐶 Nat 𝐸)(𝑂‘𝑉)))
23	21, 22	eqeltrrd 2864	. 2 ⊢ ((𝜑 ∧ (𝑏 ∈ (⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩(𝐷 Nat 𝐸)⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩) ∧ 𝑎 ∈ (⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩(𝐶 Nat 𝐷)⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩))) → (𝑥 ∈ (Base‘𝐶) ↦ ((𝑏‘((1^st ‘(2^nd ‘𝑉))‘𝑥))(⟨((1^st ‘(1^st ‘𝑈))‘((1^st ‘(2^nd ‘𝑈))‘𝑥)), ((1^st ‘(1^st ‘𝑈))‘((1^st ‘(2^nd ‘𝑉))‘𝑥))⟩(comp‘𝐸)((1^st ‘(1^st ‘𝑉))‘((1^st ‘(2^nd ‘𝑉))‘𝑥)))((((1^st ‘(2^nd ‘𝑈))‘𝑥)(2^nd ‘(1^st ‘𝑈))((1^st ‘(2^nd ‘𝑉))‘𝑥))‘(𝑎‘𝑥)))) ∈ ((𝑂‘𝑈)(𝐶 Nat 𝐸)(𝑂‘𝑉)))
24		fucof21.t	. . . 4 ⊢ 𝑇 = ((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷))
25	24	xpcfucbas 50007	. . . 4 ⊢ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = (Base‘𝑇)
26		fucof21.j	. . . 4 ⊢ 𝐽 = (Hom ‘𝑇)
27	3, 2	eleqtrd 2865	. . . 4 ⊢ (𝜑 → 𝑈 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
28	10, 2	eleqtrd 2865	. . . 4 ⊢ (𝜑 → 𝑉 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
29	24, 25, 26, 27, 28	xpcfuchom 50009	. . 3 ⊢ (𝜑 → (𝑈𝐽𝑉) = (((1^st ‘𝑈)(𝐷 Nat 𝐸)(1^st ‘𝑉)) × ((2^nd ‘𝑈)(𝐶 Nat 𝐷)(2^nd ‘𝑉))))
30	9	fveq2d 6887	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝑈) = (1^st ‘⟨⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩, ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩⟩))
31		opex 5447	. . . . . . 7 ⊢ ⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩ ∈ V
32		opex 5447	. . . . . . 7 ⊢ ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩ ∈ V
33	31, 32	op1st 7995	. . . . . 6 ⊢ (1^st ‘⟨⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩, ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩⟩) = ⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩
34	30, 33	eqtrdi 2814	. . . . 5 ⊢ (𝜑 → (1^st ‘𝑈) = ⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩)
35	14	fveq2d 6887	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝑉) = (1^st ‘⟨⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩, ⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩⟩))
36		opex 5447	. . . . . . 7 ⊢ ⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩ ∈ V
37		opex 5447	. . . . . . 7 ⊢ ⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩ ∈ V
38	36, 37	op1st 7995	. . . . . 6 ⊢ (1^st ‘⟨⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩, ⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩⟩) = ⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩
39	35, 38	eqtrdi 2814	. . . . 5 ⊢ (𝜑 → (1^st ‘𝑉) = ⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩)
40	34, 39	oveq12d 7430	. . . 4 ⊢ (𝜑 → ((1^st ‘𝑈)(𝐷 Nat 𝐸)(1^st ‘𝑉)) = (⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩(𝐷 Nat 𝐸)⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩))
41	9	fveq2d 6887	. . . . . 6 ⊢ (𝜑 → (2^nd ‘𝑈) = (2^nd ‘⟨⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩, ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩⟩))
42	31, 32	op2nd 7996	. . . . . 6 ⊢ (2^nd ‘⟨⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩, ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩⟩) = ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩
43	41, 42	eqtrdi 2814	. . . . 5 ⊢ (𝜑 → (2^nd ‘𝑈) = ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩)
44	14	fveq2d 6887	. . . . . 6 ⊢ (𝜑 → (2^nd ‘𝑉) = (2^nd ‘⟨⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩, ⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩⟩))
45	36, 37	op2nd 7996	. . . . . 6 ⊢ (2^nd ‘⟨⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩, ⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩⟩) = ⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩
46	44, 45	eqtrdi 2814	. . . . 5 ⊢ (𝜑 → (2^nd ‘𝑉) = ⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩)
47	43, 46	oveq12d 7430	. . . 4 ⊢ (𝜑 → ((2^nd ‘𝑈)(𝐶 Nat 𝐷)(2^nd ‘𝑉)) = (⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩(𝐶 Nat 𝐷)⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩))
48	40, 47	xpeq12d 5694	. . 3 ⊢ (𝜑 → (((1^st ‘𝑈)(𝐷 Nat 𝐸)(1^st ‘𝑉)) × ((2^nd ‘𝑈)(𝐶 Nat 𝐷)(2^nd ‘𝑉))) = ((⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩(𝐷 Nat 𝐸)⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩) × (⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩(𝐶 Nat 𝐷)⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩)))
49	29, 48	eqtrd 2798	. 2 ⊢ (𝜑 → (𝑈𝐽𝑉) = ((⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩(𝐷 Nat 𝐸)⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩) × (⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩(𝐶 Nat 𝐷)⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩)))
50	15, 23, 49	fmpodg 49624	1 ⊢ (𝜑 → (𝑈𝑃𝑉):(𝑈𝐽𝑉)⟶((𝑂‘𝑈)(𝐶 Nat 𝐸)(𝑂‘𝑉)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ⟨cop 4596 class class class wbr 5110 ↦ cmpt 5193 × cxp 5661 ⟶wf 6534 ‘cfv 6538 (class class class)co 7412 1^st c1st 7985 2^nd c2nd 7986 Basecbs 17270 Hom chom 17322 compcco 17323 Func cfunc 17912 Nat cnat 18002 FuncCat cfuc 18003 ×_c cxpc 18225 ∘_F cfuco 50071

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 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 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-er 8695 df-map 8827 df-ixp 8897 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-nn 12235 df-2 12304 df-3 12305 df-4 12306 df-5 12307 df-6 12308 df-7 12309 df-8 12310 df-9 12311 df-n0 12506 df-z 12593 df-dec 12713 df-uz 12864 df-fz 13537 df-struct 17208 df-slot 17243 df-ndx 17255 df-base 17271 df-hom 17335 df-cco 17336 df-cat 17725 df-cid 17726 df-func 17916 df-cofu 17918 df-nat 18004 df-fuc 18005 df-xpc 18229 df-fuco 50072

This theorem is referenced by: fucofunc 50114

W3C validator