Theorem prcofdiag 50501

Description: A diagonal functor post-composed by a pre-composition functor is another diagonal functor. (Contributed by Zhi Wang, 25-Nov-2025.)

Hypotheses

Ref	Expression
prcofdiag.l	⊢ 𝐿 = (𝐶Δfunc𝐷)
prcofdiag.m	⊢ 𝑀 = (𝐶Δfunc𝐸)
prcofdiag.f	⊢ (𝜑 → 𝐹 ∈ (𝐸 Func 𝐷))
prcofdiag.c	⊢ (𝜑 → 𝐶 ∈ Cat)
prcofdiag.g	⊢ (𝜑 → (⟨𝐷, 𝐶⟩ −∘F 𝐹) = 𝐺)

Assertion

Ref	Expression
prcofdiag	⊢ (𝜑 → (𝐺 ∘func 𝐿) = 𝑀)

Proof of Theorem prcofdiag

Dummy variables 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2761	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2761	. . . . . 6 ⊢ (Base‘(𝐸 FuncCat 𝐶)) = (Base‘(𝐸 FuncCat 𝐶))
3		prcofdiag.l	. . . . . . . . 9 ⊢ 𝐿 = (𝐶Δfunc𝐷)
4		prcofdiag.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ Cat)
5		prcofdiag.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (𝐸 Func 𝐷))
6	5	func1st2nd 50183	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐹)(𝐸 Func 𝐷)(2nd ‘𝐹))
7	6	funcrcl3 50187	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ Cat)
8		eqid 2761	. . . . . . . . 9 ⊢ (𝐷 FuncCat 𝐶) = (𝐷 FuncCat 𝐶)
9	3, 4, 7, 8	diagcl 18415	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ (𝐶 Func (𝐷 FuncCat 𝐶)))
10		prcofdiag.g	. . . . . . . . 9 ⊢ (𝜑 → (⟨𝐷, 𝐶⟩ −∘F 𝐹) = 𝐺)
11		eqid 2761	. . . . . . . . . 10 ⊢ (𝐸 FuncCat 𝐶) = (𝐸 FuncCat 𝐶)
12	8, 4, 11, 5	prcoffunca 50493	. . . . . . . . 9 ⊢ (𝜑 → (⟨𝐷, 𝐶⟩ −∘F 𝐹) ∈ ((𝐷 FuncCat 𝐶) Func (𝐸 FuncCat 𝐶)))
13	10, 12	eqeltrrd 2862	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ ((𝐷 FuncCat 𝐶) Func (𝐸 FuncCat 𝐶)))
14	9, 13	cofucl 18063	. . . . . . 7 ⊢ (𝜑 → (𝐺 ∘func 𝐿) ∈ (𝐶 Func (𝐸 FuncCat 𝐶)))
15	14	func1st2nd 50183	. . . . . 6 ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐿))(𝐶 Func (𝐸 FuncCat 𝐶))(2nd ‘(𝐺 ∘func 𝐿)))
16	1, 2, 15	funcf1 18041	. . . . 5 ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐿)):(Base‘𝐶)⟶(Base‘(𝐸 FuncCat 𝐶)))
17	16	ffnd 6710	. . . 4 ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐿)) Fn (Base‘𝐶))
18		prcofdiag.m	. . . . . . . 8 ⊢ 𝑀 = (𝐶Δfunc𝐸)
19	6	funcrcl2 50186	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Cat)
20	18, 4, 19, 11	diagcl 18415	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (𝐶 Func (𝐸 FuncCat 𝐶)))
21	20	func1st2nd 50183	. . . . . 6 ⊢ (𝜑 → (1st ‘𝑀)(𝐶 Func (𝐸 FuncCat 𝐶))(2nd ‘𝑀))
22	1, 2, 21	funcf1 18041	. . . . 5 ⊢ (𝜑 → (1st ‘𝑀):(Base‘𝐶)⟶(Base‘(𝐸 FuncCat 𝐶)))
23	22	ffnd 6710	. . . 4 ⊢ (𝜑 → (1st ‘𝑀) Fn (Base‘𝐶))
24	9	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐿 ∈ (𝐶 Func (𝐷 FuncCat 𝐶)))
25	13	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐺 ∈ ((𝐷 FuncCat 𝐶) Func (𝐸 FuncCat 𝐶)))
26		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
27	1, 24, 25, 26	cofu1 18059	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝐺 ∘func 𝐿))‘𝑥) = ((1st ‘𝐺)‘((1st ‘𝐿)‘𝑥)))
28	4	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
29	7	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
30		eqid 2761	. . . . . . 7 ⊢ ((1st ‘𝐿)‘𝑥) = ((1st ‘𝐿)‘𝑥)
31	3, 28, 29, 1, 26, 30	diag1cl 18416	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐿)‘𝑥) ∈ (𝐷 Func 𝐶))
32	10	fveq2d 6889	. . . . . . 7 ⊢ (𝜑 → (1st ‘(⟨𝐷, 𝐶⟩ −∘F 𝐹)) = (1st ‘𝐺))
33	32	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1st ‘(⟨𝐷, 𝐶⟩ −∘F 𝐹)) = (1st ‘𝐺))
34	31, 33	prcof1 50495	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐺)‘((1st ‘𝐿)‘𝑥)) = (((1st ‘𝐿)‘𝑥) ∘func 𝐹))
35	5	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐹 ∈ (𝐸 Func 𝐷))
36	3, 18, 35, 28, 1, 26	prcofdiag1 50500	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((1st ‘𝐿)‘𝑥) ∘func 𝐹) = ((1st ‘𝑀)‘𝑥))
37	27, 34, 36	3eqtrd 2800	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝐺 ∘func 𝐿))‘𝑥) = ((1st ‘𝑀)‘𝑥))
38	17, 23, 37	eqfnfvd 7032	. . 3 ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐿)) = (1st ‘𝑀))
39	1, 15	funcfn2 18044	. . . 4 ⊢ (𝜑 → (2nd ‘(𝐺 ∘func 𝐿)) Fn ((Base‘𝐶) × (Base‘𝐶)))
40	1, 21	funcfn2 18044	. . . 4 ⊢ (𝜑 → (2nd ‘𝑀) Fn ((Base‘𝐶) × (Base‘𝐶)))
41		eqid 2761	. . . . . . 7 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
42		eqid 2761	. . . . . . 7 ⊢ (Hom ‘(𝐸 FuncCat 𝐶)) = (Hom ‘(𝐸 FuncCat 𝐶))
43	15	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st ‘(𝐺 ∘func 𝐿))(𝐶 Func (𝐸 FuncCat 𝐶))(2nd ‘(𝐺 ∘func 𝐿)))
44		simprl 783	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
45		simprr 785	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
46	1, 41, 42, 43, 44, 45	funcf2 18043	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘(𝐺 ∘func 𝐿))𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st ‘(𝐺 ∘func 𝐿))‘𝑥)(Hom ‘(𝐸 FuncCat 𝐶))((1st ‘(𝐺 ∘func 𝐿))‘𝑦)))
47	46	ffnd 6710	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘(𝐺 ∘func 𝐿))𝑦) Fn (𝑥(Hom ‘𝐶)𝑦))
48	21	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st ‘𝑀)(𝐶 Func (𝐸 FuncCat 𝐶))(2nd ‘𝑀))
49	1, 41, 42, 48, 44, 45	funcf2 18043	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘𝑀)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st ‘𝑀)‘𝑥)(Hom ‘(𝐸 FuncCat 𝐶))((1st ‘𝑀)‘𝑦)))
50	49	ffnd 6710	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘𝑀)𝑦) Fn (𝑥(Hom ‘𝐶)𝑦))
51		eqid 2761	. . . . . . . 8 ⊢ (Base‘𝐸) = (Base‘𝐸)
52		eqid 2761	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
53	5	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐹 ∈ (𝐸 Func 𝐷))
54	53	func1st2nd 50183	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (1st ‘𝐹)(𝐸 Func 𝐷)(2nd ‘𝐹))
55	51, 52, 54	funcf1 18041	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (1st ‘𝐹):(Base‘𝐸)⟶(Base‘𝐷))
56		xpco2 49966	. . . . . . 7 ⊢ ((1st ‘𝐹):(Base‘𝐸)⟶(Base‘𝐷) → (((Base‘𝐷) × {𝑓}) ∘ (1st ‘𝐹)) = ((Base‘𝐸) × {𝑓}))
57	55, 56	syl 18	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (((Base‘𝐷) × {𝑓}) ∘ (1st ‘𝐹)) = ((Base‘𝐸) × {𝑓}))
58	9	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐿 ∈ (𝐶 Func (𝐷 FuncCat 𝐶)))
59	13	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐺 ∈ ((𝐷 FuncCat 𝐶) Func (𝐸 FuncCat 𝐶)))
60	44	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑥 ∈ (Base‘𝐶))
61	45	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑦 ∈ (Base‘𝐶))
62		simpr 490	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
63	1, 58, 59, 60, 61, 41, 62	cofu2 18061	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd ‘(𝐺 ∘func 𝐿))𝑦)‘𝑓) = ((((1st ‘𝐿)‘𝑥)(2nd ‘𝐺)((1st ‘𝐿)‘𝑦))‘((𝑥(2nd ‘𝐿)𝑦)‘𝑓)))
64	4	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐶 ∈ Cat)
65	7	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐷 ∈ Cat)
66	3, 1, 52, 41, 64, 65, 60, 61, 62	diag2 18419	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd ‘𝐿)𝑦)‘𝑓) = ((Base‘𝐷) × {𝑓}))
67	66	fveq2d 6889	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((((1st ‘𝐿)‘𝑥)(2nd ‘𝐺)((1st ‘𝐿)‘𝑦))‘((𝑥(2nd ‘𝐿)𝑦)‘𝑓)) = ((((1st ‘𝐿)‘𝑥)(2nd ‘𝐺)((1st ‘𝐿)‘𝑦))‘((Base‘𝐷) × {𝑓})))
68		eqid 2761	. . . . . . . 8 ⊢ (𝐷 Nat 𝐶) = (𝐷 Nat 𝐶)
69	3, 1, 52, 41, 64, 65, 60, 61, 62, 68	diag2cl 18420	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((Base‘𝐷) × {𝑓}) ∈ (((1st ‘𝐿)‘𝑥)(𝐷 Nat 𝐶)((1st ‘𝐿)‘𝑦)))
70	10	fveq2d 6889	. . . . . . . . 9 ⊢ (𝜑 → (2nd ‘(⟨𝐷, 𝐶⟩ −∘F 𝐹)) = (2nd ‘𝐺))
71	70	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (2nd ‘(⟨𝐷, 𝐶⟩ −∘F 𝐹)) = (2nd ‘𝐺))
72	68, 69, 71, 53	prcof21a 50498	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((((1st ‘𝐿)‘𝑥)(2nd ‘𝐺)((1st ‘𝐿)‘𝑦))‘((Base‘𝐷) × {𝑓})) = (((Base‘𝐷) × {𝑓}) ∘ (1st ‘𝐹)))
73	63, 67, 72	3eqtrd 2800	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd ‘(𝐺 ∘func 𝐿))𝑦)‘𝑓) = (((Base‘𝐷) × {𝑓}) ∘ (1st ‘𝐹)))
74	19	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐸 ∈ Cat)
75	18, 1, 51, 41, 64, 74, 60, 61, 62	diag2 18419	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd ‘𝑀)𝑦)‘𝑓) = ((Base‘𝐸) × {𝑓}))
76	57, 73, 75	3eqtr4d 2806	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd ‘(𝐺 ∘func 𝐿))𝑦)‘𝑓) = ((𝑥(2nd ‘𝑀)𝑦)‘𝑓))
77	47, 50, 76	eqfnfvd 7032	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘(𝐺 ∘func 𝐿))𝑦) = (𝑥(2nd ‘𝑀)𝑦))
78	39, 40, 77	eqfnovd 49975	. . 3 ⊢ (𝜑 → (2nd ‘(𝐺 ∘func 𝐿)) = (2nd ‘𝑀))
79	38, 78	opeq12d 4841	. 2 ⊢ (𝜑 → ⟨(1st ‘(𝐺 ∘func 𝐿)), (2nd ‘(𝐺 ∘func 𝐿))⟩ = ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩)
80		relfunc 18037	. . 3 ⊢ Rel (𝐶 Func (𝐸 FuncCat 𝐶))
81		1st2nd 8050	. . 3 ⊢ ((Rel (𝐶 Func (𝐸 FuncCat 𝐶)) ∧ (𝐺 ∘func 𝐿) ∈ (𝐶 Func (𝐸 FuncCat 𝐶))) → (𝐺 ∘func 𝐿) = ⟨(1st ‘(𝐺 ∘func 𝐿)), (2nd ‘(𝐺 ∘func 𝐿))⟩)
82	80, 14, 81	sylancr 599	. 2 ⊢ (𝜑 → (𝐺 ∘func 𝐿) = ⟨(1st ‘(𝐺 ∘func 𝐿)), (2nd ‘(𝐺 ∘func 𝐿))⟩)
83		1st2nd 8050	. . 3 ⊢ ((Rel (𝐶 Func (𝐸 FuncCat 𝐶)) ∧ 𝑀 ∈ (𝐶 Func (𝐸 FuncCat 𝐶))) → 𝑀 = ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩)
84	80, 20, 83	sylancr 599	. 2 ⊢ (𝜑 → 𝑀 = ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩)
85	79, 82, 84	3eqtr4d 2806	1 ⊢ (𝜑 → (𝐺 ∘func 𝐿) = 𝑀)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {csn 4584  ⟨cop 4590   class class class wbr 5103   × cxp 5649   ∘ ccom 5655  Rel wrel 5656  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  1^st c1st 7999  2^nd c2nd 8000  Basecbs 17387  Hom chom 17439  Catccat 17838   Func cfunc 18029   ∘_func ccofu 18031   Nat cnat 18119   FuncCat cfuc 18120  Δ_funccdiag 18386   −∘_F cprcof 50480

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  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 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-map 8849  df-ixp 8926  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-z 12694  df-dec 12815  df-uz 12966  df-fz 13640  df-struct 17325  df-slot 17360  df-ndx 17372  df-base 17388  df-hom 17452  df-cco 17453  df-cat 17842  df-cid 17843  df-func 18033  df-cofu 18035  df-nat 18121  df-fuc 18122  df-xpc 18346  df-1stf 18347  df-curf 18388  df-diag 18390  df-swapf 50367  df-fuco 50424  df-prcof 50481

This theorem is used by:  lmdran  50778  cmdlan  50779