Theorem prcofdiag 50020

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 2764	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2764	. . . . . 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 49702	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐹)(𝐸 Func 𝐷)(2nd ‘𝐹))
7	6	funcrcl3 49706	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ Cat)
8		eqid 2764	. . . . . . . . 9 ⊢ (𝐷 FuncCat 𝐶) = (𝐷 FuncCat 𝐶)
9	3, 4, 7, 8	diagcl 18275	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ (𝐶 Func (𝐷 FuncCat 𝐶)))
10		prcofdiag.g	. . . . . . . . 9 ⊢ (𝜑 → (⟨𝐷, 𝐶⟩ −∘F 𝐹) = 𝐺)
11		eqid 2764	. . . . . . . . . 10 ⊢ (𝐸 FuncCat 𝐶) = (𝐸 FuncCat 𝐶)
12	8, 4, 11, 5	prcoffunca 50012	. . . . . . . . 9 ⊢ (𝜑 → (⟨𝐷, 𝐶⟩ −∘F 𝐹) ∈ ((𝐷 FuncCat 𝐶) Func (𝐸 FuncCat 𝐶)))
13	10, 12	eqeltrrd 2865	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ ((𝐷 FuncCat 𝐶) Func (𝐸 FuncCat 𝐶)))
14	9, 13	cofucl 17923	. . . . . . 7 ⊢ (𝜑 → (𝐺 ∘func 𝐿) ∈ (𝐶 Func (𝐸 FuncCat 𝐶)))
15	14	func1st2nd 49702	. . . . . 6 ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐿))(𝐶 Func (𝐸 FuncCat 𝐶))(2nd ‘(𝐺 ∘func 𝐿)))
16	1, 2, 15	funcf1 17901	. . . . 5 ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐿)):(Base‘𝐶)⟶(Base‘(𝐸 FuncCat 𝐶)))
17	16	ffnd 6694	. . . 4 ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐿)) Fn (Base‘𝐶))
18		prcofdiag.m	. . . . . . . 8 ⊢ 𝑀 = (𝐶Δfunc𝐸)
19	6	funcrcl2 49705	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Cat)
20	18, 4, 19, 11	diagcl 18275	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (𝐶 Func (𝐸 FuncCat 𝐶)))
21	20	func1st2nd 49702	. . . . . 6 ⊢ (𝜑 → (1st ‘𝑀)(𝐶 Func (𝐸 FuncCat 𝐶))(2nd ‘𝑀))
22	1, 2, 21	funcf1 17901	. . . . 5 ⊢ (𝜑 → (1st ‘𝑀):(Base‘𝐶)⟶(Base‘(𝐸 FuncCat 𝐶)))
23	22	ffnd 6694	. . . 4 ⊢ (𝜑 → (1st ‘𝑀) Fn (Base‘𝐶))
24	9	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐿 ∈ (𝐶 Func (𝐷 FuncCat 𝐶)))
25	13	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐺 ∈ ((𝐷 FuncCat 𝐶) Func (𝐸 FuncCat 𝐶)))
26		simpr 488	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
27	1, 24, 25, 26	cofu1 17919	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝐺 ∘func 𝐿))‘𝑥) = ((1st ‘𝐺)‘((1st ‘𝐿)‘𝑥)))
28	4	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
29	7	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
30		eqid 2764	. . . . . . 7 ⊢ ((1st ‘𝐿)‘𝑥) = ((1st ‘𝐿)‘𝑥)
31	3, 28, 29, 1, 26, 30	diag1cl 18276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐿)‘𝑥) ∈ (𝐷 Func 𝐶))
32	10	fveq2d 6873	. . . . . . 7 ⊢ (𝜑 → (1st ‘(⟨𝐷, 𝐶⟩ −∘F 𝐹)) = (1st ‘𝐺))
33	32	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1st ‘(⟨𝐷, 𝐶⟩ −∘F 𝐹)) = (1st ‘𝐺))
34	31, 33	prcof1 50014	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐺)‘((1st ‘𝐿)‘𝑥)) = (((1st ‘𝐿)‘𝑥) ∘func 𝐹))
35	5	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐹 ∈ (𝐸 Func 𝐷))
36	3, 18, 35, 28, 1, 26	prcofdiag1 50019	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((1st ‘𝐿)‘𝑥) ∘func 𝐹) = ((1st ‘𝑀)‘𝑥))
37	27, 34, 36	3eqtrd 2803	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝐺 ∘func 𝐿))‘𝑥) = ((1st ‘𝑀)‘𝑥))
38	17, 23, 37	eqfnfvd 7016	. . 3 ⊢ (𝜑 → (1st ‘(𝐺 ∘func 𝐿)) = (1st ‘𝑀))
39	1, 15	funcfn2 17904	. . . 4 ⊢ (𝜑 → (2nd ‘(𝐺 ∘func 𝐿)) Fn ((Base‘𝐶) × (Base‘𝐶)))
40	1, 21	funcfn2 17904	. . . 4 ⊢ (𝜑 → (2nd ‘𝑀) Fn ((Base‘𝐶) × (Base‘𝐶)))
41		eqid 2764	. . . . . . 7 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
42		eqid 2764	. . . . . . 7 ⊢ (Hom ‘(𝐸 FuncCat 𝐶)) = (Hom ‘(𝐸 FuncCat 𝐶))
43	15	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st ‘(𝐺 ∘func 𝐿))(𝐶 Func (𝐸 FuncCat 𝐶))(2nd ‘(𝐺 ∘func 𝐿)))
44		simprl 780	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
45		simprr 782	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
46	1, 41, 42, 43, 44, 45	funcf2 17903	. . . . . 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 6694	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘(𝐺 ∘func 𝐿))𝑦) Fn (𝑥(Hom ‘𝐶)𝑦))
48	21	adantr 484	. . . . . . 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 17903	. . . . . 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 6694	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘𝑀)𝑦) Fn (𝑥(Hom ‘𝐶)𝑦))
51		eqid 2764	. . . . . . . 8 ⊢ (Base‘𝐸) = (Base‘𝐸)
52		eqid 2764	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
53	5	ad2antrr 736	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐹 ∈ (𝐸 Func 𝐷))
54	53	func1st2nd 49702	. . . . . . . 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 17901	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (1st ‘𝐹):(Base‘𝐸)⟶(Base‘𝐷))
56		xpco2 49483	. . . . . . 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 736	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐿 ∈ (𝐶 Func (𝐷 FuncCat 𝐶)))
59	13	ad2antrr 736	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐺 ∈ ((𝐷 FuncCat 𝐶) Func (𝐸 FuncCat 𝐶)))
60	44	adantr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑥 ∈ (Base‘𝐶))
61	45	adantr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑦 ∈ (Base‘𝐶))
62		simpr 488	. . . . . . . 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 17921	. . . . . . 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 736	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐶 ∈ Cat)
65	7	ad2antrr 736	. . . . . . . . 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 18279	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd ‘𝐿)𝑦)‘𝑓) = ((Base‘𝐷) × {𝑓}))
67	66	fveq2d 6873	. . . . . . 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 2764	. . . . . . . 8 ⊢ (𝐷 Nat 𝐶) = (𝐷 Nat 𝐶)
69	3, 1, 52, 41, 64, 65, 60, 61, 62, 68	diag2cl 18280	. . . . . . . 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 6873	. . . . . . . . 9 ⊢ (𝜑 → (2nd ‘(⟨𝐷, 𝐶⟩ −∘F 𝐹)) = (2nd ‘𝐺))
71	70	ad2antrr 736	. . . . . . . 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 50017	. . . . . . 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 2803	. . . . . 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 736	. . . . . . 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 18279	. . . . . 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 2809	. . . . 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 7016	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘(𝐺 ∘func 𝐿))𝑦) = (𝑥(2nd ‘𝑀)𝑦))
78	39, 40, 77	eqfnovd 49492	. . 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 17897	. . 3 ⊢ Rel (𝐶 Func (𝐸 FuncCat 𝐶))
81		1st2nd 8022	. . 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 596	. 2 ⊢ (𝜑 → (𝐺 ∘func 𝐿) = ⟨(1st ‘(𝐺 ∘func 𝐿)), (2nd ‘(𝐺 ∘func 𝐿))⟩)
83		1st2nd 8022	. . 3 ⊢ ((Rel (𝐶 Func (𝐸 FuncCat 𝐶)) ∧ 𝑀 ∈ (𝐶 Func (𝐸 FuncCat 𝐶))) → 𝑀 = ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩)
84	80, 20, 83	sylancr 596	. 2 ⊢ (𝜑 → 𝑀 = ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩)
85	79, 82, 84	3eqtr4d 2809	1 ⊢ (𝜑 → (𝐺 ∘func 𝐿) = 𝑀)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1562   ∈ wcel 2144  {csn 4584  ⟨cop 4590   class class class wbr 5102   × cxp 5647   ∘ ccom 5653  Rel wrel 5654  ⟶wf 6519  ‘cfv 6523  (class class class)co 7398  1^st c1st 7970  2^nd c2nd 7971  Basecbs 17247  Hom chom 17299  Catccat 17698   Func cfunc 17889   ∘_func ccofu 17891   Nat cnat 17979   FuncCat cfuc 17980  Δ_funccdiag 18246   −∘_F cprcof 49999

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  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 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-1o 8439  df-er 8680  df-map 8812  df-ixp 8882  df-en 8930  df-dom 8931  df-sdom 8932  df-fin 8933  df-pnf 11220  df-mnf 11221  df-xr 11222  df-ltxr 11223  df-le 11224  df-sub 11418  df-neg 11419  df-nn 12213  df-2 12282  df-3 12283  df-4 12284  df-5 12285  df-6 12286  df-7 12287  df-8 12288  df-9 12289  df-n0 12484  df-z 12571  df-dec 12691  df-uz 12842  df-fz 13515  df-struct 17185  df-slot 17220  df-ndx 17232  df-base 17248  df-hom 17312  df-cco 17313  df-cat 17702  df-cid 17703  df-func 17893  df-cofu 17895  df-nat 17981  df-fuc 17982  df-xpc 18206  df-1stf 18207  df-curf 18248  df-diag 18250  df-swapf 49886  df-fuco 49943  df-prcof 50000

This theorem is referenced by:  lmdran  50297  cmdlan  50298