Theorem prcofdiag 49892

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 2739	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2739	. . . . . 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 49574	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐹)(𝐸 Func 𝐷)(2nd ‘𝐹))
7	6	funcrcl3 49578	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ Cat)
8		eqid 2739	. . . . . . . . 9 ⊢ (𝐷 FuncCat 𝐶) = (𝐷 FuncCat 𝐶)
9	3, 4, 7, 8	diagcl 18199	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ (𝐶 Func (𝐷 FuncCat 𝐶)))
10		prcofdiag.g	. . . . . . . . 9 ⊢ (𝜑 → (⟨𝐷, 𝐶⟩ −∘F 𝐹) = 𝐺)
11		eqid 2739	. . . . . . . . . 10 ⊢ (𝐸 FuncCat 𝐶) = (𝐸 FuncCat 𝐶)
12	8, 4, 11, 5	prcoffunca 49884	. . . . . . . . 9 ⊢ (𝜑 → (⟨𝐷, 𝐶⟩ −∘F 𝐹) ∈ ((𝐷 FuncCat 𝐶) Func (𝐸 FuncCat 𝐶)))
13	10, 12	eqeltrrd 2840	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ ((𝐷 FuncCat 𝐶) Func (𝐸 FuncCat 𝐶)))
14	9, 13	cofucl 17847	. . . . . . 7 ⊢ (𝜑 → (𝐺 ∘func 𝐿) ∈ (𝐶 Func (𝐸 FuncCat 𝐶)))
15	14	func1st2nd 49574	. . . . . 6 ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐿))(𝐶 Func (𝐸 FuncCat 𝐶))(2nd ‘(𝐺 ∘func 𝐿)))
16	1, 2, 15	funcf1 17825	. . . . 5 ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐿)):(Base‘𝐶)⟶(Base‘(𝐸 FuncCat 𝐶)))
17	16	ffnd 6657	. . . 4 ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐿)) Fn (Base‘𝐶))
18		prcofdiag.m	. . . . . . . 8 ⊢ 𝑀 = (𝐶Δfunc𝐸)
19	6	funcrcl2 49577	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Cat)
20	18, 4, 19, 11	diagcl 18199	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (𝐶 Func (𝐸 FuncCat 𝐶)))
21	20	func1st2nd 49574	. . . . . 6 ⊢ (𝜑 → (1st ‘𝑀)(𝐶 Func (𝐸 FuncCat 𝐶))(2nd ‘𝑀))
22	1, 2, 21	funcf1 17825	. . . . 5 ⊢ (𝜑 → (1st ‘𝑀):(Base‘𝐶)⟶(Base‘(𝐸 FuncCat 𝐶)))
23	22	ffnd 6657	. . . 4 ⊢ (𝜑 → (1st ‘𝑀) Fn (Base‘𝐶))
24	9	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐿 ∈ (𝐶 Func (𝐷 FuncCat 𝐶)))
25	13	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐺 ∈ ((𝐷 FuncCat 𝐶) Func (𝐸 FuncCat 𝐶)))
26		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
27	1, 24, 25, 26	cofu1 17843	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝐺 ∘func 𝐿))‘𝑥) = ((1st ‘𝐺)‘((1st ‘𝐿)‘𝑥)))
28	4	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
29	7	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
30		eqid 2739	. . . . . . 7 ⊢ ((1st ‘𝐿)‘𝑥) = ((1st ‘𝐿)‘𝑥)
31	3, 28, 29, 1, 26, 30	diag1cl 18200	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐿)‘𝑥) ∈ (𝐷 Func 𝐶))
32	10	fveq2d 6832	. . . . . . 7 ⊢ (𝜑 → (1st ‘(⟨𝐷, 𝐶⟩ −∘F 𝐹)) = (1st ‘𝐺))
33	32	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1st ‘(⟨𝐷, 𝐶⟩ −∘F 𝐹)) = (1st ‘𝐺))
34	31, 33	prcof1 49886	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐺)‘((1st ‘𝐿)‘𝑥)) = (((1st ‘𝐿)‘𝑥) ∘func 𝐹))
35	5	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐹 ∈ (𝐸 Func 𝐷))
36	3, 18, 35, 28, 1, 26	prcofdiag1 49891	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((1st ‘𝐿)‘𝑥) ∘func 𝐹) = ((1st ‘𝑀)‘𝑥))
37	27, 34, 36	3eqtrd 2778	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝐺 ∘func 𝐿))‘𝑥) = ((1st ‘𝑀)‘𝑥))
38	17, 23, 37	eqfnfvd 6975	. . 3 ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐿)) = (1st ‘𝑀))
39	1, 15	funcfn2 17828	. . . 4 ⊢ (𝜑 → (2nd ‘(𝐺 ∘func 𝐿)) Fn ((Base‘𝐶) × (Base‘𝐶)))
40	1, 21	funcfn2 17828	. . . 4 ⊢ (𝜑 → (2nd ‘𝑀) Fn ((Base‘𝐶) × (Base‘𝐶)))
41		eqid 2739	. . . . . . 7 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
42		eqid 2739	. . . . . . 7 ⊢ (Hom ‘(𝐸 FuncCat 𝐶)) = (Hom ‘(𝐸 FuncCat 𝐶))
43	15	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st ‘(𝐺 ∘func 𝐿))(𝐶 Func (𝐸 FuncCat 𝐶))(2nd ‘(𝐺 ∘func 𝐿)))
44		simprl 776	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
45		simprr 778	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
46	1, 41, 42, 43, 44, 45	funcf2 17827	. . . . . 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 6657	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘(𝐺 ∘func 𝐿))𝑦) Fn (𝑥(Hom ‘𝐶)𝑦))
48	21	adantr 481	. . . . . . 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 17827	. . . . . 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 6657	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘𝑀)𝑦) Fn (𝑥(Hom ‘𝐶)𝑦))
51		eqid 2739	. . . . . . . 8 ⊢ (Base‘𝐸) = (Base‘𝐸)
52		eqid 2739	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
53	5	ad2antrr 732	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐹 ∈ (𝐸 Func 𝐷))
54	53	func1st2nd 49574	. . . . . . . 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 17825	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (1st ‘𝐹):(Base‘𝐸)⟶(Base‘𝐷))
56		xpco2 49355	. . . . . . 7 ⊢ ((1st ‘𝐹):(Base‘𝐸)⟶(Base‘𝐷) → (((Base‘𝐷) × {𝑓}) ∘ (1st ‘𝐹)) = ((Base‘𝐸) × {𝑓}))
57	55, 56	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (((Base‘𝐷) × {𝑓}) ∘ (1st ‘𝐹)) = ((Base‘𝐸) × {𝑓}))
58	9	ad2antrr 732	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐿 ∈ (𝐶 Func (𝐷 FuncCat 𝐶)))
59	13	ad2antrr 732	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐺 ∈ ((𝐷 FuncCat 𝐶) Func (𝐸 FuncCat 𝐶)))
60	44	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑥 ∈ (Base‘𝐶))
61	45	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑦 ∈ (Base‘𝐶))
62		simpr 485	. . . . . . . 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 17845	. . . . . . 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 732	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐶 ∈ Cat)
65	7	ad2antrr 732	. . . . . . . . 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 18203	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd ‘𝐿)𝑦)‘𝑓) = ((Base‘𝐷) × {𝑓}))
67	66	fveq2d 6832	. . . . . . 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 2739	. . . . . . . 8 ⊢ (𝐷 Nat 𝐶) = (𝐷 Nat 𝐶)
69	3, 1, 52, 41, 64, 65, 60, 61, 62, 68	diag2cl 18204	. . . . . . . 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 6832	. . . . . . . . 9 ⊢ (𝜑 → (2nd ‘(⟨𝐷, 𝐶⟩ −∘F 𝐹)) = (2nd ‘𝐺))
71	70	ad2antrr 732	. . . . . . . 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 49889	. . . . . . 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 2778	. . . . . 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 732	. . . . . . 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 18203	. . . . . 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 2784	. . . . 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 6975	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘(𝐺 ∘func 𝐿))𝑦) = (𝑥(2nd ‘𝑀)𝑦))
78	39, 40, 77	eqfnovd 49364	. . 3 ⊢ (𝜑 → (2nd ‘(𝐺 ∘func 𝐿)) = (2nd ‘𝑀))
79	38, 78	opeq12d 4813	. 2 ⊢ (𝜑 → ⟨(1st ‘(𝐺 ∘func 𝐿)), (2nd ‘(𝐺 ∘func 𝐿))⟩ = ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩)
80		relfunc 17821	. . 3 ⊢ Rel (𝐶 Func (𝐸 FuncCat 𝐶))
81		1st2nd 7982	. . 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 593	. 2 ⊢ (𝜑 → (𝐺 ∘func 𝐿) = ⟨(1st ‘(𝐺 ∘func 𝐿)), (2nd ‘(𝐺 ∘func 𝐿))⟩)
83		1st2nd 7982	. . 3 ⊢ ((Rel (𝐶 Func (𝐸 FuncCat 𝐶)) ∧ 𝑀 ∈ (𝐶 Func (𝐸 FuncCat 𝐶))) → 𝑀 = ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩)
84	80, 20, 83	sylancr 593	. 2 ⊢ (𝜑 → 𝑀 = ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩)
85	79, 82, 84	3eqtr4d 2784	1 ⊢ (𝜑 → (𝐺 ∘func 𝐿) = 𝑀)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {csn 4556  ⟨cop 4562   class class class wbr 5073   × cxp 5617   ∘ ccom 5623  Rel wrel 5624  ⟶wf 6482  ‘cfv 6486  (class class class)co 7357  1^st c1st 7930  2^nd c2nd 7931  Basecbs 17171  Hom chom 17223  Catccat 17622   Func cfunc 17813   ∘_func ccofu 17815   Nat cnat 17903   FuncCat cfuc 17904  Δ_funccdiag 18170   −∘_F cprcof 49871

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-tp 4561  df-op 4563  df-uni 4840  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-tr 5181  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7314  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7808  df-1st 7932  df-2nd 7933  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-er 8634  df-map 8766  df-ixp 8837  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-pnf 11173  df-mnf 11174  df-xr 11175  df-ltxr 11176  df-le 11177  df-sub 11371  df-neg 11372  df-nn 12167  df-2 12236  df-3 12237  df-4 12238  df-5 12239  df-6 12240  df-7 12241  df-8 12242  df-9 12243  df-n0 12430  df-z 12517  df-dec 12637  df-uz 12781  df-fz 13454  df-struct 17109  df-slot 17144  df-ndx 17156  df-base 17172  df-hom 17236  df-cco 17237  df-cat 17626  df-cid 17627  df-func 17817  df-cofu 17819  df-nat 17905  df-fuc 17906  df-xpc 18130  df-1stf 18131  df-curf 18172  df-diag 18174  df-swapf 49758  df-fuco 49815  df-prcof 49872

This theorem is referenced by:  lmdran  50169  cmdlan  50170