Theorem oppfdiag1 49446

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 2731	. . . . . . . 8 ⊢ (𝐷 FuncCat 𝐶) = (𝐷 FuncCat 𝐶)
4	3	fucbas 17865	. . . . . . 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 18142	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ (𝐶 Func (𝐷 FuncCat 𝐶)))
9	8	func1st2nd 49108	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐿)(𝐶 Func (𝐷 FuncCat 𝐶))(2nd ‘𝐿))
10	2, 4, 9	funcf1 17768	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐿):𝐴⟶(𝐷 Func 𝐶))
11		oppfdiag1.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
12	10, 11	ffvelcdmd 7013	. . . . 5 ⊢ (𝜑 → ((1st ‘𝐿)‘𝑋) ∈ (𝐷 Func 𝐶))
13	1, 12	opf11 49435	. . . 4 ⊢ (𝜑 → (1st ‘(𝐹‘((1st ‘𝐿)‘𝑋))) = (1st ‘((1st ‘𝐿)‘𝑋)))
14		oppfdiag.p	. . . . . . . . 9 ⊢ 𝑃 = (oppCat‘𝐷)
15		eqid 2731	. . . . . . . . 9 ⊢ (Base‘𝐷) = (Base‘𝐷)
16	14, 15	oppcbas 17619	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝑃)
17		oppfdiag.o	. . . . . . . . 9 ⊢ 𝑂 = (oppCat‘𝐶)
18	17, 2	oppcbas 17619	. . . . . . . 8 ⊢ 𝐴 = (Base‘𝑂)
19		eqid 2731	. . . . . . . . . . . . 13 ⊢ (oppCat‘(𝐷 FuncCat 𝐶)) = (oppCat‘(𝐷 FuncCat 𝐶))
20	17, 19, 8	oppfoppc2 49174	. . . . . . . . . . . 12 ⊢ (𝜑 → ( oppFunc ‘𝐿) ∈ (𝑂 Func (oppCat‘(𝐷 FuncCat 𝐶))))
21		eqid 2731	. . . . . . . . . . . . . 14 ⊢ (𝑃 FuncCat 𝑂) = (𝑃 FuncCat 𝑂)
22		eqid 2731	. . . . . . . . . . . . . 14 ⊢ (𝐷 Nat 𝐶) = (𝐷 Nat 𝐶)
23		eqidd 2732	. . . . . . . . . . . . . 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 49444	. . . . . . . . . . . . 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 5087	. . . . . . . . . . . . 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 218	. . . . . . . . . . . 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 17786	. . . . . . . . . . 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 49109	. . . . . . . . . . . 12 ⊢ (𝜑 → (1st ‘⟨𝐹, (𝑚 ∈ (𝐷 Func 𝐶), 𝑛 ∈ (𝐷 Func 𝐶) ↦ ( I ↾ (𝑛(𝐷 Nat 𝐶)𝑚)))⟩) = 𝐹)
29	8	oppf1 49171	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1st ‘( oppFunc ‘𝐿)) = (1st ‘𝐿))
30	29	fveq1d 6819	. . . . . . . . . . . 12 ⊢ (𝜑 → ((1st ‘( oppFunc ‘𝐿))‘𝑋) = ((1st ‘𝐿)‘𝑋))
31	28, 30	fveq12d 6824	. . . . . . . . . . 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 2766	. . . . . . . . . 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 17865	. . . . . . . . . . . 12 ⊢ (𝑃 Func 𝑂) = (Base‘(𝑃 FuncCat 𝑂))
34	20, 26	cofucl 17790	. . . . . . . . . . . . 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 49108	. . . . . . . . . . . 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 17768	. . . . . . . . . . 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 7013	. . . . . . . . . 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 2832	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘((1st ‘𝐿)‘𝑋)) ∈ (𝑃 Func 𝑂))
39	38	func1st2nd 49108	. . . . . . . 8 ⊢ (𝜑 → (1st ‘(𝐹‘((1st ‘𝐿)‘𝑋)))(𝑃 Func 𝑂)(2nd ‘(𝐹‘((1st ‘𝐿)‘𝑋))))
40	16, 18, 39	funcf1 17768	. . . . . . 7 ⊢ (𝜑 → (1st ‘(𝐹‘((1st ‘𝐿)‘𝑋))):(Base‘𝐷)⟶𝐴)
41	13, 40	feq1dd 6629	. . . . . 6 ⊢ (𝜑 → (1st ‘((1st ‘𝐿)‘𝑋)):(Base‘𝐷)⟶𝐴)
42	41	ffnd 6647	. . . . 5 ⊢ (𝜑 → (1st ‘((1st ‘𝐿)‘𝑋)) Fn (Base‘𝐷))
43		eqid 2731	. . . . . . . . . . . 12 ⊢ (𝑂Δfunc𝑃) = (𝑂Δfunc𝑃)
44	17	oppccat 17623	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
45	6, 44	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑂 ∈ Cat)
46	14	oppccat 17623	. . . . . . . . . . . . 13 ⊢ (𝐷 ∈ Cat → 𝑃 ∈ Cat)
47	7, 46	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ Cat)
48	43, 45, 47, 21	diagcl 18142	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑂Δfunc𝑃) ∈ (𝑂 Func (𝑃 FuncCat 𝑂)))
49	48	func1st2nd 49108	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘(𝑂Δfunc𝑃))(𝑂 Func (𝑃 FuncCat 𝑂))(2nd ‘(𝑂Δfunc𝑃)))
50	18, 33, 49	funcf1 17768	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘(𝑂Δfunc𝑃)):𝐴⟶(𝑃 Func 𝑂))
51	50, 11	ffvelcdmd 7013	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘(𝑂Δfunc𝑃))‘𝑋) ∈ (𝑃 Func 𝑂))
52	51	func1st2nd 49108	. . . . . . 7 ⊢ (𝜑 → (1st ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋))(𝑃 Func 𝑂)(2nd ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋)))
53	16, 18, 52	funcf1 17768	. . . . . 6 ⊢ (𝜑 → (1st ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋)):(Base‘𝐷)⟶𝐴)
54	53	ffnd 6647	. . . . 5 ⊢ (𝜑 → (1st ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋)) Fn (Base‘𝐷))
55	6	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐷)) → 𝐶 ∈ Cat)
56	7	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐷)) → 𝐷 ∈ Cat)
57	11	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐷)) → 𝑋 ∈ 𝐴)
58		eqid 2731	. . . . . . 7 ⊢ ((1st ‘𝐿)‘𝑋) = ((1st ‘𝐿)‘𝑋)
59		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐷)) → 𝑦 ∈ (Base‘𝐷))
60	5, 55, 56, 2, 57, 58, 15, 59	diag11 18144	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐷)) → ((1st ‘((1st ‘𝐿)‘𝑋))‘𝑦) = 𝑋)
61	45	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐷)) → 𝑂 ∈ Cat)
62	47	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐷)) → 𝑃 ∈ Cat)
63		eqid 2731	. . . . . . 7 ⊢ ((1st ‘(𝑂Δfunc𝑃))‘𝑋) = ((1st ‘(𝑂Δfunc𝑃))‘𝑋)
64	43, 61, 62, 18, 57, 63, 16, 59	diag11 18144	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐷)) → ((1st ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋))‘𝑦) = 𝑋)
65	60, 64	eqtr4d 2769	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐷)) → ((1st ‘((1st ‘𝐿)‘𝑋))‘𝑦) = ((1st ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋))‘𝑦))
66	42, 54, 65	eqfnfvd 6962	. . . 4 ⊢ (𝜑 → (1st ‘((1st ‘𝐿)‘𝑋)) = (1st ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋)))
67	13, 66	eqtrd 2766	. . 3 ⊢ (𝜑 → (1st ‘(𝐹‘((1st ‘𝐿)‘𝑋))) = (1st ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋)))
68	16, 39	funcfn2 17771	. . . 4 ⊢ (𝜑 → (2nd ‘(𝐹‘((1st ‘𝐿)‘𝑋))) Fn ((Base‘𝐷) × (Base‘𝐷)))
69	16, 52	funcfn2 17771	. . . 4 ⊢ (𝜑 → (2nd ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋)) Fn ((Base‘𝐷) × (Base‘𝐷)))
70	1, 12	opf12 49436	. . . . . 6 ⊢ (𝜑 → (𝑦(2nd ‘(𝐹‘((1st ‘𝐿)‘𝑋)))𝑧) = (𝑧(2nd ‘((1st ‘𝐿)‘𝑋))𝑦))
71	70	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → (𝑦(2nd ‘(𝐹‘((1st ‘𝐿)‘𝑋)))𝑧) = (𝑧(2nd ‘((1st ‘𝐿)‘𝑋))𝑦))
72		eqid 2731	. . . . . . . . . . 11 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
73	72, 14	oppchom 17616	. . . . . . . . . 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 2731	. . . . . . . . . 10 ⊢ (Hom ‘𝑃) = (Hom ‘𝑃)
76		eqid 2731	. . . . . . . . . 10 ⊢ (Hom ‘𝑂) = (Hom ‘𝑂)
77	39	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → (1st ‘(𝐹‘((1st ‘𝐿)‘𝑋)))(𝑃 Func 𝑂)(2nd ‘(𝐹‘((1st ‘𝐿)‘𝑋))))
78		simprl 770	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → 𝑦 ∈ (Base‘𝐷))
79		simprr 772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → 𝑧 ∈ (Base‘𝐷))
80	16, 75, 76, 77, 78, 79	funcf2 17770	. . . . . . . . 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 6632	. . . . . . . 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 6629	. . . . . . 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 6647	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → (𝑧(2nd ‘((1st ‘𝐿)‘𝑋))𝑦) Fn (𝑧(Hom ‘𝐷)𝑦))
84	52	adantr 480	. . . . . . . . 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 17770	. . . . . . . 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 6632	. . . . . . 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 6647	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → (𝑦(2nd ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋))𝑧) Fn (𝑧(Hom ‘𝐷)𝑦))
88		eqid 2731	. . . . . . . . . . 11 ⊢ (Id‘𝐶) = (Id‘𝐶)
89	17, 88	oppcid 17622	. . . . . . . . . 10 ⊢ (𝐶 ∈ Cat → (Id‘𝑂) = (Id‘𝐶))
90	6, 89	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (Id‘𝑂) = (Id‘𝐶))
91	90	fveq1d 6819	. . . . . . . 8 ⊢ (𝜑 → ((Id‘𝑂)‘𝑋) = ((Id‘𝐶)‘𝑋))
92	91	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑦)) → ((Id‘𝑂)‘𝑋) = ((Id‘𝐶)‘𝑋))
93	6	ad2antrr 726	. . . . . . . . 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 726	. . . . . . . . 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 726	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑦)) → 𝑋 ∈ 𝐴)
98	78	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑦)) → 𝑦 ∈ (Base‘𝐷))
99		eqid 2731	. . . . . . . 8 ⊢ (Id‘𝑂) = (Id‘𝑂)
100	79	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑦)) → 𝑧 ∈ (Base‘𝐷))
101		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) ∧ 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑦)) → 𝑓 ∈ (𝑧(Hom ‘𝐷)𝑦))
102	101, 73	eleqtrrdi 2842	. . . . . . . 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 18145	. . . . . . 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 18145	. . . . . . 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 2777	. . . . . 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 6962	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝐷) ∧ 𝑧 ∈ (Base‘𝐷))) → (𝑧(2nd ‘((1st ‘𝐿)‘𝑋))𝑦) = (𝑦(2nd ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋))𝑧))
107	71, 106	eqtrd 2766	. . . 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 48897	. . 3 ⊢ (𝜑 → (2nd ‘(𝐹‘((1st ‘𝐿)‘𝑋))) = (2nd ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋)))
109	67, 108	opeq12d 4828	. 2 ⊢ (𝜑 → ⟨(1st ‘(𝐹‘((1st ‘𝐿)‘𝑋))), (2nd ‘(𝐹‘((1st ‘𝐿)‘𝑋)))⟩ = ⟨(1st ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋)), (2nd ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋))⟩)
110		relfunc 17764	. . 3 ⊢ Rel (𝑃 Func 𝑂)
111		1st2nd 7966	. . 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 587	. 2 ⊢ (𝜑 → (𝐹‘((1st ‘𝐿)‘𝑋)) = ⟨(1st ‘(𝐹‘((1st ‘𝐿)‘𝑋))), (2nd ‘(𝐹‘((1st ‘𝐿)‘𝑋)))⟩)
113		1st2nd 7966	. . 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 587	. 2 ⊢ (𝜑 → ((1st ‘(𝑂Δfunc𝑃))‘𝑋) = ⟨(1st ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋)), (2nd ‘((1st ‘(𝑂Δfunc𝑃))‘𝑋))⟩)
115	109, 112, 114	3eqtr4d 2776	1 ⊢ (𝜑 → (𝐹‘((1st ‘𝐿)‘𝑋)) = ((1st ‘(𝑂Δfunc𝑃))‘𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ⟨cop 4577   class class class wbr 5086   I cid 5505   ↾ cres 5613  Rel wrel 5616  ‘cfv 6476  (class class class)co 7341   ∈ cmpo 7343  1^st c1st 7914  2^nd c2nd 7915  Basecbs 17115  Hom chom 17167  Catccat 17565  Idccid 17566  oppCatcoppc 17612   Func cfunc 17756   ∘_func ccofu 17758   Nat cnat 17846   FuncCat cfuc 17847  Δ_funccdiag 18113   oppFunc coppf 49154

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663  ax-cnex 11057  ax-resscn 11058  ax-1cn 11059  ax-icn 11060  ax-addcl 11061  ax-addrcl 11062  ax-mulcl 11063  ax-mulrcl 11064  ax-mulcom 11065  ax-addass 11066  ax-mulass 11067  ax-distr 11068  ax-i2m1 11069  ax-1ne0 11070  ax-1rid 11071  ax-rnegex 11072  ax-rrecex 11073  ax-cnre 11074  ax-pre-lttri 11075  ax-pre-lttrn 11076  ax-pre-ltadd 11077  ax-pre-mulgt0 11078

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-tp 4576  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-tpos 8151  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-er 8617  df-map 8747  df-ixp 8817  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-pnf 11143  df-mnf 11144  df-xr 11145  df-ltxr 11146  df-le 11147  df-sub 11341  df-neg 11342  df-nn 12121  df-2 12183  df-3 12184  df-4 12185  df-5 12186  df-6 12187  df-7 12188  df-8 12189  df-9 12190  df-n0 12377  df-z 12464  df-dec 12584  df-uz 12728  df-fz 13403  df-struct 17053  df-sets 17070  df-slot 17088  df-ndx 17100  df-base 17116  df-hom 17180  df-cco 17181  df-cat 17569  df-cid 17570  df-homf 17571  df-comf 17572  df-oppc 17613  df-sect 17649  df-inv 17650  df-iso 17651  df-func 17760  df-idfu 17761  df-cofu 17762  df-full 17808  df-fth 17809  df-nat 17848  df-fuc 17849  df-catc 18001  df-xpc 18073  df-1stf 18074  df-curf 18115  df-diag 18117  df-oppf 49155

This theorem is referenced by:  oppfdiag1a  49447  oppfdiag  49448