Theorem prcofdiag1 49392

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 49075	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐹)(𝐸 Func 𝐷)(2nd ‘𝐹))
7	6	funcrcl3 49079	. . . . . . . . 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 18135	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘𝐿)‘𝑋) ∈ (𝐷 Func 𝐶))
11	3, 10	cofucl 17782	. . . . . . 7 ⊢ (𝜑 → (((1st ‘𝐿)‘𝑋) ∘func 𝐹) ∈ (𝐸 Func 𝐶))
12	11	func1st2nd 49075	. . . . . 6 ⊢ (𝜑 → (1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹))(𝐸 Func 𝐶)(2nd ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)))
13	1, 2, 12	funcf1 17760	. . . . 5 ⊢ (𝜑 → (1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)):(Base‘𝐸)⟶𝐵)
14	13	ffnd 6647	. . . 4 ⊢ (𝜑 → (1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)) Fn (Base‘𝐸))
15		prcofdiag.m	. . . . . . . 8 ⊢ 𝑀 = (𝐶Δfunc𝐸)
16	6	funcrcl2 49078	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Cat)
17		eqid 2729	. . . . . . . 8 ⊢ ((1st ‘𝑀)‘𝑋) = ((1st ‘𝑀)‘𝑋)
18	15, 5, 16, 2, 8, 17	diag1cl 18135	. . . . . . 7 ⊢ (𝜑 → ((1st ‘𝑀)‘𝑋) ∈ (𝐸 Func 𝐶))
19	18	func1st2nd 49075	. . . . . 6 ⊢ (𝜑 → (1st ‘((1st ‘𝑀)‘𝑋))(𝐸 Func 𝐶)(2nd ‘((1st ‘𝑀)‘𝑋)))
20	1, 2, 19	funcf1 17760	. . . . 5 ⊢ (𝜑 → (1st ‘((1st ‘𝑀)‘𝑋)):(Base‘𝐸)⟶𝐵)
21	20	ffnd 6647	. . . 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 17760	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐹):(Base‘𝐸)⟶(Base‘𝐷))
27	26	ffvelcdmda 7011	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
28	4, 22, 23, 2, 24, 9, 25, 27	diag11 18136	. . . . 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 17778	. . . . 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 18136	. . . . 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 6961	. . 3 ⊢ (𝜑 → (1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)) = (1st ‘((1st ‘𝑀)‘𝑋)))
37	1, 12	funcfn2 17763	. . . 4 ⊢ (𝜑 → (2nd ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)) Fn ((Base‘𝐸) × (Base‘𝐸)))
38	1, 19	funcfn2 17763	. . . 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 17762	. . . . . 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 6647	. . . . 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 17762	. . . . . 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 6647	. . . . 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 17760	. . . . . . . 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 7012	. . . . . . 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 7012	. . . . . . 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 17762	. . . . . . . 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 7012	. . . . . . 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 18137	. . . . . 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 17780	. . . . . 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 18137	. . . . . 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 6961	. . . 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 48864	. . 3 ⊢ (𝜑 → (2nd ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)) = (2nd ‘((1st ‘𝑀)‘𝑋)))
72	36, 71	opeq12d 4830	. 2 ⊢ (𝜑 → ⟨(1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)), (2nd ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹))⟩ = ⟨(1st ‘((1st ‘𝑀)‘𝑋)), (2nd ‘((1st ‘𝑀)‘𝑋))⟩)
73		relfunc 17756	. . 3 ⊢ Rel (𝐸 Func 𝐶)
74		1st2nd 7965	. . 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 7965	. . 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 4579   class class class wbr 5088  Rel wrel 5618  ‘cfv 6476  (class class class)co 7340  1^st c1st 7913  2^nd c2nd 7914  Basecbs 17107  Hom chom 17159  Catccat 17557  Idccid 17558   Func cfunc 17748   ∘_func ccofu 17750  Δ_funccdiag 18105

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 5214  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5367  ax-un 7662  ax-cnex 11053  ax-resscn 11054  ax-1cn 11055  ax-icn 11056  ax-addcl 11057  ax-addrcl 11058  ax-mulcl 11059  ax-mulrcl 11060  ax-mulcom 11061  ax-addass 11062  ax-mulass 11063  ax-distr 11064  ax-i2m1 11065  ax-1ne0 11066  ax-1rid 11067  ax-rnegex 11068  ax-rrecex 11069  ax-cnre 11070  ax-pre-lttri 11071  ax-pre-lttrn 11072  ax-pre-ltadd 11073  ax-pre-mulgt0 11074

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 3393  df-v 3435  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4857  df-iun 4940  df-br 5089  df-opab 5151  df-mpt 5170  df-tr 5196  df-id 5508  df-eprel 5513  df-po 5521  df-so 5522  df-fr 5566  df-we 5568  df-xp 5619  df-rel 5620  df-cnv 5621  df-co 5622  df-dm 5623  df-rn 5624  df-res 5625  df-ima 5626  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7297  df-ov 7343  df-oprab 7344  df-mpo 7345  df-om 7791  df-1st 7915  df-2nd 7916  df-frecs 8205  df-wrecs 8236  df-recs 8285  df-rdg 8323  df-1o 8379  df-er 8616  df-map 8746  df-ixp 8816  df-en 8864  df-dom 8865  df-sdom 8866  df-fin 8867  df-pnf 11139  df-mnf 11140  df-xr 11141  df-ltxr 11142  df-le 11143  df-sub 11337  df-neg 11338  df-nn 12117  df-2 12179  df-3 12180  df-4 12181  df-5 12182  df-6 12183  df-7 12184  df-8 12185  df-9 12186  df-n0 12373  df-z 12460  df-dec 12580  df-uz 12724  df-fz 13399  df-struct 17045  df-slot 17080  df-ndx 17092  df-base 17108  df-hom 17172  df-cco 17173  df-cat 17561  df-cid 17562  df-func 17752  df-cofu 17754  df-nat 17840  df-fuc 17841  df-xpc 18065  df-1stf 18066  df-curf 18107  df-diag 18109

This theorem is referenced by:  prcofdiag  49393