Theorem oppfdiag1 50225

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 2765	. . . . . . . 8 ⊢ (𝐷 FuncCat 𝐶) = (𝐷 FuncCat 𝐶)
4	3	fucbas 18038	. . . . . . 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 18315	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ (𝐶 Func (𝐷 FuncCat 𝐶)))
9	8	func1st2nd 49887	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐿)(𝐶 Func (𝐷 FuncCat 𝐶))(2nd ‘𝐿))
10	2, 4, 9	funcf1 17941	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐿):𝐴⟶(𝐷 Func 𝐶))
11		oppfdiag1.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
12	10, 11	ffvelcdmd 7084	. . . . 5 ⊢ (𝜑 → ((1st ‘𝐿)‘𝑋) ∈ (𝐷 Func 𝐶))
13	1, 12	opf11 50214	. . . 4 ⊢ (𝜑 → (1st ‘(𝐹‘((1st ‘𝐿)‘𝑋))) = (1st ‘((1st ‘𝐿)‘𝑋)))
14		oppfdiag.p	. . . . . . . . 9 ⊢ 𝑃 = (oppCat‘𝐷)
15		eqid 2765	. . . . . . . . 9 ⊢ (Base‘𝐷) = (Base‘𝐷)
16	14, 15	oppcbas 17792	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝑃)
17		oppfdiag.o	. . . . . . . . 9 ⊢ 𝑂 = (oppCat‘𝐶)
18	17, 2	oppcbas 17792	. . . . . . . 8 ⊢ 𝐴 = (Base‘𝑂)
19		eqid 2765	. . . . . . . . . . . . 13 ⊢ (oppCat‘(𝐷 FuncCat 𝐶)) = (oppCat‘(𝐷 FuncCat 𝐶))
20	17, 19, 8	oppfoppc2 49953	. . . . . . . . . . . 12 ⊢ (𝜑 → ( oppFunc ‘𝐿) ∈ (𝑂 Func (oppCat‘(𝐷 FuncCat 𝐶))))
21		eqid 2765	. . . . . . . . . . . . . 14 ⊢ (𝑃 FuncCat 𝑂) = (𝑃 FuncCat 𝑂)
22		eqid 2765	. . . . . . . . . . . . . 14 ⊢ (𝐷 Nat 𝐶) = (𝐷 Nat 𝐶)
23		eqidd 2766	. . . . . . . . . . . . . 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 50223	. . . . . . . . . . . . 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 5112	. . . . . . . . . . . . 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 221	. . . . . . . . . . . 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 17959	. . . . . . . . . . 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 49888	. . . . . . . . . . . 12 ⊢ (𝜑 → (1st ‘⟨𝐹, (𝑚 ∈ (𝐷 Func 𝐶), 𝑛 ∈ (𝐷 Func 𝐶) ↦ ( I ↾ (𝑛(𝐷 Nat 𝐶)𝑚)))⟩) = 𝐹)
29	8	oppf1 49950	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1st ‘( oppFunc ‘𝐿)) = (1st ‘𝐿))
30	29	fveq1d 6887	. . . . . . . . . . . 12 ⊢ (𝜑 → ((1st ‘( oppFunc ‘𝐿))‘𝑋) = ((1st ‘𝐿)‘𝑋))
31	28, 30	fveq12d 6892	. . . . . . . . . . 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 2800	. . . . . . . . . 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 18038	. . . . . . . . . . . 12 ⊢ (𝑃 Func 𝑂) = (Base‘(𝑃 FuncCat 𝑂))
34	20, 26	cofucl 17963	. . . . . . . . . . . . 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 49887	. . . . . . . . . . . 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 17941	. . . . . . . . . . 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 7084	. . . . . . . . . 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 2866	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘((1st ‘𝐿)‘𝑋)) ∈ (𝑃 Func 𝑂))
39	38	func1st2nd 49887	. . . . . . . 8 ⊢ (𝜑 → (1st ‘(𝐹‘((1st ‘𝐿)‘𝑋)))(𝑃 Func 𝑂)(2nd ‘(𝐹‘((1st ‘𝐿)‘𝑋))))
40	16, 18, 39	funcf1 17941	. . . . . . 7 ⊢ (𝜑 → (1st ‘(𝐹‘((1st ‘𝐿)‘𝑋))):(Base‘𝐷)⟶𝐴)
41	13, 40	feq1dd 6692	. . . . . 6 ⊢ (𝜑 → (1st ‘((1st ‘𝐿)‘𝑋)):(Base‘𝐷)⟶𝐴)
42	41	ffnd 6710	. . . . 5 ⊢ (𝜑 → (1st ‘((1st ‘𝐿)‘𝑋)) Fn (Base‘𝐷))
43		eqid 2765	. . . . . . . . . . . 12 ⊢ (𝑂Δfunc𝑃) = (𝑂Δfunc𝑃)
44	17	oppccat 17796	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
45	6, 44	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑂 ∈ Cat)
46	14	oppccat 17796	. . . . . . . . . . . . 13 ⊢ (𝐷 ∈ Cat → 𝑃 ∈ Cat)
47	7, 46	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ Cat)
48	43, 45, 47, 21	diagcl 18315	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑂Δfunc𝑃) ∈ (𝑂 Func (𝑃 FuncCat 𝑂)))
49	48	func1st2nd 49887	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘(𝑂Δfunc𝑃))(𝑂 Func (𝑃 FuncCat 𝑂))(2nd ‘(𝑂Δfunc𝑃)))
50	18, 33, 49	funcf1 17941	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘(𝑂Δfunc𝑃)):𝐴⟶(𝑃 Func 𝑂))
51	50, 11	ffvelcdmd 7084	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘(𝑂Δfunc𝑃))‘𝑋) ∈ (𝑃 Func 𝑂))
52	51	func1st2nd 49887	. . . . . . 7 ⊢ (𝜑 → (1st ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋))(𝑃 Func 𝑂)(2nd ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋)))
53	16, 18, 52	funcf1 17941	. . . . . 6 ⊢ (𝜑 → (1st ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋)):(Base‘𝐷)⟶𝐴)
54	53	ffnd 6710	. . . . 5 ⊢ (𝜑 → (1st ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋)) Fn (Base‘𝐷))
55	6	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐷)) → 𝐶 ∈ Cat)
56	7	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐷)) → 𝐷 ∈ Cat)
57	11	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐷)) → 𝑋 ∈ 𝐴)
58		eqid 2765	. . . . . . 7 ⊢ ((1st ‘𝐿)‘𝑋) = ((1st ‘𝐿)‘𝑋)
59		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐷)) → 𝑦 ∈ (Base‘𝐷))
60	5, 55, 56, 2, 57, 58, 15, 59	diag11 18317	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐷)) → ((1st ‘((1st ‘𝐿)‘𝑋))‘𝑦) = 𝑋)
61	45	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐷)) → 𝑂 ∈ Cat)
62	47	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐷)) → 𝑃 ∈ Cat)
63		eqid 2765	. . . . . . 7 ⊢ ((1st ‘(𝑂Δfunc𝑃))‘𝑋) = ((1st ‘(𝑂Δfunc𝑃))‘𝑋)
64	43, 61, 62, 18, 57, 63, 16, 59	diag11 18317	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐷)) → ((1st ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋))‘𝑦) = 𝑋)
65	60, 64	eqtr4d 2803	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐷)) → ((1st ‘((1st ‘𝐿)‘𝑋))‘𝑦) = ((1st ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋))‘𝑦))
66	42, 54, 65	eqfnfvd 7032	. . . 4 ⊢ (𝜑 → (1st ‘((1st ‘𝐿)‘𝑋)) = (1st ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋)))
67	13, 66	eqtrd 2800	. . 3 ⊢ (𝜑 → (1st ‘(𝐹‘((1st ‘𝐿)‘𝑋))) = (1st ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋)))
68	16, 39	funcfn2 17944	. . . 4 ⊢ (𝜑 → (2nd ‘(𝐹‘((1st ‘𝐿)‘𝑋))) Fn ((Base‘𝐷) × (Base‘𝐷)))
69	16, 52	funcfn2 17944	. . . 4 ⊢ (𝜑 → (2nd ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋)) Fn ((Base‘𝐷) × (Base‘𝐷)))
70	1, 12	opf12 50215	. . . . . 6 ⊢ (𝜑 → (𝑦(2nd ‘(𝐹‘((1st ‘𝐿)‘𝑋)))𝑧) = (𝑧(2nd ‘((1st ‘𝐿)‘𝑋))𝑦))
71	70	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → (𝑦(2nd ‘(𝐹‘((1st ‘𝐿)‘𝑋)))𝑧) = (𝑧(2nd ‘((1st ‘𝐿)‘𝑋))𝑦))
72		eqid 2765	. . . . . . . . . . 11 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
73	72, 14	oppchom 17789	. . . . . . . . . 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 2765	. . . . . . . . . 10 ⊢ (Hom ‘𝑃) = (Hom ‘𝑃)
76		eqid 2765	. . . . . . . . . 10 ⊢ (Hom ‘𝑂) = (Hom ‘𝑂)
77	39	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → (1st ‘(𝐹‘((1st ‘𝐿)‘𝑋)))(𝑃 Func 𝑂)(2nd ‘(𝐹‘((1st ‘𝐿)‘𝑋))))
78		simprl 783	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷))
79		simprr 785	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → 𝑧 ∈ (Base‘𝐷))
80	16, 75, 76, 77, 78, 79	funcf2 17943	. . . . . . . . 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 6695	. . . . . . . 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 6692	. . . . . . 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 6710	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → (𝑧(2nd ‘((1st ‘𝐿)‘𝑋))𝑦) Fn (𝑧(Hom ‘𝐷)𝑦))
84	52	adantr 486	. . . . . . . . 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 17943	. . . . . . . 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 6695	. . . . . . 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 6710	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → (𝑦(2nd ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋))𝑧) Fn (𝑧(Hom ‘𝐷)𝑦))
88		eqid 2765	. . . . . . . . . . 11 ⊢ (Id‘𝐶) = (Id‘𝐶)
89	17, 88	oppcid 17795	. . . . . . . . . 10 ⊢ (𝐶 ∈ Cat → (Id‘𝑂) = (Id‘𝐶))
90	6, 89	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (Id‘𝑂) = (Id‘𝐶))
91	90	fveq1d 6887	. . . . . . . 8 ⊢ (𝜑 → ((Id‘𝑂)‘𝑋) = ((Id‘𝐶)‘𝑋))
92	91	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑦)) → ((Id‘𝑂)‘𝑋) = ((Id‘𝐶)‘𝑋))
93	6	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑦)) → 𝐶 ∈ Cat)
94	93, 44	syl 18	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑦)) → 𝑂 ∈ Cat)
95	7	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑦)) → 𝐷 ∈ Cat)
96	95, 46	syl 18	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑦)) → 𝑃 ∈ Cat)
97	11	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑦)) → 𝑋 ∈ 𝐴)
98	78	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑦)) → 𝑦 ∈ (Base‘𝐷))
99		eqid 2765	. . . . . . . 8 ⊢ (Id‘𝑂) = (Id‘𝑂)
100	79	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑦)) → 𝑧 ∈ (Base‘𝐷))
101		simpr 490	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑦)) → 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑦))
102	101, 73	eleqtrrdi 2876	. . . . . . . 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 18318	. . . . . . 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 18318	. . . . . . 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 2811	. . . . . 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 7032	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → (𝑧(2nd ‘((1st ‘𝐿)‘𝑋))𝑦) = (𝑦(2nd ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋))𝑧))
107	71, 106	eqtrd 2800	. . . 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 49677	. . 3 ⊢ (𝜑 → (2nd ‘(𝐹‘((1st ‘𝐿)‘𝑋))) = (2nd ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋)))
109	67, 108	opeq12d 4848	. 2 ⊢ (𝜑 → ⟨(1st ‘(𝐹‘((1st ‘𝐿)‘𝑋))), (2nd ‘(𝐹‘((1st ‘𝐿)‘𝑋)))⟩ = ⟨(1st ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋)), (2nd ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋))⟩)
110		relfunc 17937	. . 3 ⊢ Rel (𝑃 Func 𝑂)
111		1st2nd 8038	. . 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 599	. 2 ⊢ (𝜑 → (𝐹‘((1st ‘𝐿)‘𝑋)) = ⟨(1st ‘(𝐹‘((1st ‘𝐿)‘𝑋))), (2nd ‘(𝐹‘((1st ‘𝐿)‘𝑋)))⟩)
113		1st2nd 8038	. . 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 599	. 2 ⊢ (𝜑 → ((1st ‘(𝑂Δfunc𝑃))‘𝑋) = ⟨(1st ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋)), (2nd ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋))⟩)
115	109, 112, 114	3eqtr4d 2810	1 ⊢ (𝜑 → (𝐹‘((1st ‘𝐿)‘𝑋)) = ((1st ‘(𝑂Δfunc𝑃))‘𝑋))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ⟨cop 4597   class class class wbr 5111   I cid 5557   ↾ cres 5665  Rel wrel 5668  ‘cfv 6540  (class class class)co 7416   ∈ cmpo 7418  1^st c1st 7986  2^nd c2nd 7987  Basecbs 17287  Hom chom 17339  Catccat 17738  Idccid 17739  oppCatcoppc 17785   Func cfunc 17929   ∘_func ccofu 17931   Nat cnat 18019   FuncCat cfuc 18020  Δ_funccdiag 18286   oppFunc coppf 49933

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738  ax-cnex 11167  ax-resscn 11168  ax-1cn 11169  ax-icn 11170  ax-addcl 11171  ax-addrcl 11172  ax-mulcl 11173  ax-mulrcl 11174  ax-mulcom 11175  ax-addass 11176  ax-mulass 11177  ax-distr 11178  ax-i2m1 11179  ax-1ne0 11180  ax-1rid 11181  ax-rnegex 11182  ax-rrecex 11183  ax-cnre 11184  ax-pre-lttri 11185  ax-pre-lttrn 11186  ax-pre-ltadd 11187  ax-pre-mulgt0 11188

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7865  df-1st 7988  df-2nd 7989  df-tpos 8224  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-er 8696  df-map 8828  df-ixp 8898  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11454  df-neg 11455  df-nn 12245  df-2 12314  df-3 12315  df-4 12316  df-5 12317  df-6 12318  df-7 12319  df-8 12320  df-9 12321  df-n0 12516  df-z 12603  df-dec 12724  df-uz 12875  df-fz 13548  df-struct 17225  df-sets 17242  df-slot 17260  df-ndx 17272  df-base 17288  df-hom 17352  df-cco 17353  df-cat 17742  df-cid 17743  df-homf 17744  df-comf 17745  df-oppc 17786  df-sect 17822  df-inv 17823  df-iso 17824  df-func 17933  df-idfu 17934  df-cofu 17935  df-full 17981  df-fth 17982  df-nat 18021  df-fuc 18022  df-catc 18174  df-xpc 18246  df-1stf 18247  df-curf 18288  df-diag 18290  df-oppf 49934

This theorem is used by:  oppfdiag1a  50226  oppfdiag  50227