Theorem prcofdiag1 50199

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 2763	. . . . . 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 49882	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐹)(𝐸 Func 𝐷)(2nd ‘𝐹))
7	6	funcrcl3 49886	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ Cat)
8		prcofdiag1.x	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9		eqid 2763	. . . . . . . . 9 ⊢ ((1st ‘𝐿)‘𝑋) = ((1st ‘𝐿)‘𝑋)
10	4, 5, 7, 2, 8, 9	diag1cl 18302	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘𝐿)‘𝑋) ∈ (𝐷 Func 𝐶))
11	3, 10	cofucl 17949	. . . . . . 7 ⊢ (𝜑 → (((1st ‘𝐿)‘𝑋) ∘func 𝐹) ∈ (𝐸 Func 𝐶))
12	11	func1st2nd 49882	. . . . . 6 ⊢ (𝜑 → (1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹))(𝐸 Func 𝐶)(2nd ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)))
13	1, 2, 12	funcf1 17927	. . . . 5 ⊢ (𝜑 → (1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)):(Base‘𝐸)⟶𝐵)
14	13	ffnd 6706	. . . 4 ⊢ (𝜑 → (1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)) Fn (Base‘𝐸))
15		prcofdiag.m	. . . . . . . 8 ⊢ 𝑀 = (𝐶Δfunc𝐸)
16	6	funcrcl2 49885	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Cat)
17		eqid 2763	. . . . . . . 8 ⊢ ((1st ‘𝑀)‘𝑋) = ((1st ‘𝑀)‘𝑋)
18	15, 5, 16, 2, 8, 17	diag1cl 18302	. . . . . . 7 ⊢ (𝜑 → ((1st ‘𝑀)‘𝑋) ∈ (𝐸 Func 𝐶))
19	18	func1st2nd 49882	. . . . . 6 ⊢ (𝜑 → (1st ‘((1st ‘𝑀)‘𝑋))(𝐸 Func 𝐶)(2nd ‘((1st ‘𝑀)‘𝑋)))
20	1, 2, 19	funcf1 17927	. . . . 5 ⊢ (𝜑 → (1st ‘((1st ‘𝑀)‘𝑋)):(Base‘𝐸)⟶𝐵)
21	20	ffnd 6706	. . . 4 ⊢ (𝜑 → (1st ‘((1st ‘𝑀)‘𝑋)) Fn (Base‘𝐸))
22	5	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → 𝐶 ∈ Cat)
23	7	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → 𝐷 ∈ Cat)
24	8	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → 𝑋 ∈ 𝐵)
25		eqid 2763	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
26	1, 25, 6	funcf1 17927	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐹):(Base‘𝐸)⟶(Base‘𝐷))
27	26	ffvelcdmda 7079	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
28	4, 22, 23, 2, 24, 9, 25, 27	diag11 18303	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → ((1st ‘((1st ‘𝐿)‘𝑋))‘((1st ‘𝐹)‘𝑥)) = 𝑋)
29	3	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → 𝐹 ∈ (𝐸 Func 𝐷))
30	10	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → ((1st ‘𝐿)‘𝑋) ∈ (𝐷 Func 𝐶))
31		simpr 489	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → 𝑥 ∈ (Base‘𝐸))
32	1, 29, 30, 31	cofu1 17945	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → ((1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹))‘𝑥) = ((1st ‘((1st ‘𝐿)‘𝑋))‘((1st ‘𝐹)‘𝑥)))
33	16	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → 𝐸 ∈ Cat)
34	15, 22, 33, 2, 24, 17, 1, 31	diag11 18303	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → ((1st ‘((1st ‘𝑀)‘𝑋))‘𝑥) = 𝑋)
35	28, 32, 34	3eqtr4d 2808	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → ((1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹))‘𝑥) = ((1st ‘((1st ‘𝑀)‘𝑋))‘𝑥))
36	14, 21, 35	eqfnfvd 7028	. . 3 ⊢ (𝜑 → (1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)) = (1st ‘((1st ‘𝑀)‘𝑋)))
37	1, 12	funcfn2 17930	. . . 4 ⊢ (𝜑 → (2nd ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)) Fn ((Base‘𝐸) × (Base‘𝐸)))
38	1, 19	funcfn2 17930	. . . 4 ⊢ (𝜑 → (2nd ‘((1st ‘𝑀)‘𝑋)) Fn ((Base‘𝐸) × (Base‘𝐸)))
39		eqid 2763	. . . . . . 7 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
40		eqid 2763	. . . . . . 7 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
41	12	adantr 485	. . . . . . 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 782	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) → 𝑥 ∈ (Base‘𝐸))
43		simprr 784	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) → 𝑦 ∈ (Base‘𝐸))
44	1, 39, 40, 41, 42, 43	funcf2 17929	. . . . . 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 6706	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) → (𝑥(2nd ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹))𝑦) Fn (𝑥(Hom ‘𝐸)𝑦))
46	19	adantr 485	. . . . . . 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 17929	. . . . . 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 6706	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) → (𝑥(2nd ‘((1st ‘𝑀)‘𝑋))𝑦) Fn (𝑥(Hom ‘𝐸)𝑦))
49	5	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → 𝐶 ∈ Cat)
50	7	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → 𝐷 ∈ Cat)
51	8	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → 𝑋 ∈ 𝐵)
52	6	ad2antrr 738	. . . . . . . . 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 17927	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → (1st ‘𝐹):(Base‘𝐸)⟶(Base‘𝐷))
54	42	adantr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → 𝑥 ∈ (Base‘𝐸))
55	53, 54	ffvelcdmd 7080	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
56		eqid 2763	. . . . . . 7 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
57		eqid 2763	. . . . . . 7 ⊢ (Id‘𝐶) = (Id‘𝐶)
58	43	adantr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → 𝑦 ∈ (Base‘𝐸))
59	53, 58	ffvelcdmd 7080	. . . . . . 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 17929	. . . . . . . 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 489	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦))
62	60, 61	ffvelcdmd 7080	. . . . . . 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 18304	. . . . . 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 738	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐸) ∧ 𝑦 ∈ (Base‘𝐸))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐸)𝑦)) → 𝐹 ∈ (𝐸 Func 𝐷))
65	10	ad2antrr 738	. . . . . . 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 17947	. . . . . 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 738	. . . . . . 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 18304	. . . . . 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 2808	. . . . 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 7028	. . . 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 49672	. . 3 ⊢ (𝜑 → (2nd ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)) = (2nd ‘((1st ‘𝑀)‘𝑋)))
72	36, 71	opeq12d 4846	. 2 ⊢ (𝜑 → ⟨(1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)), (2nd ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹))⟩ = ⟨(1st ‘((1st ‘𝑀)‘𝑋)), (2nd ‘((1st ‘𝑀)‘𝑋))⟩)
73		relfunc 17923	. . 3 ⊢ Rel (𝐸 Func 𝐶)
74		1st2nd 8032	. . 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 598	. 2 ⊢ (𝜑 → (((1st ‘𝐿)‘𝑋) ∘func 𝐹) = ⟨(1st ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹)), (2nd ‘(((1st ‘𝐿)‘𝑋) ∘func 𝐹))⟩)
76		1st2nd 8032	. . 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 598	. 2 ⊢ (𝜑 → ((1st ‘𝑀)‘𝑋) = ⟨(1st ‘((1st ‘𝑀)‘𝑋)), (2nd ‘((1st ‘𝑀)‘𝑋))⟩)
78	72, 75, 77	3eqtr4d 2808	1 ⊢ (𝜑 → (((1st ‘𝐿)‘𝑋) ∘func 𝐹) = ((1st ‘𝑀)‘𝑋))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ⟨cop 4595   class class class wbr 5109  Rel wrel 5666  ‘cfv 6536  (class class class)co 7410  1^st c1st 7980  2^nd c2nd 7981  Basecbs 17273  Hom chom 17325  Catccat 17724  Idccid 17725   Func cfunc 17915   ∘_func ccofu 17917  Δ_funccdiag 18272

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11160  ax-resscn 11161  ax-1cn 11162  ax-icn 11163  ax-addcl 11164  ax-addrcl 11165  ax-mulcl 11166  ax-mulrcl 11167  ax-mulcom 11168  ax-addass 11169  ax-mulass 11170  ax-distr 11171  ax-i2m1 11172  ax-1ne0 11173  ax-1rid 11174  ax-rnegex 11175  ax-rrecex 11176  ax-cnre 11177  ax-pre-lttri 11178  ax-pre-lttrn 11179  ax-pre-ltadd 11180  ax-pre-mulgt0 11181

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-map 8822  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11447  df-neg 11448  df-nn 12238  df-2 12307  df-3 12308  df-4 12309  df-5 12310  df-6 12311  df-7 12312  df-8 12313  df-9 12314  df-n0 12509  df-z 12596  df-dec 12716  df-uz 12867  df-fz 13540  df-struct 17211  df-slot 17246  df-ndx 17258  df-base 17274  df-hom 17338  df-cco 17339  df-cat 17728  df-cid 17729  df-func 17919  df-cofu 17921  df-nat 18007  df-fuc 18008  df-xpc 18232  df-1stf 18233  df-curf 18274  df-diag 18276

This theorem is used by:  prcofdiag  50200