fucof21 - Mathbox for Zhi Wang

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Theorem fucof21 50166

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 17944	. . . . 5 ⊢ Rel (𝐷 Func 𝐸)
5		relfunc 17944	. . . . 5 ⊢ Rel (𝐶 Func 𝐷)
6	2, 3, 4, 5	fuco2eld3 50134	. . . 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 50133	. . 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 50134	. . . 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 50133	. . 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 50155	. 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 50158	. . 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 50165	. . 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 2867	. 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 50071	. . . 4 ⊢ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = (Base‘𝑇)
26		fucof21.j	. . . 4 ⊢ 𝐽 = (Hom ‘𝑇)
27	3, 2	eleqtrd 2868	. . . 4 ⊢ (𝜑 → 𝑈 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
28	10, 2	eleqtrd 2868	. . . 4 ⊢ (𝜑 → 𝑉 ∈ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
29	24, 25, 26, 27, 28	xpcfuchom 50073	. . 3 ⊢ (𝜑 → (𝑈𝐽𝑉) = (((1^st ‘𝑈)(𝐷 Nat 𝐸)(1^st ‘𝑉)) × ((2^nd ‘𝑈)(𝐶 Nat 𝐷)(2^nd ‘𝑉))))
30	9	fveq2d 6892	. . . . . 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 5450	. . . . . . 7 ⊢ ⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩ ∈ V
32		opex 5450	. . . . . . 7 ⊢ ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩ ∈ V
33	31, 32	op1st 8003	. . . . . 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 2817	. . . . 5 ⊢ (𝜑 → (1^st ‘𝑈) = ⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩)
35	14	fveq2d 6892	. . . . . 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 5450	. . . . . . 7 ⊢ ⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩ ∈ V
37		opex 5450	. . . . . . 7 ⊢ ⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩ ∈ V
38	36, 37	op1st 8003	. . . . . 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 2817	. . . . 5 ⊢ (𝜑 → (1^st ‘𝑉) = ⟨(1^st ‘(1^st ‘𝑉)), (2^nd ‘(1^st ‘𝑉))⟩)
40	34, 39	oveq12d 7441	. . . 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 6892	. . . . . 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 8004	. . . . . 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 2817	. . . . 5 ⊢ (𝜑 → (2^nd ‘𝑈) = ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩)
44	14	fveq2d 6892	. . . . . 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 8004	. . . . . 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 2817	. . . . 5 ⊢ (𝜑 → (2^nd ‘𝑉) = ⟨(1^st ‘(2^nd ‘𝑉)), (2^nd ‘(2^nd ‘𝑉))⟩)
47	43, 46	oveq12d 7441	. . . 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 5697	. . 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 2801	. 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 49688	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 2146 ⟨cop 4600 class class class wbr 5114 ↦ cmpt 5197 × cxp 5664 ⟶wf 6539 ‘cfv 6543 (class class class)co 7423 1^st c1st 7993 2^nd c2nd 7994 Basecbs 17294 Hom chom 17346 compcco 17347 Func cfunc 17936 Nat cnat 18026 FuncCat cfuc 18027 ×_c cxpc 18249 ∘_F cfuco 50135

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 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195

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 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4878 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7872 df-1st 7995 df-2nd 7996 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8703 df-map 8835 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-nn 12252 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12523 df-z 12610 df-dec 12730 df-uz 12881 df-fz 13554 df-struct 17232 df-slot 17267 df-ndx 17279 df-base 17295 df-hom 17359 df-cco 17360 df-cat 17749 df-cid 17750 df-func 17940 df-cofu 17942 df-nat 18028 df-fuc 18029 df-xpc 18253 df-fuco 50136

This theorem is used by: fucofunc 50178

W3C validator