Theorem prcofdiag1 50322

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 2760	. . . . . 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 50005	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐹)(𝐸 Func 𝐷)(2nd ‘𝐹))
7	6	funcrcl3 50009	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ Cat)
8		prcofdiag1.x	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9		eqid 2760	. . . . . . . . 9 ⊢ ((1st ‘𝐿)‘𝑋) = ((1st ‘𝐿)‘𝑋)
10	4, 5, 7, 2, 8, 9	diag1cl 18333	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘𝐿)‘𝑋) ∈ (𝐷 Func 𝐶))
11	3, 10	cofucl 17980	. . . . . . 7 ⊢ (𝜑 → (((1st ‘𝐿)‘𝑋) ∘func 𝐹) ∈ (𝐸 Func 𝐶))
12	11	func1st2nd 50005	. . . . . 6 ⊢ (𝜑 → (1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹))(𝐸 Func 𝐶)(2nd ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)))
13	1, 2, 12	funcf1 17958	. . . . 5 ⊢ (𝜑 → (1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)):(Base‘𝐸)⟶𝐵)
14	13	ffnd 6704	. . . 4 ⊢ (𝜑 → (1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)) Fn (Base‘𝐸))
15		prcofdiag.m	. . . . . . . 8 ⊢ 𝑀 = (𝐶Δfunc𝐸)
16	6	funcrcl2 50008	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Cat)
17		eqid 2760	. . . . . . . 8 ⊢ ((1st ‘𝑀)‘𝑋) = ((1st ‘𝑀)‘𝑋)
18	15, 5, 16, 2, 8, 17	diag1cl 18333	. . . . . . 7 ⊢ (𝜑 → ((1st ‘𝑀)‘𝑋) ∈ (𝐸 Func 𝐶))
19	18	func1st2nd 50005	. . . . . 6 ⊢ (𝜑 → (1st ‘((1st ‘𝑀)‘𝑋))(𝐸 Func 𝐶)(2nd ‘((1st ‘𝑀)‘𝑋)))
20	1, 2, 19	funcf1 17958	. . . . 5 ⊢ (𝜑 → (1st ‘((1st ‘𝑀)‘𝑋)):(Base‘𝐸)⟶𝐵)
21	20	ffnd 6704	. . . 4 ⊢ (𝜑 → (1st ‘((1st ‘𝑀)‘𝑋)) Fn (Base‘𝐸))
22	5	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → 𝐶 ∈ Cat)
23	7	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → 𝐷 ∈ Cat)
24	8	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → 𝑋 ∈ 𝐵)
25		eqid 2760	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
26	1, 25, 6	funcf1 17958	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐹):(Base‘𝐸)⟶(Base‘𝐷))
27	26	ffvelcdmda 7078	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
28	4, 22, 23, 2, 24, 9, 25, 27	diag11 18334	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → ((1st ‘((1st ‘𝐿)‘𝑋))‘((1st ‘𝐹)‘𝑥)) = 𝑋)
29	3	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → 𝐹 ∈ (𝐸 Func 𝐷))
30	10	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → ((1st ‘𝐿)‘𝑋) ∈ (𝐷 Func 𝐶))
31		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → 𝑥 ∈ (Base‘𝐸))
32	1, 29, 30, 31	cofu1 17976	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → ((1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹))‘𝑥) = ((1st ‘((1st ‘𝐿)‘𝑋))‘((1st ‘𝐹)‘𝑥)))
33	16	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → 𝐸 ∈ Cat)
34	15, 22, 33, 2, 24, 17, 1, 31	diag11 18334	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → ((1st ‘((1st ‘𝑀)‘𝑋))‘𝑥) = 𝑋)
35	28, 32, 34	3eqtr4d 2805	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → ((1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹))‘𝑥) = ((1st ‘((1st ‘𝑀)‘𝑋))‘𝑥))
36	14, 21, 35	eqfnfvd 7026	. . 3 ⊢ (𝜑 → (1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)) = (1st ‘((1st ‘𝑀)‘𝑋)))
37	1, 12	funcfn2 17961	. . . 4 ⊢ (𝜑 → (2nd ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)) Fn ((Base‘𝐸) × (Base‘𝐸)))
38	1, 19	funcfn2 17961	. . . 4 ⊢ (𝜑 → (2nd ‘((1st ‘𝑀)‘𝑋)) Fn ((Base‘𝐸) × (Base‘𝐸)))
39		eqid 2760	. . . . . . 7 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
40		eqid 2760	. . . . . . 7 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
41	12	adantr 486	. . . . . . 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 783	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) → 𝑥 ∈ (Base‘𝐸))
43		simprr 785	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) → 𝑦 ∈ (Base‘𝐸))
44	1, 39, 40, 41, 42, 43	funcf2 17960	. . . . . 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 6704	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) → (𝑥(2nd ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹))𝑦) Fn (𝑥(Hom ‘𝐸)𝑦))
46	19	adantr 486	. . . . . . 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 17960	. . . . . 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 6704	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) → (𝑥(2nd ‘((1st ‘𝑀)‘𝑋))𝑦) Fn (𝑥(Hom ‘𝐸)𝑦))
49	5	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → 𝐶 ∈ Cat)
50	7	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → 𝐷 ∈ Cat)
51	8	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → 𝑋 ∈ 𝐵)
52	6	ad2antrr 739	. . . . . . . . 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 17958	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → (1st ‘𝐹):(Base‘𝐸)⟶(Base‘𝐷))
54	42	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → 𝑥 ∈ (Base‘𝐸))
55	53, 54	ffvelcdmd 7079	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
56		eqid 2760	. . . . . . 7 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
57		eqid 2760	. . . . . . 7 ⊢ (Id‘𝐶) = (Id‘𝐶)
58	43	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → 𝑦 ∈ (Base‘𝐸))
59	53, 58	ffvelcdmd 7079	. . . . . . 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 17960	. . . . . . . 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 490	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦))
62	60, 61	ffvelcdmd 7079	. . . . . . 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 18335	. . . . . 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 739	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → 𝐹 ∈ (𝐸 Func 𝐷))
65	10	ad2antrr 739	. . . . . . 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 17978	. . . . . 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 739	. . . . . . 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 18335	. . . . . 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 2805	. . . . 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 7026	. . . 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 49797	. . 3 ⊢ (𝜑 → (2nd ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)) = (2nd ‘((1st ‘𝑀)‘𝑋)))
72	36, 71	opeq12d 4841	. 2 ⊢ (𝜑 → ⟨(1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)), (2nd ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹))⟩ = ⟨(1st ‘((1st ‘𝑀)‘𝑋)), (2nd ‘((1st ‘𝑀)‘𝑋))⟩)
73		relfunc 17954	. . 3 ⊢ Rel (𝐸 Func 𝐶)
74		1st2nd 8037	. . 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 599	. 2 ⊢ (𝜑 → (((1st ‘𝐿)‘𝑋) ∘func 𝐹) = ⟨(1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)), (2nd ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹))⟩)
76		1st2nd 8037	. . 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 599	. 2 ⊢ (𝜑 → ((1st ‘𝑀)‘𝑋) = ⟨(1st ‘((1st ‘𝑀)‘𝑋)), (2nd ‘((1st ‘𝑀)‘𝑋))⟩)
78	72, 75, 77	3eqtr4d 2805	1 ⊢ (𝜑 → (((1st ‘𝐿)‘𝑋) ∘func 𝐹) = ((1st ‘𝑀)‘𝑋))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ⟨cop 4590   class class class wbr 5103  Rel wrel 5660  ‘cfv 6533  (class class class)co 7414  1^st c1st 7985  2^nd c2nd 7986  Basecbs 17304  Hom chom 17356  Catccat 17755  Idccid 17756   Func cfunc 17946   ∘_func ccofu 17948  Δ_funccdiag 18303

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 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  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 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-1o 8458  df-er 8699  df-map 8831  df-ixp 8908  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-nn 12261  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12532  df-z 12619  df-dec 12740  df-uz 12891  df-fz 13565  df-struct 17242  df-slot 17277  df-ndx 17289  df-base 17305  df-hom 17369  df-cco 17370  df-cat 17759  df-cid 17760  df-func 17950  df-cofu 17952  df-nat 18038  df-fuc 18039  df-xpc 18263  df-1stf 18264  df-curf 18305  df-diag 18307

This theorem is used by:  prcofdiag  50323