Theorem oppfdiag 49998

Description: A diagonal functor for opposite categories is the opposite functor of the diagonal functor for original categories post-composed by an isomorphism (fucoppc 49992). (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)
oppfdiag.f	⊢ (𝜑 → 𝐹 = ( oppFunc ↾ (𝐷 Func 𝐶)))
oppfdiag.n	⊢ 𝑁 = (𝐷 Nat 𝐶)
oppfdiag.g	⊢ (𝜑 → 𝐺 = (𝑚 ∈ (𝐷 Func 𝐶), 𝑛 ∈ (𝐷 Func 𝐶) ↦ ( I ↾ (𝑛𝑁𝑚))))

Assertion

Ref	Expression
oppfdiag	⊢ (𝜑 → (⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿)) = (𝑂Δfunc𝑃))

Distinct variable groups:   𝐶,𝑚,𝑛   𝐷,𝑚,𝑛   𝑚,𝐿,𝑛   𝑚,𝑁,𝑛   𝜑,𝑚,𝑛

Allowed substitution hints:   𝑃(𝑚,𝑛)   𝐹(𝑚,𝑛)   𝐺(𝑚,𝑛)   𝑂(𝑚,𝑛)

Proof of Theorem oppfdiag

Dummy variables 𝑓 𝑦 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oppfdiag.o	. . . . . . 7 ⊢ 𝑂 = (oppCat‘𝐶)
2		eqid 2761	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
3	1, 2	oppcbas 17741	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝑂)
4		eqid 2761	. . . . . . 7 ⊢ (𝑃 FuncCat 𝑂) = (𝑃 FuncCat 𝑂)
5	4	fucbas 17987	. . . . . 6 ⊢ (𝑃 Func 𝑂) = (Base‘(𝑃 FuncCat 𝑂))
6		eqid 2761	. . . . . . . . 9 ⊢ (oppCat‘(𝐷 FuncCat 𝐶)) = (oppCat‘(𝐷 FuncCat 𝐶))
7		oppfdiag.l	. . . . . . . . . 10 ⊢ 𝐿 = (𝐶Δfunc𝐷)
8		oppfdiag.c	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ Cat)
9		oppfdiag.d	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ Cat)
10		eqid 2761	. . . . . . . . . 10 ⊢ (𝐷 FuncCat 𝐶) = (𝐷 FuncCat 𝐶)
11	7, 8, 9, 10	diagcl 18264	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ (𝐶 Func (𝐷 FuncCat 𝐶)))
12	1, 6, 11	oppfoppc2 49724	. . . . . . . 8 ⊢ (𝜑 → ( oppFunc ‘𝐿) ∈ (𝑂 Func (oppCat‘(𝐷 FuncCat 𝐶))))
13		oppfdiag.p	. . . . . . . . . 10 ⊢ 𝑃 = (oppCat‘𝐷)
14		oppfdiag.n	. . . . . . . . . 10 ⊢ 𝑁 = (𝐷 Nat 𝐶)
15		oppfdiag.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 = ( oppFunc ↾ (𝐷 Func 𝐶)))
16		oppfdiag.g	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 = (𝑚 ∈ (𝐷 Func 𝐶), 𝑛 ∈ (𝐷 Func 𝐶) ↦ ( I ↾ (𝑛𝑁𝑚))))
17	13, 1, 10, 6, 4, 14, 15, 16, 9, 8	fucoppcfunc 49994	. . . . . . . . 9 ⊢ (𝜑 → 𝐹((oppCat‘(𝐷 FuncCat 𝐶)) Func (𝑃 FuncCat 𝑂))𝐺)
18		df-br 5098	. . . . . . . . 9 ⊢ (𝐹((oppCat‘(𝐷 FuncCat 𝐶)) Func (𝑃 FuncCat 𝑂))𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ ((oppCat‘(𝐷 FuncCat 𝐶)) Func (𝑃 FuncCat 𝑂)))
19	17, 18	sylib 220	. . . . . . . 8 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ ((oppCat‘(𝐷 FuncCat 𝐶)) Func (𝑃 FuncCat 𝑂)))
20	12, 19	cofucl 17912	. . . . . . 7 ⊢ (𝜑 → (⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿)) ∈ (𝑂 Func (𝑃 FuncCat 𝑂)))
21	20	func1st2nd 49658	. . . . . 6 ⊢ (𝜑 → (1st ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿)))(𝑂 Func (𝑃 FuncCat 𝑂))(2nd ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿))))
22	3, 5, 21	funcf1 17890	. . . . 5 ⊢ (𝜑 → (1st ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿))):(Base‘𝐶)⟶(𝑃 Func 𝑂))
23	22	ffnd 6687	. . . 4 ⊢ (𝜑 → (1st ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿))) Fn (Base‘𝐶))
24		eqid 2761	. . . . . . . 8 ⊢ (𝑂Δfunc𝑃) = (𝑂Δfunc𝑃)
25	1	oppccat 17745	. . . . . . . . 9 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
26	8, 25	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑂 ∈ Cat)
27	13	oppccat 17745	. . . . . . . . 9 ⊢ (𝐷 ∈ Cat → 𝑃 ∈ Cat)
28	9, 27	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ Cat)
29	24, 26, 28, 4	diagcl 18264	. . . . . . 7 ⊢ (𝜑 → (𝑂Δfunc𝑃) ∈ (𝑂 Func (𝑃 FuncCat 𝑂)))
30	29	func1st2nd 49658	. . . . . 6 ⊢ (𝜑 → (1st ‘(𝑂Δfunc𝑃))(𝑂 Func (𝑃 FuncCat 𝑂))(2nd ‘(𝑂Δfunc𝑃)))
31	3, 5, 30	funcf1 17890	. . . . 5 ⊢ (𝜑 → (1st ‘(𝑂Δfunc𝑃)):(Base‘𝐶)⟶(𝑃 Func 𝑂))
32	31	ffnd 6687	. . . 4 ⊢ (𝜑 → (1st ‘(𝑂Δfunc𝑃)) Fn (Base‘𝐶))
33	12	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ( oppFunc ‘𝐿) ∈ (𝑂 Func (oppCat‘(𝐷 FuncCat 𝐶))))
34	19	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ⟨𝐹, 𝐺⟩ ∈ ((oppCat‘(𝐷 FuncCat 𝐶)) Func (𝑃 FuncCat 𝑂)))
35		simpr 488	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
36	3, 33, 34, 35	cofu1 17908	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿)))‘𝑥) = ((1st ‘⟨𝐹, 𝐺⟩)‘((1st ‘( oppFunc ‘𝐿))‘𝑥)))
37	17	func1st 49659	. . . . . . 7 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
38	11	oppf1 49721	. . . . . . . 8 ⊢ (𝜑 → (1st ‘( oppFunc ‘𝐿)) = (1st ‘𝐿))
39	38	fveq1d 6864	. . . . . . 7 ⊢ (𝜑 → ((1st ‘( oppFunc ‘𝐿))‘𝑥) = ((1st ‘𝐿)‘𝑥))
40	37, 39	fveq12d 6869	. . . . . 6 ⊢ (𝜑 → ((1st ‘⟨𝐹, 𝐺⟩)‘((1st ‘( oppFunc ‘𝐿))‘𝑥)) = (𝐹‘((1st ‘𝐿)‘𝑥)))
41	40	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘⟨𝐹, 𝐺⟩)‘((1st ‘( oppFunc ‘𝐿))‘𝑥)) = (𝐹‘((1st ‘𝐿)‘𝑥)))
42	8	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
43	9	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
44	15	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐹 = ( oppFunc ↾ (𝐷 Func 𝐶)))
45	1, 13, 7, 42, 43, 44, 2, 35	oppfdiag1 49996	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝐹‘((1st ‘𝐿)‘𝑥)) = ((1st ‘(𝑂Δfunc𝑃))‘𝑥))
46	36, 41, 45	3eqtrd 2800	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿)))‘𝑥) = ((1st ‘(𝑂Δfunc𝑃))‘𝑥))
47	23, 32, 46	eqfnfvd 7009	. . 3 ⊢ (𝜑 → (1st ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿))) = (1st ‘(𝑂Δfunc𝑃)))
48	3, 21	funcfn2 17893	. . . 4 ⊢ (𝜑 → (2nd ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿))) Fn ((Base‘𝐶) × (Base‘𝐶)))
49	3, 30	funcfn2 17893	. . . 4 ⊢ (𝜑 → (2nd ‘(𝑂Δfunc𝑃)) Fn ((Base‘𝐶) × (Base‘𝐶)))
50		eqid 2761	. . . . . . . . 9 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
51	50, 1	oppchom 17738	. . . . . . . 8 ⊢ (𝑥(Hom ‘𝑂)𝑦) = (𝑦(Hom ‘𝐶)𝑥)
52	51	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(Hom ‘𝑂)𝑦) = (𝑦(Hom ‘𝐶)𝑥))
53		eqid 2761	. . . . . . . 8 ⊢ (Hom ‘𝑂) = (Hom ‘𝑂)
54		eqid 2761	. . . . . . . . 9 ⊢ (𝑃 Nat 𝑂) = (𝑃 Nat 𝑂)
55	4, 54	fuchom 17988	. . . . . . . 8 ⊢ (𝑃 Nat 𝑂) = (Hom ‘(𝑃 FuncCat 𝑂))
56	21	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿)))(𝑂 Func (𝑃 FuncCat 𝑂))(2nd ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿))))
57		simprl 780	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
58		simprr 782	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
59	3, 53, 55, 56, 57, 58	funcf2 17892	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿)))𝑦):(𝑥(Hom ‘𝑂)𝑦)⟶(((1st ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿)))‘𝑥)(𝑃 Nat 𝑂)((1st ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿)))‘𝑦)))
60	52, 59	feq2dd 6672	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿)))𝑦):(𝑦(Hom ‘𝐶)𝑥)⟶(((1st ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿)))‘𝑥)(𝑃 Nat 𝑂)((1st ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿)))‘𝑦)))
61	60	ffnd 6687	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿)))𝑦) Fn (𝑦(Hom ‘𝐶)𝑥))
62	30	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st ‘(𝑂Δfunc𝑃))(𝑂 Func (𝑃 FuncCat 𝑂))(2nd ‘(𝑂Δfunc𝑃)))
63	3, 53, 55, 62, 57, 58	funcf2 17892	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘(𝑂Δfunc𝑃))𝑦):(𝑥(Hom ‘𝑂)𝑦)⟶(((1st ‘(𝑂Δfunc𝑃))‘𝑥)(𝑃 Nat 𝑂)((1st ‘(𝑂Δfunc𝑃))‘𝑦)))
64	52, 63	feq2dd 6672	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘(𝑂Δfunc𝑃))𝑦):(𝑦(Hom ‘𝐶)𝑥)⟶(((1st ‘(𝑂Δfunc𝑃))‘𝑥)(𝑃 Nat 𝑂)((1st ‘(𝑂Δfunc𝑃))‘𝑦)))
65	64	ffnd 6687	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘(𝑂Δfunc𝑃))𝑦) Fn (𝑦(Hom ‘𝐶)𝑥))
66	12	ad2antrr 736	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → ( oppFunc ‘𝐿) ∈ (𝑂 Func (oppCat‘(𝐷 FuncCat 𝐶))))
67	19	ad2antrr 736	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → ⟨𝐹, 𝐺⟩ ∈ ((oppCat‘(𝐷 FuncCat 𝐶)) Func (𝑃 FuncCat 𝑂)))
68	57	adantr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → 𝑥 ∈ (Base‘𝐶))
69	58	adantr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → 𝑦 ∈ (Base‘𝐶))
70		simpr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥))
71	70, 51	eleqtrrdi 2872	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → 𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦))
72	3, 66, 67, 68, 69, 53, 71	cofu2 17910	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → ((𝑥(2nd ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿)))𝑦)‘𝑓) = ((((1st ‘( oppFunc ‘𝐿))‘𝑥)(2nd ‘⟨𝐹, 𝐺⟩)((1st ‘( oppFunc ‘𝐿))‘𝑦))‘((𝑥(2nd ‘( oppFunc ‘𝐿))𝑦)‘𝑓)))
73	17	func2nd 49660	. . . . . . . . . 10 ⊢ (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
74	38	fveq1d 6864	. . . . . . . . . 10 ⊢ (𝜑 → ((1st ‘( oppFunc ‘𝐿))‘𝑦) = ((1st ‘𝐿)‘𝑦))
75	73, 39, 74	oveq123d 7412	. . . . . . . . 9 ⊢ (𝜑 → (((1st ‘( oppFunc ‘𝐿))‘𝑥)(2nd ‘⟨𝐹, 𝐺⟩)((1st ‘( oppFunc ‘𝐿))‘𝑦)) = (((1st ‘𝐿)‘𝑥)𝐺((1st ‘𝐿)‘𝑦)))
76	11	oppf2 49722	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥(2nd ‘( oppFunc ‘𝐿))𝑦) = (𝑦(2nd ‘𝐿)𝑥))
77	76	fveq1d 6864	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥(2nd ‘( oppFunc ‘𝐿))𝑦)‘𝑓) = ((𝑦(2nd ‘𝐿)𝑥)‘𝑓))
78	75, 77	fveq12d 6869	. . . . . . . 8 ⊢ (𝜑 → ((((1st ‘( oppFunc ‘𝐿))‘𝑥)(2nd ‘⟨𝐹, 𝐺⟩)((1st ‘( oppFunc ‘𝐿))‘𝑦))‘((𝑥(2nd ‘( oppFunc ‘𝐿))𝑦)‘𝑓)) = ((((1st ‘𝐿)‘𝑥)𝐺((1st ‘𝐿)‘𝑦))‘((𝑦(2nd ‘𝐿)𝑥)‘𝑓)))
79	78	ad2antrr 736	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → ((((1st ‘( oppFunc ‘𝐿))‘𝑥)(2nd ‘⟨𝐹, 𝐺⟩)((1st ‘( oppFunc ‘𝐿))‘𝑦))‘((𝑥(2nd ‘( oppFunc ‘𝐿))𝑦)‘𝑓)) = ((((1st ‘𝐿)‘𝑥)𝐺((1st ‘𝐿)‘𝑦))‘((𝑦(2nd ‘𝐿)𝑥)‘𝑓)))
80	16	ad2antrr 736	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → 𝐺 = (𝑚 ∈ (𝐷 Func 𝐶), 𝑛 ∈ (𝐷 Func 𝐶) ↦ ( I ↾ (𝑛𝑁𝑚))))
81	8	ad2antrr 736	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → 𝐶 ∈ Cat)
82	9	ad2antrr 736	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → 𝐷 ∈ Cat)
83		eqid 2761	. . . . . . . . 9 ⊢ ((1st ‘𝐿)‘𝑥) = ((1st ‘𝐿)‘𝑥)
84	7, 81, 82, 2, 68, 83	diag1cl 18265	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → ((1st ‘𝐿)‘𝑥) ∈ (𝐷 Func 𝐶))
85		eqid 2761	. . . . . . . . 9 ⊢ ((1st ‘𝐿)‘𝑦) = ((1st ‘𝐿)‘𝑦)
86	7, 81, 82, 2, 69, 85	diag1cl 18265	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → ((1st ‘𝐿)‘𝑦) ∈ (𝐷 Func 𝐶))
87		eqid 2761	. . . . . . . . 9 ⊢ (Base‘𝐷) = (Base‘𝐷)
88	7, 2, 87, 50, 81, 82, 69, 68, 70	diag2 18268	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → ((𝑦(2nd ‘𝐿)𝑥)‘𝑓) = ((Base‘𝐷) × {𝑓}))
89	7, 2, 87, 50, 81, 82, 69, 68, 70, 14	diag2cl 18269	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → ((Base‘𝐷) × {𝑓}) ∈ (((1st ‘𝐿)‘𝑦)𝑁((1st ‘𝐿)‘𝑥)))
90	80, 84, 86, 88, 89	opf2 49988	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → ((((1st ‘𝐿)‘𝑥)𝐺((1st ‘𝐿)‘𝑦))‘((𝑦(2nd ‘𝐿)𝑥)‘𝑓)) = ((Base‘𝐷) × {𝑓}))
91	72, 79, 90	3eqtrd 2800	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → ((𝑥(2nd ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿)))𝑦)‘𝑓) = ((Base‘𝐷) × {𝑓}))
92	13, 87	oppcbas 17741	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝑃)
93	81, 25	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → 𝑂 ∈ Cat)
94	82, 27	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → 𝑃 ∈ Cat)
95	24, 3, 92, 53, 93, 94, 68, 69, 71	diag2 18268	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → ((𝑥(2nd ‘(𝑂Δfunc𝑃))𝑦)‘𝑓) = ((Base‘𝐷) × {𝑓}))
96	91, 95	eqtr4d 2799	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → ((𝑥(2nd ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿)))𝑦)‘𝑓) = ((𝑥(2nd ‘(𝑂Δfunc𝑃))𝑦)‘𝑓))
97	61, 65, 96	eqfnfvd 7009	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿)))𝑦) = (𝑥(2nd ‘(𝑂Δfunc𝑃))𝑦))
98	48, 49, 97	eqfnovd 49448	. . 3 ⊢ (𝜑 → (2nd ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿))) = (2nd ‘(𝑂Δfunc𝑃)))
99	47, 98	opeq12d 4836	. 2 ⊢ (𝜑 → ⟨(1st ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿))), (2nd ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿)))⟩ = ⟨(1st ‘(𝑂Δfunc𝑃)), (2nd ‘(𝑂Δfunc𝑃))⟩)
100		relfunc 17886	. . 3 ⊢ Rel (𝑂 Func (𝑃 FuncCat 𝑂))
101		1st2nd 8015	. . 3 ⊢ ((Rel (𝑂 Func (𝑃 FuncCat 𝑂)) ∧ (⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿)) ∈ (𝑂 Func (𝑃 FuncCat 𝑂))) → (⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿)) = ⟨(1st ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿))), (2nd ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿)))⟩)
102	100, 20, 101	sylancr 596	. 2 ⊢ (𝜑 → (⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿)) = ⟨(1st ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿))), (2nd ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿)))⟩)
103		1st2nd 8015	. . 3 ⊢ ((Rel (𝑂 Func (𝑃 FuncCat 𝑂)) ∧ (𝑂Δfunc𝑃) ∈ (𝑂 Func (𝑃 FuncCat 𝑂))) → (𝑂Δfunc𝑃) = ⟨(1st ‘(𝑂Δfunc𝑃)), (2nd ‘(𝑂Δfunc𝑃))⟩)
104	100, 29, 103	sylancr 596	. 2 ⊢ (𝜑 → (𝑂Δfunc𝑃) = ⟨(1st ‘(𝑂Δfunc𝑃)), (2nd ‘(𝑂Δfunc𝑃))⟩)
105	99, 102, 104	3eqtr4d 2806	1 ⊢ (𝜑 → (⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿)) = (𝑂Δfunc𝑃))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  {csn 4579  ⟨cop 4585   class class class wbr 5097   I cid 5537   × cxp 5641   ↾ 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  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:  lmddu  50249