Theorem prcofdiag1 49884

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 2737	. . . . . 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 49567	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐹)(𝐸 Func 𝐷)(2nd ‘𝐹))
7	6	funcrcl3 49571	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ Cat)
8		prcofdiag1.x	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9		eqid 2737	. . . . . . . . 9 ⊢ ((1st ‘𝐿)‘𝑋) = ((1st ‘𝐿)‘𝑋)
10	4, 5, 7, 2, 8, 9	diag1cl 18203	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘𝐿)‘𝑋) ∈ (𝐷 Func 𝐶))
11	3, 10	cofucl 17850	. . . . . . 7 ⊢ (𝜑 → (((1st ‘𝐿)‘𝑋) ∘func 𝐹) ∈ (𝐸 Func 𝐶))
12	11	func1st2nd 49567	. . . . . 6 ⊢ (𝜑 → (1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹))(𝐸 Func 𝐶)(2nd ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)))
13	1, 2, 12	funcf1 17828	. . . . 5 ⊢ (𝜑 → (1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)):(Base‘𝐸)⟶𝐵)
14	13	ffnd 6665	. . . 4 ⊢ (𝜑 → (1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)) Fn (Base‘𝐸))
15		prcofdiag.m	. . . . . . . 8 ⊢ 𝑀 = (𝐶Δfunc𝐸)
16	6	funcrcl2 49570	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Cat)
17		eqid 2737	. . . . . . . 8 ⊢ ((1st ‘𝑀)‘𝑋) = ((1st ‘𝑀)‘𝑋)
18	15, 5, 16, 2, 8, 17	diag1cl 18203	. . . . . . 7 ⊢ (𝜑 → ((1st ‘𝑀)‘𝑋) ∈ (𝐸 Func 𝐶))
19	18	func1st2nd 49567	. . . . . 6 ⊢ (𝜑 → (1st ‘((1st ‘𝑀)‘𝑋))(𝐸 Func 𝐶)(2nd ‘((1st ‘𝑀)‘𝑋)))
20	1, 2, 19	funcf1 17828	. . . . 5 ⊢ (𝜑 → (1st ‘((1st ‘𝑀)‘𝑋)):(Base‘𝐸)⟶𝐵)
21	20	ffnd 6665	. . . 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 2737	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
26	1, 25, 6	funcf1 17828	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐹):(Base‘𝐸)⟶(Base‘𝐷))
27	26	ffvelcdmda 7032	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
28	4, 22, 23, 2, 24, 9, 25, 27	diag11 18204	. . . . 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 17846	. . . . 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 18204	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → ((1st ‘((1st ‘𝑀)‘𝑋))‘𝑥) = 𝑋)
35	28, 32, 34	3eqtr4d 2782	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → ((1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹))‘𝑥) = ((1st ‘((1st ‘𝑀)‘𝑋))‘𝑥))
36	14, 21, 35	eqfnfvd 6982	. . 3 ⊢ (𝜑 → (1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)) = (1st ‘((1st ‘𝑀)‘𝑋)))
37	1, 12	funcfn2 17831	. . . 4 ⊢ (𝜑 → (2nd ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)) Fn ((Base‘𝐸) × (Base‘𝐸)))
38	1, 19	funcfn2 17831	. . . 4 ⊢ (𝜑 → (2nd ‘((1st ‘𝑀)‘𝑋)) Fn ((Base‘𝐸) × (Base‘𝐸)))
39		eqid 2737	. . . . . . 7 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
40		eqid 2737	. . . . . . 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 771	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) → 𝑥 ∈ (Base‘𝐸))
43		simprr 773	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) → 𝑦 ∈ (Base‘𝐸))
44	1, 39, 40, 41, 42, 43	funcf2 17830	. . . . . 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 6665	. . . . 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 17830	. . . . . 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 6665	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) → (𝑥(2nd ‘((1st ‘𝑀)‘𝑋))𝑦) Fn (𝑥(Hom ‘𝐸)𝑦))
49	5	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → 𝐶 ∈ Cat)
50	7	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → 𝐷 ∈ Cat)
51	8	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → 𝑋 ∈ 𝐵)
52	6	ad2antrr 727	. . . . . . . . 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 17828	. . . . . . . 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 7033	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
56		eqid 2737	. . . . . . 7 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
57		eqid 2737	. . . . . . 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 7033	. . . . . . 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 17830	. . . . . . . 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 7033	. . . . . . 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 18205	. . . . . 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 727	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → 𝐹 ∈ (𝐸 Func 𝐷))
65	10	ad2antrr 727	. . . . . . 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 17848	. . . . . 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 727	. . . . . . 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 18205	. . . . . 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 2782	. . . . 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 6982	. . . 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 49357	. . 3 ⊢ (𝜑 → (2nd ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)) = (2nd ‘((1st ‘𝑀)‘𝑋)))
72	36, 71	opeq12d 4825	. 2 ⊢ (𝜑 → ⟨(1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)), (2nd ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹))⟩ = ⟨(1st ‘((1st ‘𝑀)‘𝑋)), (2nd ‘((1st ‘𝑀)‘𝑋))⟩)
73		relfunc 17824	. . 3 ⊢ Rel (𝐸 Func 𝐶)
74		1st2nd 7987	. . 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 588	. 2 ⊢ (𝜑 → (((1st ‘𝐿)‘𝑋) ∘func 𝐹) = ⟨(1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)), (2nd ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹))⟩)
76		1st2nd 7987	. . 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 588	. 2 ⊢ (𝜑 → ((1st ‘𝑀)‘𝑋) = ⟨(1st ‘((1st ‘𝑀)‘𝑋)), (2nd ‘((1st ‘𝑀)‘𝑋))⟩)
78	72, 75, 77	3eqtr4d 2782	1 ⊢ (𝜑 → (((1st ‘𝐿)‘𝑋) ∘func 𝐹) = ((1st ‘𝑀)‘𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ⟨cop 4574   class class class wbr 5086  Rel wrel 5631  ‘cfv 6494  (class class class)co 7362  1^st c1st 7935  2^nd c2nd 7936  Basecbs 17174  Hom chom 17226  Catccat 17625  Idccid 17626   Func cfunc 17816   ∘_func ccofu 17818  Δ_funccdiag 18173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7813  df-1st 7937  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-er 8638  df-map 8770  df-ixp 8841  df-en 8889  df-dom 8890  df-sdom 8891  df-fin 8892  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-nn 12170  df-2 12239  df-3 12240  df-4 12241  df-5 12242  df-6 12243  df-7 12244  df-8 12245  df-9 12246  df-n0 12433  df-z 12520  df-dec 12640  df-uz 12784  df-fz 13457  df-struct 17112  df-slot 17147  df-ndx 17159  df-base 17175  df-hom 17239  df-cco 17240  df-cat 17629  df-cid 17630  df-func 17820  df-cofu 17822  df-nat 17908  df-fuc 17909  df-xpc 18133  df-1stf 18134  df-curf 18175  df-diag 18177

This theorem is referenced by:  prcofdiag  49885