Theorem fuco22natlem2 49238

Description: Lemma for fuco22nat 49241. The commutative square of natural transformation 𝐵 in category 𝐸, combined with the commutative square of fuco22natlem1 49237. (Contributed by Zhi Wang, 30-Sep-2025.)

Hypotheses

Ref	Expression
fuco22natlem1.x	⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
fuco22natlem1.y	⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
fuco22natlem1.a	⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝑀, 𝑁⟩))
fuco22natlem1.h	⊢ (𝜑 → 𝐻 ∈ (𝑋(Hom ‘𝐶)𝑌))
fuco22natlem2.b	⊢ (𝜑 → 𝐵 ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝑅, 𝑆⟩))

Assertion

Ref	Expression
fuco22natlem2	⊢ (𝜑 → (((𝐵‘(𝑀‘𝑌))(⟨(𝐾‘(𝐹‘𝑌)), (𝐾‘(𝑀‘𝑌))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌)))(((𝐹‘𝑌)𝐿(𝑀‘𝑌))‘(𝐴‘𝑌)))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝐹‘𝑌))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌)))(((𝐹‘𝑋)𝐿(𝐹‘𝑌))‘((𝑋𝐺𝑌)‘𝐻))) = ((((𝑀‘𝑋)𝑆(𝑀‘𝑌))‘((𝑋𝑁𝑌)‘𝐻))(⟨(𝐾‘(𝐹‘𝑋)), (𝑅‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌)))((𝐵‘(𝑀‘𝑋))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑋)))(((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋)))))

Proof of Theorem fuco22natlem2

Step	Hyp	Ref	Expression
1		eqid 2730	. . 3 ⊢ (Base‘𝐸) = (Base‘𝐸)
2		eqid 2730	. . 3 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
3		eqid 2730	. . 3 ⊢ (comp‘𝐸) = (comp‘𝐸)
4		eqid 2730	. . . . 5 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
5		fuco22natlem2.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝑅, 𝑆⟩))
6	4, 5	natrcl2 49128	. . . 4 ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)
7	6	funcrcl3 48997	. . 3 ⊢ (𝜑 → 𝐸 ∈ Cat)
8		eqid 2730	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
9	8, 1, 6	funcf1 17834	. . . 4 ⊢ (𝜑 → 𝐾:(Base‘𝐷)⟶(Base‘𝐸))
10		eqid 2730	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
11		eqid 2730	. . . . . . 7 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
12		fuco22natlem1.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝑀, 𝑁⟩))
13	11, 12	natrcl2 49128	. . . . . 6 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
14	10, 8, 13	funcf1 17834	. . . . 5 ⊢ (𝜑 → 𝐹:(Base‘𝐶)⟶(Base‘𝐷))
15		fuco22natlem1.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
16	14, 15	ffvelcdmd 7064	. . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (Base‘𝐷))
17	9, 16	ffvelcdmd 7064	. . 3 ⊢ (𝜑 → (𝐾‘(𝐹‘𝑋)) ∈ (Base‘𝐸))
18		fuco22natlem1.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
19	14, 18	ffvelcdmd 7064	. . . 4 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (Base‘𝐷))
20	9, 19	ffvelcdmd 7064	. . 3 ⊢ (𝜑 → (𝐾‘(𝐹‘𝑌)) ∈ (Base‘𝐸))
21	11, 12	natrcl3 49129	. . . . . 6 ⊢ (𝜑 → 𝑀(𝐶 Func 𝐷)𝑁)
22	10, 8, 21	funcf1 17834	. . . . 5 ⊢ (𝜑 → 𝑀:(Base‘𝐶)⟶(Base‘𝐷))
23	22, 18	ffvelcdmd 7064	. . . 4 ⊢ (𝜑 → (𝑀‘𝑌) ∈ (Base‘𝐷))
24	9, 23	ffvelcdmd 7064	. . 3 ⊢ (𝜑 → (𝐾‘(𝑀‘𝑌)) ∈ (Base‘𝐸))
25		eqid 2730	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
26	8, 25, 2, 6, 16, 19	funcf2 17836	. . . 4 ⊢ (𝜑 → ((𝐹‘𝑋)𝐿(𝐹‘𝑌)):((𝐹‘𝑋)(Hom ‘𝐷)(𝐹‘𝑌))⟶((𝐾‘(𝐹‘𝑋))(Hom ‘𝐸)(𝐾‘(𝐹‘𝑌))))
27		eqid 2730	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
28	10, 27, 25, 13, 15, 18	funcf2 17836	. . . . 5 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋(Hom ‘𝐶)𝑌)⟶((𝐹‘𝑋)(Hom ‘𝐷)(𝐹‘𝑌)))
29		fuco22natlem1.h	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ (𝑋(Hom ‘𝐶)𝑌))
30	28, 29	ffvelcdmd 7064	. . . 4 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝐻) ∈ ((𝐹‘𝑋)(Hom ‘𝐷)(𝐹‘𝑌)))
31	26, 30	ffvelcdmd 7064	. . 3 ⊢ (𝜑 → (((𝐹‘𝑋)𝐿(𝐹‘𝑌))‘((𝑋𝐺𝑌)‘𝐻)) ∈ ((𝐾‘(𝐹‘𝑋))(Hom ‘𝐸)(𝐾‘(𝐹‘𝑌))))
32	8, 25, 2, 6, 19, 23	funcf2 17836	. . . 4 ⊢ (𝜑 → ((𝐹‘𝑌)𝐿(𝑀‘𝑌)):((𝐹‘𝑌)(Hom ‘𝐷)(𝑀‘𝑌))⟶((𝐾‘(𝐹‘𝑌))(Hom ‘𝐸)(𝐾‘(𝑀‘𝑌))))
33	11, 12, 10, 25, 18	natcl 17924	. . . 4 ⊢ (𝜑 → (𝐴‘𝑌) ∈ ((𝐹‘𝑌)(Hom ‘𝐷)(𝑀‘𝑌)))
34	32, 33	ffvelcdmd 7064	. . 3 ⊢ (𝜑 → (((𝐹‘𝑌)𝐿(𝑀‘𝑌))‘(𝐴‘𝑌)) ∈ ((𝐾‘(𝐹‘𝑌))(Hom ‘𝐸)(𝐾‘(𝑀‘𝑌))))
35	4, 5	natrcl3 49129	. . . . 5 ⊢ (𝜑 → 𝑅(𝐷 Func 𝐸)𝑆)
36	8, 1, 35	funcf1 17834	. . . 4 ⊢ (𝜑 → 𝑅:(Base‘𝐷)⟶(Base‘𝐸))
37	36, 23	ffvelcdmd 7064	. . 3 ⊢ (𝜑 → (𝑅‘(𝑀‘𝑌)) ∈ (Base‘𝐸))
38	4, 5, 8, 2, 23	natcl 17924	. . 3 ⊢ (𝜑 → (𝐵‘(𝑀‘𝑌)) ∈ ((𝐾‘(𝑀‘𝑌))(Hom ‘𝐸)(𝑅‘(𝑀‘𝑌))))
39	1, 2, 3, 7, 17, 20, 24, 31, 34, 37, 38	catass 17653	. 2 ⊢ (𝜑 → (((𝐵‘(𝑀‘𝑌))(⟨(𝐾‘(𝐹‘𝑌)), (𝐾‘(𝑀‘𝑌))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌)))(((𝐹‘𝑌)𝐿(𝑀‘𝑌))‘(𝐴‘𝑌)))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝐹‘𝑌))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌)))(((𝐹‘𝑋)𝐿(𝐹‘𝑌))‘((𝑋𝐺𝑌)‘𝐻))) = ((𝐵‘(𝑀‘𝑌))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑌))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌)))((((𝐹‘𝑌)𝐿(𝑀‘𝑌))‘(𝐴‘𝑌))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝐹‘𝑌))⟩(comp‘𝐸)(𝐾‘(𝑀‘𝑌)))(((𝐹‘𝑋)𝐿(𝐹‘𝑌))‘((𝑋𝐺𝑌)‘𝐻)))))
40	15, 18, 12, 29, 6	fuco22natlem1 49237	. . 3 ⊢ (𝜑 → ((((𝐹‘𝑌)𝐿(𝑀‘𝑌))‘(𝐴‘𝑌))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝐹‘𝑌))⟩(comp‘𝐸)(𝐾‘(𝑀‘𝑌)))(((𝐹‘𝑋)𝐿(𝐹‘𝑌))‘((𝑋𝐺𝑌)‘𝐻))) = ((((𝑀‘𝑋)𝐿(𝑀‘𝑌))‘((𝑋𝑁𝑌)‘𝐻))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝐾‘(𝑀‘𝑌)))(((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋))))
41	40	oveq2d 7410	. 2 ⊢ (𝜑 → ((𝐵‘(𝑀‘𝑌))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑌))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌)))((((𝐹‘𝑌)𝐿(𝑀‘𝑌))‘(𝐴‘𝑌))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝐹‘𝑌))⟩(comp‘𝐸)(𝐾‘(𝑀‘𝑌)))(((𝐹‘𝑋)𝐿(𝐹‘𝑌))‘((𝑋𝐺𝑌)‘𝐻)))) = ((𝐵‘(𝑀‘𝑌))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑌))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌)))((((𝑀‘𝑋)𝐿(𝑀‘𝑌))‘((𝑋𝑁𝑌)‘𝐻))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝐾‘(𝑀‘𝑌)))(((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋)))))
42	22, 15	ffvelcdmd 7064	. . . . 5 ⊢ (𝜑 → (𝑀‘𝑋) ∈ (Base‘𝐷))
43	10, 27, 25, 21, 15, 18	funcf2 17836	. . . . . 6 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋(Hom ‘𝐶)𝑌)⟶((𝑀‘𝑋)(Hom ‘𝐷)(𝑀‘𝑌)))
44	43, 29	ffvelcdmd 7064	. . . . 5 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐻) ∈ ((𝑀‘𝑋)(Hom ‘𝐷)(𝑀‘𝑌)))
45	4, 5, 8, 25, 3, 42, 23, 44	nati 17926	. . . 4 ⊢ (𝜑 → ((𝐵‘(𝑀‘𝑌))(⟨(𝐾‘(𝑀‘𝑋)), (𝐾‘(𝑀‘𝑌))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌)))(((𝑀‘𝑋)𝐿(𝑀‘𝑌))‘((𝑋𝑁𝑌)‘𝐻))) = ((((𝑀‘𝑋)𝑆(𝑀‘𝑌))‘((𝑋𝑁𝑌)‘𝐻))(⟨(𝐾‘(𝑀‘𝑋)), (𝑅‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌)))(𝐵‘(𝑀‘𝑋))))
46	45	oveq1d 7409	. . 3 ⊢ (𝜑 → (((𝐵‘(𝑀‘𝑌))(⟨(𝐾‘(𝑀‘𝑋)), (𝐾‘(𝑀‘𝑌))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌)))(((𝑀‘𝑋)𝐿(𝑀‘𝑌))‘((𝑋𝑁𝑌)‘𝐻)))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌)))(((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋))) = (((((𝑀‘𝑋)𝑆(𝑀‘𝑌))‘((𝑋𝑁𝑌)‘𝐻))(⟨(𝐾‘(𝑀‘𝑋)), (𝑅‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌)))(𝐵‘(𝑀‘𝑋)))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌)))(((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋))))
47	9, 42	ffvelcdmd 7064	. . . 4 ⊢ (𝜑 → (𝐾‘(𝑀‘𝑋)) ∈ (Base‘𝐸))
48	8, 25, 2, 6, 16, 42	funcf2 17836	. . . . 5 ⊢ (𝜑 → ((𝐹‘𝑋)𝐿(𝑀‘𝑋)):((𝐹‘𝑋)(Hom ‘𝐷)(𝑀‘𝑋))⟶((𝐾‘(𝐹‘𝑋))(Hom ‘𝐸)(𝐾‘(𝑀‘𝑋))))
49	11, 12, 10, 25, 15	natcl 17924	. . . . 5 ⊢ (𝜑 → (𝐴‘𝑋) ∈ ((𝐹‘𝑋)(Hom ‘𝐷)(𝑀‘𝑋)))
50	48, 49	ffvelcdmd 7064	. . . 4 ⊢ (𝜑 → (((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋)) ∈ ((𝐾‘(𝐹‘𝑋))(Hom ‘𝐸)(𝐾‘(𝑀‘𝑋))))
51	8, 25, 2, 6, 42, 23	funcf2 17836	. . . . 5 ⊢ (𝜑 → ((𝑀‘𝑋)𝐿(𝑀‘𝑌)):((𝑀‘𝑋)(Hom ‘𝐷)(𝑀‘𝑌))⟶((𝐾‘(𝑀‘𝑋))(Hom ‘𝐸)(𝐾‘(𝑀‘𝑌))))
52	51, 44	ffvelcdmd 7064	. . . 4 ⊢ (𝜑 → (((𝑀‘𝑋)𝐿(𝑀‘𝑌))‘((𝑋𝑁𝑌)‘𝐻)) ∈ ((𝐾‘(𝑀‘𝑋))(Hom ‘𝐸)(𝐾‘(𝑀‘𝑌))))
53	1, 2, 3, 7, 17, 47, 24, 50, 52, 37, 38	catass 17653	. . 3 ⊢ (𝜑 → (((𝐵‘(𝑀‘𝑌))(⟨(𝐾‘(𝑀‘𝑋)), (𝐾‘(𝑀‘𝑌))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌)))(((𝑀‘𝑋)𝐿(𝑀‘𝑌))‘((𝑋𝑁𝑌)‘𝐻)))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌)))(((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋))) = ((𝐵‘(𝑀‘𝑌))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑌))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌)))((((𝑀‘𝑋)𝐿(𝑀‘𝑌))‘((𝑋𝑁𝑌)‘𝐻))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝐾‘(𝑀‘𝑌)))(((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋)))))
54	36, 42	ffvelcdmd 7064	. . . 4 ⊢ (𝜑 → (𝑅‘(𝑀‘𝑋)) ∈ (Base‘𝐸))
55	4, 5, 8, 2, 42	natcl 17924	. . . 4 ⊢ (𝜑 → (𝐵‘(𝑀‘𝑋)) ∈ ((𝐾‘(𝑀‘𝑋))(Hom ‘𝐸)(𝑅‘(𝑀‘𝑋))))
56	8, 25, 2, 35, 42, 23	funcf2 17836	. . . . 5 ⊢ (𝜑 → ((𝑀‘𝑋)𝑆(𝑀‘𝑌)):((𝑀‘𝑋)(Hom ‘𝐷)(𝑀‘𝑌))⟶((𝑅‘(𝑀‘𝑋))(Hom ‘𝐸)(𝑅‘(𝑀‘𝑌))))
57	56, 44	ffvelcdmd 7064	. . . 4 ⊢ (𝜑 → (((𝑀‘𝑋)𝑆(𝑀‘𝑌))‘((𝑋𝑁𝑌)‘𝐻)) ∈ ((𝑅‘(𝑀‘𝑋))(Hom ‘𝐸)(𝑅‘(𝑀‘𝑌))))
58	1, 2, 3, 7, 17, 47, 54, 50, 55, 37, 57	catass 17653	. . 3 ⊢ (𝜑 → (((((𝑀‘𝑋)𝑆(𝑀‘𝑌))‘((𝑋𝑁𝑌)‘𝐻))(⟨(𝐾‘(𝑀‘𝑋)), (𝑅‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌)))(𝐵‘(𝑀‘𝑋)))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌)))(((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋))) = ((((𝑀‘𝑋)𝑆(𝑀‘𝑌))‘((𝑋𝑁𝑌)‘𝐻))(⟨(𝐾‘(𝐹‘𝑋)), (𝑅‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌)))((𝐵‘(𝑀‘𝑋))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑋)))(((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋)))))
59	46, 53, 58	3eqtr3d 2773	. 2 ⊢ (𝜑 → ((𝐵‘(𝑀‘𝑌))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑌))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌)))((((𝑀‘𝑋)𝐿(𝑀‘𝑌))‘((𝑋𝑁𝑌)‘𝐻))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝐾‘(𝑀‘𝑌)))(((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋)))) = ((((𝑀‘𝑋)𝑆(𝑀‘𝑌))‘((𝑋𝑁𝑌)‘𝐻))(⟨(𝐾‘(𝐹‘𝑋)), (𝑅‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌)))((𝐵‘(𝑀‘𝑋))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑋)))(((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋)))))
60	39, 41, 59	3eqtrd 2769	1 ⊢ (𝜑 → (((𝐵‘(𝑀‘𝑌))(⟨(𝐾‘(𝐹‘𝑌)), (𝐾‘(𝑀‘𝑌))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌)))(((𝐹‘𝑌)𝐿(𝑀‘𝑌))‘(𝐴‘𝑌)))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝐹‘𝑌))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌)))(((𝐹‘𝑋)𝐿(𝐹‘𝑌))‘((𝑋𝐺𝑌)‘𝐻))) = ((((𝑀‘𝑋)𝑆(𝑀‘𝑌))‘((𝑋𝑁𝑌)‘𝐻))(⟨(𝐾‘(𝐹‘𝑋)), (𝑅‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑌)))((𝐵‘(𝑀‘𝑋))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑋)))(((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋)))))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ⟨cop 4603  ‘cfv 6519  (class class class)co 7394  Basecbs 17185  Hom chom 17237  compcco 17238   Nat cnat 17912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5242  ax-sep 5259  ax-nul 5269  ax-pow 5328  ax-pr 5395  ax-un 7718

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ne 2928  df-ral 3047  df-rex 3056  df-reu 3358  df-rab 3412  df-v 3457  df-sbc 3762  df-csb 3871  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-iun 4965  df-br 5116  df-opab 5178  df-mpt 5197  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-iota 6472  df-fun 6521  df-fn 6522  df-f 6523  df-f1 6524  df-fo 6525  df-f1o 6526  df-fv 6527  df-ov 7397  df-oprab 7398  df-mpo 7399  df-1st 7977  df-2nd 7978  df-map 8805  df-ixp 8875  df-cat 17635  df-func 17826  df-nat 17914

This theorem is referenced by:  fuco22natlem3  49239