fucof21 - Mathbox for Zhi Wang

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Theorem fucof21 49834

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 17820	. . . . 5 ⊢ Rel (𝐷 Func 𝐸)
5		relfunc 17820	. . . . 5 ⊢ Rel (𝐶 Func 𝐷)
6	2, 3, 4, 5	fuco2eld3 49802	. . . 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 495	. . 3 ⊢ (𝜑 → (1^st ‘(2^nd ‘𝑈))(𝐶 Func 𝐷)(2^nd ‘(2^nd ‘𝑈)))
8	6	simpld 494	. . 3 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑈))(𝐷 Func 𝐸)(2^nd ‘(1^st ‘𝑈)))
9	2, 3, 4, 5	fuco2eld2 49801	. . 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 49802	. . . 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 495	. . 3 ⊢ (𝜑 → (1^st ‘(2^nd ‘𝑉))(𝐶 Func 𝐷)(2^nd ‘(2^nd ‘𝑉)))
13	11	simpld 494	. . 3 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑉))(𝐷 Func 𝐸)(2^nd ‘(1^st ‘𝑉)))
14	2, 10, 4, 5	fuco2eld2 49801	. . 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 49823	. 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 480	. . . 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 480	. . . 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 480	. . . 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 773	. . . 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 771	. . . 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 49826	. . 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 49833	. . 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 2838	. 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 49739	. . . 4 ⊢ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = (Base‘𝑇)
26		fucof21.j	. . . 4 ⊢ 𝐽 = (Hom ‘𝑇)
27	3, 2	eleqtrd 2839	. . . 4 ⊢ (𝜑 → 𝑈 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
28	10, 2	eleqtrd 2839	. . . 4 ⊢ (𝜑 → 𝑉 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
29	24, 25, 26, 27, 28	xpcfuchom 49741	. . 3 ⊢ (𝜑 → (𝑈𝐽𝑉) = (((1^st ‘𝑈)(𝐷 Nat 𝐸)(1^st ‘𝑉)) × ((2^nd ‘𝑈)(𝐶 Nat 𝐷)(2^nd ‘𝑉))))
30	9	fveq2d 6838	. . . . . 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 5411	. . . . . . 7 ⊢ ⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩ ∈ V
32		opex 5411	. . . . . . 7 ⊢ ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩ ∈ V
33	31, 32	op1st 7943	. . . . . 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 2788	. . . . 5 ⊢ (𝜑 → (1^st ‘𝑈) = ⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩)
35	14	fveq2d 6838	. . . . . 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 5411	. . . . . . 7 ⊢ ⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩ ∈ V
37		opex 5411	. . . . . . 7 ⊢ ⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩ ∈ V
38	36, 37	op1st 7943	. . . . . 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 2788	. . . . 5 ⊢ (𝜑 → (1^st ‘𝑉) = ⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩)
40	34, 39	oveq12d 7378	. . . 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 6838	. . . . . 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 7944	. . . . . 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 2788	. . . . 5 ⊢ (𝜑 → (2^nd ‘𝑈) = ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩)
44	14	fveq2d 6838	. . . . . 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 7944	. . . . . 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 2788	. . . . 5 ⊢ (𝜑 → (2^nd ‘𝑉) = ⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩)
47	43, 46	oveq12d 7378	. . . 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 5655	. . 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 2772	. 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 49356	1 ⊢ (𝜑 → (𝑈𝑃𝑉):(𝑈𝐽𝑉)⟶((𝑂‘𝑈)(𝐶 Nat 𝐸)(𝑂‘𝑉)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ⟨cop 4574 class class class wbr 5086 ↦ cmpt 5167 × cxp 5622 ⟶wf 6488 ‘cfv 6492 (class class class)co 7360 1^st c1st 7933 2^nd c2nd 7934 Basecbs 17170 Hom chom 17222 compcco 17223 Func cfunc 17812 Nat cnat 17902 FuncCat cfuc 17903 ×_c cxpc 18125 ∘_F cfuco 49803

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 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-er 8636 df-map 8768 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-fz 13453 df-struct 17108 df-slot 17143 df-ndx 17155 df-base 17171 df-hom 17235 df-cco 17236 df-cat 17625 df-cid 17626 df-func 17816 df-cofu 17818 df-nat 17904 df-fuc 17905 df-xpc 18129 df-fuco 49804

This theorem is referenced by: fucofunc 49846

W3C validator