fucof21 - Mathbox for Zhi Wang

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Theorem fucof21 50684

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 18069	. . . . 5 ⊢ Rel (𝐷 Func 𝐸)
5		relfunc 18069	. . . . 5 ⊢ Rel (𝐶 Func 𝐷)
6	2, 3, 4, 5	fuco2eld3 50652	. . . 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 501	. . 3 ⊢ (𝜑 → (1^st ‘(2^nd ‘𝑈))(𝐶 Func 𝐷)(2^nd ‘(2^nd ‘𝑈)))
8	6	simpld 500	. . 3 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑈))(𝐷 Func 𝐸)(2^nd ‘(1^st ‘𝑈)))
9	2, 3, 4, 5	fuco2eld2 50651	. . 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 50652	. . . 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 501	. . 3 ⊢ (𝜑 → (1^st ‘(2^nd ‘𝑉))(𝐶 Func 𝐷)(2^nd ‘(2^nd ‘𝑉)))
13	11	simpld 500	. . 3 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑉))(𝐷 Func 𝐸)(2^nd ‘(1^st ‘𝑉)))
14	2, 10, 4, 5	fuco2eld2 50651	. . 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 50673	. 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 486	. . . 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 486	. . . 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 486	. . . 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 785	. . . 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 783	. . . 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 50676	. . 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 50683	. . 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 2865	. 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 50589	. . . 4 ⊢ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = (Base‘𝑇)
26		fucof21.j	. . . 4 ⊢ 𝐽 = (Hom ‘𝑇)
27	3, 2	eleqtrd 2866	. . . 4 ⊢ (𝜑 → 𝑈 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
28	10, 2	eleqtrd 2866	. . . 4 ⊢ (𝜑 → 𝑉 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
29	24, 25, 26, 27, 28	xpcfuchom 50591	. . 3 ⊢ (𝜑 → (𝑈𝐽𝑉) = (((1^st ‘𝑈)(𝐷 Nat 𝐸)(1^st ‘𝑉)) × ((2^nd ‘𝑈)(𝐶 Nat 𝐷)(2^nd ‘𝑉))))
30	9	fveq2d 6915	. . . . . 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 5437	. . . . . . 7 ⊢ ⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩ ∈ V
32		opex 5437	. . . . . . 7 ⊢ ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩ ∈ V
33	31, 32	op1st 8036	. . . . . 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 6915	. . . . . 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 5437	. . . . . . 7 ⊢ ⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩ ∈ V
37		opex 5437	. . . . . . 7 ⊢ ⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩ ∈ V
38	36, 37	op1st 8036	. . . . . 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 7464	. . . 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 6915	. . . . . 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 8037	. . . . . 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 6915	. . . . . 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 8037	. . . . . 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 7464	. . . 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 5687	. . 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 8111	1 ⊢ (𝜑 → (𝑈𝑃𝑉):(𝑈𝐽𝑉)⟶((𝑂‘𝑈)(𝐶 Nat 𝐸)(𝑂‘𝑉)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ⟨cop 4838 class class class wbr 5106 ↦ cmpt 5190 × cxp 5654 ⟶wf 6557 ‘cfv 6561 (class class class)co 7446 1^st c1st 8026 2^nd c2nd 8027 Basecbs 17419 Hom chom 17471 compcco 17472 Func cfunc 18061 Nat cnat 18151 FuncCat cfuc 18152 ×_c cxpc 18374 ∘_F cfuco 50653

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5236 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7778 ax-cnex 11286 ax-resscn 11287 ax-1cn 11288 ax-icn 11289 ax-addcl 11290 ax-addrcl 11291 ax-mulcl 11292 ax-mulrcl 11293 ax-mulcom 11294 ax-addass 11295 ax-mulass 11296 ax-distr 11297 ax-i2m1 11298 ax-1ne0 11299 ax-1rid 11300 ax-rnegex 11301 ax-rrecex 11302 ax-cnre 11303 ax-pre-lttri 11304 ax-pre-lttrn 11305 ax-pre-ltadd 11306 ax-pre-mulgt0 11307

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5551 df-eprel 5556 df-po 5564 df-so 5565 df-fr 5609 df-we 5611 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-pred 6325 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6515 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7403 df-ov 7449 df-oprab 7450 df-mpo 7451 df-om 7905 df-1st 8028 df-2nd 8029 df-frecs 8330 df-wrecs 8361 df-recs 8410 df-rdg 8449 df-1o 8507 df-er 8748 df-map 8880 df-ixp 8957 df-en 9005 df-dom 9006 df-sdom 9007 df-fin 9008 df-pnf 11375 df-mnf 11376 df-xr 11377 df-ltxr 11378 df-le 11379 df-sub 11573 df-neg 11574 df-nn 12366 df-2 12435 df-3 12436 df-4 12437 df-5 12438 df-6 12439 df-7 12440 df-8 12441 df-9 12442 df-n0 12637 df-z 12724 df-dec 12845 df-uz 12996 df-fz 13670 df-struct 17357 df-slot 17392 df-ndx 17404 df-base 17420 df-hom 17484 df-cco 17485 df-cat 17874 df-cid 17875 df-func 18065 df-cofu 18067 df-nat 18153 df-fuc 18154 df-xpc 18378 df-fuco 50654

This theorem is used by: fucofunc 50696

W3C validator