Theorem oppfdiag1 49996

Description: A constant functor for opposite categories is the opposite functor of the constant functor for original categories. (Contributed by Zhi Wang, 19-Nov-2025.)

Hypotheses

Ref	Expression
oppfdiag.o	⊢ 𝑂 = (oppCat‘𝐶)
oppfdiag.p	⊢ 𝑃 = (oppCat‘𝐷)
oppfdiag.l	⊢ 𝐿 = (𝐶Δfunc𝐷)
oppfdiag.c	⊢ (𝜑 → 𝐶 ∈ Cat)
oppfdiag.d	⊢ (𝜑 → 𝐷 ∈ Cat)
oppfdiag1.f	⊢ (𝜑 → 𝐹 = ( oppFunc ↾ (𝐷 Func 𝐶)))
oppfdiag1.a	⊢ 𝐴 = (Base‘𝐶)
oppfdiag1.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)

Assertion

Ref	Expression
oppfdiag1	⊢ (𝜑 → (𝐹‘((1st ‘𝐿)‘𝑋)) = ((1st ‘(𝑂Δfunc𝑃))‘𝑋))

Proof of Theorem oppfdiag1

Dummy variables 𝑓 𝑦 𝑧 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oppfdiag1.f	. . . . 5 ⊢ (𝜑 → 𝐹 = ( oppFunc ↾ (𝐷 Func 𝐶)))
2		oppfdiag1.a	. . . . . . 7 ⊢ 𝐴 = (Base‘𝐶)
3		eqid 2761	. . . . . . . 8 ⊢ (𝐷 FuncCat 𝐶) = (𝐷 FuncCat 𝐶)
4	3	fucbas 17987	. . . . . . 7 ⊢ (𝐷 Func 𝐶) = (Base‘(𝐷 FuncCat 𝐶))
5		oppfdiag.l	. . . . . . . . 9 ⊢ 𝐿 = (𝐶Δfunc𝐷)
6		oppfdiag.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		oppfdiag.d	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ Cat)
8	5, 6, 7, 3	diagcl 18264	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ (𝐶 Func (𝐷 FuncCat 𝐶)))
9	8	func1st2nd 49658	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐿)(𝐶 Func (𝐷 FuncCat 𝐶))(2nd ‘𝐿))
10	2, 4, 9	funcf1 17890	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐿):𝐴⟶(𝐷 Func 𝐶))
11		oppfdiag1.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
12	10, 11	ffvelcdmd 7061	. . . . 5 ⊢ (𝜑 → ((1st ‘𝐿)‘𝑋) ∈ (𝐷 Func 𝐶))
13	1, 12	opf11 49985	. . . 4 ⊢ (𝜑 → (1st ‘(𝐹‘((1st ‘𝐿)‘𝑋))) = (1st ‘((1st ‘𝐿)‘𝑋)))
14		oppfdiag.p	. . . . . . . . 9 ⊢ 𝑃 = (oppCat‘𝐷)
15		eqid 2761	. . . . . . . . 9 ⊢ (Base‘𝐷) = (Base‘𝐷)
16	14, 15	oppcbas 17741	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝑃)
17		oppfdiag.o	. . . . . . . . 9 ⊢ 𝑂 = (oppCat‘𝐶)
18	17, 2	oppcbas 17741	. . . . . . . 8 ⊢ 𝐴 = (Base‘𝑂)
19		eqid 2761	. . . . . . . . . . . . 13 ⊢ (oppCat‘(𝐷 FuncCat 𝐶)) = (oppCat‘(𝐷 FuncCat 𝐶))
20	17, 19, 8	oppfoppc2 49724	. . . . . . . . . . . 12 ⊢ (𝜑 → ( oppFunc ‘𝐿) ∈ (𝑂 Func (oppCat‘(𝐷 FuncCat 𝐶))))
21		eqid 2761	. . . . . . . . . . . . . 14 ⊢ (𝑃 FuncCat 𝑂) = (𝑃 FuncCat 𝑂)
22		eqid 2761	. . . . . . . . . . . . . 14 ⊢ (𝐷 Nat 𝐶) = (𝐷 Nat 𝐶)
23		eqidd 2762	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑚 ∈ (𝐷 Func 𝐶), 𝑛 ∈ (𝐷 Func 𝐶) ↦ ( I ↾ (𝑛(𝐷 Nat 𝐶)𝑚))) = (𝑚 ∈ (𝐷 Func 𝐶), 𝑛 ∈ (𝐷 Func 𝐶) ↦ ( I ↾ (𝑛(𝐷 Nat 𝐶)𝑚))))
24	14, 17, 3, 19, 21, 22, 1, 23, 7, 6	fucoppcfunc 49994	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹((oppCat‘(𝐷 FuncCat 𝐶)) Func (𝑃 FuncCat 𝑂))(𝑚 ∈ (𝐷 Func 𝐶), 𝑛 ∈ (𝐷 Func 𝐶) ↦ ( I ↾ (𝑛(𝐷 Nat 𝐶)𝑚))))
25		df-br 5098	. . . . . . . . . . . . 13 ⊢ (𝐹((oppCat‘(𝐷 FuncCat 𝐶)) Func (𝑃 FuncCat 𝑂))(𝑚 ∈ (𝐷 Func 𝐶), 𝑛 ∈ (𝐷 Func 𝐶) ↦ ( I ↾ (𝑛(𝐷 Nat 𝐶)𝑚))) ↔ ⟨𝐹, (𝑚 ∈ (𝐷 Func 𝐶), 𝑛 ∈ (𝐷 Func 𝐶) ↦ ( I ↾ (𝑛(𝐷 Nat 𝐶)𝑚)))⟩ ∈ ((oppCat‘(𝐷 FuncCat 𝐶)) Func (𝑃 FuncCat 𝑂)))
26	24, 25	sylib 220	. . . . . . . . . . . 12 ⊢ (𝜑 → ⟨𝐹, (𝑚 ∈ (𝐷 Func 𝐶), 𝑛 ∈ (𝐷 Func 𝐶) ↦ ( I ↾ (𝑛(𝐷 Nat 𝐶)𝑚)))⟩ ∈ ((oppCat‘(𝐷 FuncCat 𝐶)) Func (𝑃 FuncCat 𝑂)))
27	18, 20, 26, 11	cofu1 17908	. . . . . . . . . . 11 ⊢ (𝜑 → ((1st ‘(⟨𝐹, (𝑚 ∈ (𝐷 Func 𝐶), 𝑛 ∈ (𝐷 Func 𝐶) ↦ ( I ↾ (𝑛(𝐷 Nat 𝐶)𝑚)))⟩ ∘func ( oppFunc ‘𝐿)))‘𝑋) = ((1st ‘⟨𝐹, (𝑚 ∈ (𝐷 Func 𝐶), 𝑛 ∈ (𝐷 Func 𝐶) ↦ ( I ↾ (𝑛(𝐷 Nat 𝐶)𝑚)))⟩)‘((1st ‘( oppFunc ‘𝐿))‘𝑋)))
28	24	func1st 49659	. . . . . . . . . . . 12 ⊢ (𝜑 → (1st ‘⟨𝐹, (𝑚 ∈ (𝐷 Func 𝐶), 𝑛 ∈ (𝐷 Func 𝐶) ↦ ( I ↾ (𝑛(𝐷 Nat 𝐶)𝑚)))⟩) = 𝐹)
29	8	oppf1 49721	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1st ‘( oppFunc ‘𝐿)) = (1st ‘𝐿))
30	29	fveq1d 6864	. . . . . . . . . . . 12 ⊢ (𝜑 → ((1st ‘( oppFunc ‘𝐿))‘𝑋) = ((1st ‘𝐿)‘𝑋))
31	28, 30	fveq12d 6869	. . . . . . . . . . 11 ⊢ (𝜑 → ((1st ‘⟨𝐹, (𝑚 ∈ (𝐷 Func 𝐶), 𝑛 ∈ (𝐷 Func 𝐶) ↦ ( I ↾ (𝑛(𝐷 Nat 𝐶)𝑚)))⟩)‘((1st ‘( oppFunc ‘𝐿))‘𝑋)) = (𝐹‘((1st ‘𝐿)‘𝑋)))
32	27, 31	eqtrd 2796	. . . . . . . . . 10 ⊢ (𝜑 → ((1st ‘(⟨𝐹, (𝑚 ∈ (𝐷 Func 𝐶), 𝑛 ∈ (𝐷 Func 𝐶) ↦ ( I ↾ (𝑛(𝐷 Nat 𝐶)𝑚)))⟩ ∘func ( oppFunc ‘𝐿)))‘𝑋) = (𝐹‘((1st ‘𝐿)‘𝑋)))
33	21	fucbas 17987	. . . . . . . . . . . 12 ⊢ (𝑃 Func 𝑂) = (Base‘(𝑃 FuncCat 𝑂))
34	20, 26	cofucl 17912	. . . . . . . . . . . . 13 ⊢ (𝜑 → (⟨𝐹, (𝑚 ∈ (𝐷 Func 𝐶), 𝑛 ∈ (𝐷 Func 𝐶) ↦ ( I ↾ (𝑛(𝐷 Nat 𝐶)𝑚)))⟩ ∘func ( oppFunc ‘𝐿)) ∈ (𝑂 Func (𝑃 FuncCat 𝑂)))
35	34	func1st2nd 49658	. . . . . . . . . . . 12 ⊢ (𝜑 → (1st ‘(⟨𝐹, (𝑚 ∈ (𝐷 Func 𝐶), 𝑛 ∈ (𝐷 Func 𝐶) ↦ ( I ↾ (𝑛(𝐷 Nat 𝐶)𝑚)))⟩ ∘func ( oppFunc ‘𝐿)))(𝑂 Func (𝑃 FuncCat 𝑂))(2nd ‘(⟨𝐹, (𝑚 ∈ (𝐷 Func 𝐶), 𝑛 ∈ (𝐷 Func 𝐶) ↦ ( I ↾ (𝑛(𝐷 Nat 𝐶)𝑚)))⟩ ∘func ( oppFunc ‘𝐿))))
36	18, 33, 35	funcf1 17890	. . . . . . . . . . 11 ⊢ (𝜑 → (1st ‘(⟨𝐹, (𝑚 ∈ (𝐷 Func 𝐶), 𝑛 ∈ (𝐷 Func 𝐶) ↦ ( I ↾ (𝑛(𝐷 Nat 𝐶)𝑚)))⟩ ∘func ( oppFunc ‘𝐿))):𝐴⟶(𝑃 Func 𝑂))
37	36, 11	ffvelcdmd 7061	. . . . . . . . . 10 ⊢ (𝜑 → ((1st ‘(⟨𝐹, (𝑚 ∈ (𝐷 Func 𝐶), 𝑛 ∈ (𝐷 Func 𝐶) ↦ ( I ↾ (𝑛(𝐷 Nat 𝐶)𝑚)))⟩ ∘func ( oppFunc ‘𝐿)))‘𝑋) ∈ (𝑃 Func 𝑂))
38	32, 37	eqeltrrd 2862	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘((1st ‘𝐿)‘𝑋)) ∈ (𝑃 Func 𝑂))
39	38	func1st2nd 49658	. . . . . . . 8 ⊢ (𝜑 → (1st ‘(𝐹‘((1st ‘𝐿)‘𝑋)))(𝑃 Func 𝑂)(2nd ‘(𝐹‘((1st ‘𝐿)‘𝑋))))
40	16, 18, 39	funcf1 17890	. . . . . . 7 ⊢ (𝜑 → (1st ‘(𝐹‘((1st ‘𝐿)‘𝑋))):(Base‘𝐷)⟶𝐴)
41	13, 40	feq1dd 6669	. . . . . 6 ⊢ (𝜑 → (1st ‘((1st ‘𝐿)‘𝑋)):(Base‘𝐷)⟶𝐴)
42	41	ffnd 6687	. . . . 5 ⊢ (𝜑 → (1st ‘((1st ‘𝐿)‘𝑋)) Fn (Base‘𝐷))
43		eqid 2761	. . . . . . . . . . . 12 ⊢ (𝑂Δfunc𝑃) = (𝑂Δfunc𝑃)
44	17	oppccat 17745	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
45	6, 44	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑂 ∈ Cat)
46	14	oppccat 17745	. . . . . . . . . . . . 13 ⊢ (𝐷 ∈ Cat → 𝑃 ∈ Cat)
47	7, 46	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ Cat)
48	43, 45, 47, 21	diagcl 18264	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑂Δfunc𝑃) ∈ (𝑂 Func (𝑃 FuncCat 𝑂)))
49	48	func1st2nd 49658	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘(𝑂Δfunc𝑃))(𝑂 Func (𝑃 FuncCat 𝑂))(2nd ‘(𝑂Δfunc𝑃)))
50	18, 33, 49	funcf1 17890	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘(𝑂Δfunc𝑃)):𝐴⟶(𝑃 Func 𝑂))
51	50, 11	ffvelcdmd 7061	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘(𝑂Δfunc𝑃))‘𝑋) ∈ (𝑃 Func 𝑂))
52	51	func1st2nd 49658	. . . . . . 7 ⊢ (𝜑 → (1st ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋))(𝑃 Func 𝑂)(2nd ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋)))
53	16, 18, 52	funcf1 17890	. . . . . 6 ⊢ (𝜑 → (1st ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋)):(Base‘𝐷)⟶𝐴)
54	53	ffnd 6687	. . . . 5 ⊢ (𝜑 → (1st ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋)) Fn (Base‘𝐷))
55	6	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐷)) → 𝐶 ∈ Cat)
56	7	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐷)) → 𝐷 ∈ Cat)
57	11	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐷)) → 𝑋 ∈ 𝐴)
58		eqid 2761	. . . . . . 7 ⊢ ((1st ‘𝐿)‘𝑋) = ((1st ‘𝐿)‘𝑋)
59		simpr 488	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐷)) → 𝑦 ∈ (Base‘𝐷))
60	5, 55, 56, 2, 57, 58, 15, 59	diag11 18266	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐷)) → ((1st ‘((1st ‘𝐿)‘𝑋))‘𝑦) = 𝑋)
61	45	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐷)) → 𝑂 ∈ Cat)
62	47	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐷)) → 𝑃 ∈ Cat)
63		eqid 2761	. . . . . . 7 ⊢ ((1st ‘(𝑂Δfunc𝑃))‘𝑋) = ((1st ‘(𝑂Δfunc𝑃))‘𝑋)
64	43, 61, 62, 18, 57, 63, 16, 59	diag11 18266	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐷)) → ((1st ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋))‘𝑦) = 𝑋)
65	60, 64	eqtr4d 2799	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐷)) → ((1st ‘((1st ‘𝐿)‘𝑋))‘𝑦) = ((1st ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋))‘𝑦))
66	42, 54, 65	eqfnfvd 7009	. . . 4 ⊢ (𝜑 → (1st ‘((1st ‘𝐿)‘𝑋)) = (1st ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋)))
67	13, 66	eqtrd 2796	. . 3 ⊢ (𝜑 → (1st ‘(𝐹‘((1st ‘𝐿)‘𝑋))) = (1st ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋)))
68	16, 39	funcfn2 17893	. . . 4 ⊢ (𝜑 → (2nd ‘(𝐹‘((1st ‘𝐿)‘𝑋))) Fn ((Base‘𝐷) × (Base‘𝐷)))
69	16, 52	funcfn2 17893	. . . 4 ⊢ (𝜑 → (2nd ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋)) Fn ((Base‘𝐷) × (Base‘𝐷)))
70	1, 12	opf12 49986	. . . . . 6 ⊢ (𝜑 → (𝑦(2nd ‘(𝐹‘((1st ‘𝐿)‘𝑋)))𝑧) = (𝑧(2nd ‘((1st ‘𝐿)‘𝑋))𝑦))
71	70	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → (𝑦(2nd ‘(𝐹‘((1st ‘𝐿)‘𝑋)))𝑧) = (𝑧(2nd ‘((1st ‘𝐿)‘𝑋))𝑦))
72		eqid 2761	. . . . . . . . . . 11 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
73	72, 14	oppchom 17738	. . . . . . . . . 10 ⊢ (𝑦(Hom ‘𝑃)𝑧) = (𝑧(Hom ‘𝐷)𝑦)
74	73	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → (𝑦(Hom ‘𝑃)𝑧) = (𝑧(Hom ‘𝐷)𝑦))
75		eqid 2761	. . . . . . . . . 10 ⊢ (Hom ‘𝑃) = (Hom ‘𝑃)
76		eqid 2761	. . . . . . . . . 10 ⊢ (Hom ‘𝑂) = (Hom ‘𝑂)
77	39	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → (1st ‘(𝐹‘((1st ‘𝐿)‘𝑋)))(𝑃 Func 𝑂)(2nd ‘(𝐹‘((1st ‘𝐿)‘𝑋))))
78		simprl 780	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷))
79		simprr 782	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → 𝑧 ∈ (Base‘𝐷))
80	16, 75, 76, 77, 78, 79	funcf2 17892	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → (𝑦(2nd ‘(𝐹‘((1st ‘𝐿)‘𝑋)))𝑧):(𝑦(Hom ‘𝑃)𝑧)⟶(((1st ‘(𝐹‘((1st ‘𝐿)‘𝑋)))‘𝑦)(Hom ‘𝑂)((1st ‘(𝐹‘((1st ‘𝐿)‘𝑋)))‘𝑧)))
81	74, 80	feq2dd 6672	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → (𝑦(2nd ‘(𝐹‘((1st ‘𝐿)‘𝑋)))𝑧):(𝑧(Hom ‘𝐷)𝑦)⟶(((1st ‘(𝐹‘((1st ‘𝐿)‘𝑋)))‘𝑦)(Hom ‘𝑂)((1st ‘(𝐹‘((1st ‘𝐿)‘𝑋)))‘𝑧)))
82	71, 81	feq1dd 6669	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → (𝑧(2nd ‘((1st ‘𝐿)‘𝑋))𝑦):(𝑧(Hom ‘𝐷)𝑦)⟶(((1st ‘(𝐹‘((1st ‘𝐿)‘𝑋)))‘𝑦)(Hom ‘𝑂)((1st ‘(𝐹‘((1st ‘𝐿)‘𝑋)))‘𝑧)))
83	82	ffnd 6687	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → (𝑧(2nd ‘((1st ‘𝐿)‘𝑋))𝑦) Fn (𝑧(Hom ‘𝐷)𝑦))
84	52	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → (1st ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋))(𝑃 Func 𝑂)(2nd ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋)))
85	16, 75, 76, 84, 78, 79	funcf2 17892	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → (𝑦(2nd ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋))𝑧):(𝑦(Hom ‘𝑃)𝑧)⟶(((1st ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋))‘𝑦)(Hom ‘𝑂)((1st ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋))‘𝑧)))
86	74, 85	feq2dd 6672	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → (𝑦(2nd ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋))𝑧):(𝑧(Hom ‘𝐷)𝑦)⟶(((1st ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋))‘𝑦)(Hom ‘𝑂)((1st ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋))‘𝑧)))
87	86	ffnd 6687	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → (𝑦(2nd ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋))𝑧) Fn (𝑧(Hom ‘𝐷)𝑦))
88		eqid 2761	. . . . . . . . . . 11 ⊢ (Id‘𝐶) = (Id‘𝐶)
89	17, 88	oppcid 17744	. . . . . . . . . 10 ⊢ (𝐶 ∈ Cat → (Id‘𝑂) = (Id‘𝐶))
90	6, 89	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (Id‘𝑂) = (Id‘𝐶))
91	90	fveq1d 6864	. . . . . . . 8 ⊢ (𝜑 → ((Id‘𝑂)‘𝑋) = ((Id‘𝐶)‘𝑋))
92	91	ad2antrr 736	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑦)) → ((Id‘𝑂)‘𝑋) = ((Id‘𝐶)‘𝑋))
93	6	ad2antrr 736	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑦)) → 𝐶 ∈ Cat)
94	93, 44	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑦)) → 𝑂 ∈ Cat)
95	7	ad2antrr 736	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑦)) → 𝐷 ∈ Cat)
96	95, 46	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑦)) → 𝑃 ∈ Cat)
97	11	ad2antrr 736	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑦)) → 𝑋 ∈ 𝐴)
98	78	adantr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑦)) → 𝑦 ∈ (Base‘𝐷))
99		eqid 2761	. . . . . . . 8 ⊢ (Id‘𝑂) = (Id‘𝑂)
100	79	adantr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑦)) → 𝑧 ∈ (Base‘𝐷))
101		simpr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑦)) → 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑦))
102	101, 73	eleqtrrdi 2872	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑦)) → 𝑓 ∈ (𝑦(Hom ‘𝑃)𝑧))
103	43, 94, 96, 18, 97, 63, 16, 98, 75, 99, 100, 102	diag12 18267	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑦)) → ((𝑦(2nd ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋))𝑧)‘𝑓) = ((Id‘𝑂)‘𝑋))
104	5, 93, 95, 2, 97, 58, 15, 100, 72, 88, 98, 101	diag12 18267	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑦)) → ((𝑧(2nd ‘((1st ‘𝐿)‘𝑋))𝑦)‘𝑓) = ((Id‘𝐶)‘𝑋))
105	92, 103, 104	3eqtr4rd 2807	. . . . . 6 ⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑦)) → ((𝑧(2nd ‘((1st ‘𝐿)‘𝑋))𝑦)‘𝑓) = ((𝑦(2nd ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋))𝑧)‘𝑓))
106	83, 87, 105	eqfnfvd 7009	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → (𝑧(2nd ‘((1st ‘𝐿)‘𝑋))𝑦) = (𝑦(2nd ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋))𝑧))
107	71, 106	eqtrd 2796	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → (𝑦(2nd ‘(𝐹‘((1st ‘𝐿)‘𝑋)))𝑧) = (𝑦(2nd ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋))𝑧))
108	68, 69, 107	eqfnovd 49448	. . 3 ⊢ (𝜑 → (2nd ‘(𝐹‘((1st ‘𝐿)‘𝑋))) = (2nd ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋)))
109	67, 108	opeq12d 4836	. 2 ⊢ (𝜑 → ⟨(1st ‘(𝐹‘((1st ‘𝐿)‘𝑋))), (2nd ‘(𝐹‘((1st ‘𝐿)‘𝑋)))⟩ = ⟨(1st ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋)), (2nd ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋))⟩)
110		relfunc 17886	. . 3 ⊢ Rel (𝑃 Func 𝑂)
111		1st2nd 8015	. . 3 ⊢ ((Rel (𝑃 Func 𝑂) ∧ (𝐹‘((1st ‘𝐿)‘𝑋)) ∈ (𝑃 Func 𝑂)) → (𝐹‘((1st ‘𝐿)‘𝑋)) = ⟨(1st ‘(𝐹‘((1st ‘𝐿)‘𝑋))), (2nd ‘(𝐹‘((1st ‘𝐿)‘𝑋)))⟩)
112	110, 38, 111	sylancr 596	. 2 ⊢ (𝜑 → (𝐹‘((1st ‘𝐿)‘𝑋)) = ⟨(1st ‘(𝐹‘((1st ‘𝐿)‘𝑋))), (2nd ‘(𝐹‘((1st ‘𝐿)‘𝑋)))⟩)
113		1st2nd 8015	. . 3 ⊢ ((Rel (𝑃 Func 𝑂) ∧ ((1st ‘(𝑂Δfunc𝑃))‘𝑋) ∈ (𝑃 Func 𝑂)) → ((1st ‘(𝑂Δfunc𝑃))‘𝑋) = ⟨(1st ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋)), (2nd ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋))⟩)
114	110, 51, 113	sylancr 596	. 2 ⊢ (𝜑 → ((1st ‘(𝑂Δfunc𝑃))‘𝑋) = ⟨(1st ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋)), (2nd ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋))⟩)
115	109, 112, 114	3eqtr4d 2806	1 ⊢ (𝜑 → (𝐹‘((1st ‘𝐿)‘𝑋)) = ((1st ‘(𝑂Δfunc𝑃))‘𝑋))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ⟨cop 4585   class class class wbr 5097   I cid 5537   ↾ cres 5645  Rel wrel 5648  ‘cfv 6516  (class class class)co 7391   ∈ cmpo 7393  1^st c1st 7963  2^nd c2nd 7964  Basecbs 17236  Hom chom 17288  Catccat 17687  Idccid 17688  oppCatcoppc 17734   Func cfunc 17878   ∘_func ccofu 17880   Nat cnat 17968   FuncCat cfuc 17969  Δ_funccdiag 18235   oppFunc coppf 49704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713  ax-cnex 11123  ax-resscn 11124  ax-1cn 11125  ax-icn 11126  ax-addcl 11127  ax-addrcl 11128  ax-mulcl 11129  ax-mulrcl 11130  ax-mulcom 11131  ax-addass 11132  ax-mulass 11133  ax-distr 11134  ax-i2m1 11135  ax-1ne0 11136  ax-1rid 11137  ax-rnegex 11138  ax-rrecex 11139  ax-cnre 11140  ax-pre-lttri 11141  ax-pre-lttrn 11142  ax-pre-ltadd 11143  ax-pre-mulgt0 11144

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6283  df-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7348  df-ov 7394  df-oprab 7395  df-mpo 7396  df-om 7842  df-1st 7965  df-2nd 7966  df-tpos 8200  df-frecs 8256  df-wrecs 8287  df-recs 8336  df-rdg 8375  df-1o 8431  df-er 8672  df-map 8804  df-ixp 8874  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-pnf 11212  df-mnf 11213  df-xr 11214  df-ltxr 11215  df-le 11216  df-sub 11410  df-neg 11411  df-nn 12205  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-6 12278  df-7 12279  df-8 12280  df-9 12281  df-n0 12476  df-z 12563  df-dec 12683  df-uz 12834  df-fz 13507  df-struct 17174  df-sets 17191  df-slot 17209  df-ndx 17221  df-base 17237  df-hom 17301  df-cco 17302  df-cat 17691  df-cid 17692  df-homf 17693  df-comf 17694  df-oppc 17735  df-sect 17771  df-inv 17772  df-iso 17773  df-func 17882  df-idfu 17883  df-cofu 17884  df-full 17930  df-fth 17931  df-nat 17970  df-fuc 17971  df-catc 18123  df-xpc 18195  df-1stf 18196  df-curf 18237  df-diag 18239  df-oppf 49705

This theorem is referenced by:  oppfdiag1a  49997  oppfdiag  49998