Theorem prcofdiag1 49388

Description: A constant functor pre-composed by a functor is another constant 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)
prcofdiag1.b	⊢ 𝐵 = (Base‘𝐶)
prcofdiag1.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

Assertion

Ref	Expression
prcofdiag1	⊢ (𝜑 → (((1st ‘𝐿)‘𝑋) ∘func 𝐹) = ((1st ‘𝑀)‘𝑋))

Proof of Theorem prcofdiag1

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

Step	Hyp	Ref	Expression
1		eqid 2729	. . . . . 6 ⊢ (Base‘𝐸) = (Base‘𝐸)
2		prcofdiag1.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
3		prcofdiag.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (𝐸 Func 𝐷))
4		prcofdiag.l	. . . . . . . . 9 ⊢ 𝐿 = (𝐶Δfunc𝐷)
5		prcofdiag.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ Cat)
6	3	func1st2nd 49071	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐹)(𝐸 Func 𝐷)(2nd ‘𝐹))
7	6	funcrcl3 49075	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ Cat)
8		prcofdiag1.x	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9		eqid 2729	. . . . . . . . 9 ⊢ ((1st ‘𝐿)‘𝑋) = ((1st ‘𝐿)‘𝑋)
10	4, 5, 7, 2, 8, 9	diag1cl 18148	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘𝐿)‘𝑋) ∈ (𝐷 Func 𝐶))
11	3, 10	cofucl 17795	. . . . . . 7 ⊢ (𝜑 → (((1st ‘𝐿)‘𝑋) ∘func 𝐹) ∈ (𝐸 Func 𝐶))
12	11	func1st2nd 49071	. . . . . 6 ⊢ (𝜑 → (1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹))(𝐸 Func 𝐶)(2nd ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)))
13	1, 2, 12	funcf1 17773	. . . . 5 ⊢ (𝜑 → (1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)):(Base‘𝐸)⟶𝐵)
14	13	ffnd 6653	. . . 4 ⊢ (𝜑 → (1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)) Fn (Base‘𝐸))
15		prcofdiag.m	. . . . . . . 8 ⊢ 𝑀 = (𝐶Δfunc𝐸)
16	6	funcrcl2 49074	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Cat)
17		eqid 2729	. . . . . . . 8 ⊢ ((1st ‘𝑀)‘𝑋) = ((1st ‘𝑀)‘𝑋)
18	15, 5, 16, 2, 8, 17	diag1cl 18148	. . . . . . 7 ⊢ (𝜑 → ((1st ‘𝑀)‘𝑋) ∈ (𝐸 Func 𝐶))
19	18	func1st2nd 49071	. . . . . 6 ⊢ (𝜑 → (1st ‘((1st ‘𝑀)‘𝑋))(𝐸 Func 𝐶)(2nd ‘((1st ‘𝑀)‘𝑋)))
20	1, 2, 19	funcf1 17773	. . . . 5 ⊢ (𝜑 → (1st ‘((1st ‘𝑀)‘𝑋)):(Base‘𝐸)⟶𝐵)
21	20	ffnd 6653	. . . 4 ⊢ (𝜑 → (1st ‘((1st ‘𝑀)‘𝑋)) Fn (Base‘𝐸))
22	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → 𝐶 ∈ Cat)
23	7	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → 𝐷 ∈ Cat)
24	8	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → 𝑋 ∈ 𝐵)
25		eqid 2729	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
26	1, 25, 6	funcf1 17773	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐹):(Base‘𝐸)⟶(Base‘𝐷))
27	26	ffvelcdmda 7018	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
28	4, 22, 23, 2, 24, 9, 25, 27	diag11 18149	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → ((1st ‘((1st ‘𝐿)‘𝑋))‘((1st ‘𝐹)‘𝑥)) = 𝑋)
29	3	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → 𝐹 ∈ (𝐸 Func 𝐷))
30	10	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → ((1st ‘𝐿)‘𝑋) ∈ (𝐷 Func 𝐶))
31		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → 𝑥 ∈ (Base‘𝐸))
32	1, 29, 30, 31	cofu1 17791	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → ((1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹))‘𝑥) = ((1st ‘((1st ‘𝐿)‘𝑋))‘((1st ‘𝐹)‘𝑥)))
33	16	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → 𝐸 ∈ Cat)
34	15, 22, 33, 2, 24, 17, 1, 31	diag11 18149	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → ((1st ‘((1st ‘𝑀)‘𝑋))‘𝑥) = 𝑋)
35	28, 32, 34	3eqtr4d 2774	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → ((1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹))‘𝑥) = ((1st ‘((1st ‘𝑀)‘𝑋))‘𝑥))
36	14, 21, 35	eqfnfvd 6968	. . 3 ⊢ (𝜑 → (1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)) = (1st ‘((1st ‘𝑀)‘𝑋)))
37	1, 12	funcfn2 17776	. . . 4 ⊢ (𝜑 → (2nd ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)) Fn ((Base‘𝐸) × (Base‘𝐸)))
38	1, 19	funcfn2 17776	. . . 4 ⊢ (𝜑 → (2nd ‘((1st ‘𝑀)‘𝑋)) Fn ((Base‘𝐸) × (Base‘𝐸)))
39		eqid 2729	. . . . . . 7 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
40		eqid 2729	. . . . . . 7 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
41	12	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) → (1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹))(𝐸 Func 𝐶)(2nd ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)))
42		simprl 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) → 𝑥 ∈ (Base‘𝐸))
43		simprr 772	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) → 𝑦 ∈ (Base‘𝐸))
44	1, 39, 40, 41, 42, 43	funcf2 17775	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) → (𝑥(2nd ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹))𝑦):(𝑥(Hom ‘𝐸)𝑦)⟶(((1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹))‘𝑥)(Hom ‘𝐶)((1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹))‘𝑦)))
45	44	ffnd 6653	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) → (𝑥(2nd ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹))𝑦) Fn (𝑥(Hom ‘𝐸)𝑦))
46	19	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) → (1st ‘((1st ‘𝑀)‘𝑋))(𝐸 Func 𝐶)(2nd ‘((1st ‘𝑀)‘𝑋)))
47	1, 39, 40, 46, 42, 43	funcf2 17775	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) → (𝑥(2nd ‘((1st ‘𝑀)‘𝑋))𝑦):(𝑥(Hom ‘𝐸)𝑦)⟶(((1st ‘((1st ‘𝑀)‘𝑋))‘𝑥)(Hom ‘𝐶)((1st ‘((1st ‘𝑀)‘𝑋))‘𝑦)))
48	47	ffnd 6653	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) → (𝑥(2nd ‘((1st ‘𝑀)‘𝑋))𝑦) Fn (𝑥(Hom ‘𝐸)𝑦))
49	5	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → 𝐶 ∈ Cat)
50	7	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → 𝐷 ∈ Cat)
51	8	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → 𝑋 ∈ 𝐵)
52	6	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → (1st ‘𝐹)(𝐸 Func 𝐷)(2nd ‘𝐹))
53	1, 25, 52	funcf1 17773	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → (1st ‘𝐹):(Base‘𝐸)⟶(Base‘𝐷))
54	42	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → 𝑥 ∈ (Base‘𝐸))
55	53, 54	ffvelcdmd 7019	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
56		eqid 2729	. . . . . . 7 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
57		eqid 2729	. . . . . . 7 ⊢ (Id‘𝐶) = (Id‘𝐶)
58	43	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → 𝑦 ∈ (Base‘𝐸))
59	53, 58	ffvelcdmd 7019	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → ((1st ‘𝐹)‘𝑦) ∈ (Base‘𝐷))
60	1, 39, 56, 52, 54, 58	funcf2 17775	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → (𝑥(2nd ‘𝐹)𝑦):(𝑥(Hom ‘𝐸)𝑦)⟶(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦)))
61		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦))
62	60, 61	ffvelcdmd 7019	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → ((𝑥(2nd ‘𝐹)𝑦)‘𝑓) ∈ (((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦)))
63	4, 49, 50, 2, 51, 9, 25, 55, 56, 57, 59, 62	diag12 18150	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → ((((1st ‘𝐹)‘𝑥)(2nd ‘((1st ‘𝐿)‘𝑋))((1st ‘𝐹)‘𝑦))‘((𝑥(2nd ‘𝐹)𝑦)‘𝑓)) = ((Id‘𝐶)‘𝑋))
64	3	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → 𝐹 ∈ (𝐸 Func 𝐷))
65	10	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → ((1st ‘𝐿)‘𝑋) ∈ (𝐷 Func 𝐶))
66	1, 64, 65, 54, 58, 39, 61	cofu2 17793	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → ((𝑥(2nd ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹))𝑦)‘𝑓) = ((((1st ‘𝐹)‘𝑥)(2nd ‘((1st ‘𝐿)‘𝑋))((1st ‘𝐹)‘𝑦))‘((𝑥(2nd ‘𝐹)𝑦)‘𝑓)))
67	16	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → 𝐸 ∈ Cat)
68	15, 49, 67, 2, 51, 17, 1, 54, 39, 57, 58, 61	diag12 18150	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → ((𝑥(2nd ‘((1st ‘𝑀)‘𝑋))𝑦)‘𝑓) = ((Id‘𝐶)‘𝑋))
69	63, 66, 68	3eqtr4d 2774	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → ((𝑥(2nd ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹))𝑦)‘𝑓) = ((𝑥(2nd ‘((1st ‘𝑀)‘𝑋))𝑦)‘𝑓))
70	45, 48, 69	eqfnfvd 6968	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) → (𝑥(2nd ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹))𝑦) = (𝑥(2nd ‘((1st ‘𝑀)‘𝑋))𝑦))
71	37, 38, 70	eqfnovd 48860	. . 3 ⊢ (𝜑 → (2nd ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)) = (2nd ‘((1st ‘𝑀)‘𝑋)))
72	36, 71	opeq12d 4832	. 2 ⊢ (𝜑 → ⟨(1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)), (2nd ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹))⟩ = ⟨(1st ‘((1st ‘𝑀)‘𝑋)), (2nd ‘((1st ‘𝑀)‘𝑋))⟩)
73		relfunc 17769	. . 3 ⊢ Rel (𝐸 Func 𝐶)
74		1st2nd 7974	. . 3 ⊢ ((Rel (𝐸 Func 𝐶) ∧ (((1st ‘𝐿)‘𝑋) ∘func 𝐹) ∈ (𝐸 Func 𝐶)) → (((1st ‘𝐿)‘𝑋) ∘func 𝐹) = ⟨(1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)), (2nd ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹))⟩)
75	73, 11, 74	sylancr 587	. 2 ⊢ (𝜑 → (((1st ‘𝐿)‘𝑋) ∘func 𝐹) = ⟨(1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)), (2nd ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹))⟩)
76		1st2nd 7974	. . 3 ⊢ ((Rel (𝐸 Func 𝐶) ∧ ((1st ‘𝑀)‘𝑋) ∈ (𝐸 Func 𝐶)) → ((1st ‘𝑀)‘𝑋) = ⟨(1st ‘((1st ‘𝑀)‘𝑋)), (2nd ‘((1st ‘𝑀)‘𝑋))⟩)
77	73, 18, 76	sylancr 587	. 2 ⊢ (𝜑 → ((1st ‘𝑀)‘𝑋) = ⟨(1st ‘((1st ‘𝑀)‘𝑋)), (2nd ‘((1st ‘𝑀)‘𝑋))⟩)
78	72, 75, 77	3eqtr4d 2774	1 ⊢ (𝜑 → (((1st ‘𝐿)‘𝑋) ∘func 𝐹) = ((1st ‘𝑀)‘𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ⟨cop 4583   class class class wbr 5092  Rel wrel 5624  ‘cfv 6482  (class class class)co 7349  1^st c1st 7922  2^nd c2nd 7923  Basecbs 17120  Hom chom 17172  Catccat 17570  Idccid 17571   Func cfunc 17761   ∘_func ccofu 17763  Δ_funccdiag 18118

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 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  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 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-er 8625  df-map 8755  df-ixp 8825  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-5 12194  df-6 12195  df-7 12196  df-8 12197  df-9 12198  df-n0 12385  df-z 12472  df-dec 12592  df-uz 12736  df-fz 13411  df-struct 17058  df-slot 17093  df-ndx 17105  df-base 17121  df-hom 17185  df-cco 17186  df-cat 17574  df-cid 17575  df-func 17765  df-cofu 17767  df-nat 17853  df-fuc 17854  df-xpc 18078  df-1stf 18079  df-curf 18120  df-diag 18122

This theorem is referenced by:  prcofdiag  49389