Theorem oppfdiag 50882

Description: A diagonal functor for opposite categories is the opposite functor of the diagonal functor for original categories post-composed by an isomorphism (fucoppc 50876). (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 2763	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
3	1, 2	oppcbas 17942	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝑂)
4		eqid 2763	. . . . . . 7 ⊢ (𝑃 FuncCat 𝑂) = (𝑃 FuncCat 𝑂)
5	4	fucbas 18188	. . . . . 6 ⊢ (𝑃 Func 𝑂) = (Base‘(𝑃 FuncCat 𝑂))
6		eqid 2763	. . . . . . . . 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 2763	. . . . . . . . . 10 ⊢ (𝐷 FuncCat 𝐶) = (𝐷 FuncCat 𝐶)
11	7, 8, 9, 10	diagcl 18465	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ (𝐶 Func (𝐷 FuncCat 𝐶)))
12	1, 6, 11	oppfoppc2 50589	. . . . . . . 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 50878	. . . . . . . . 9 ⊢ (𝜑 → 𝐹((oppCat‘(𝐷 FuncCat 𝐶)) Func (𝑃 FuncCat 𝑂))𝐺)
18		df-br 5109	. . . . . . . . 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 221	. . . . . . . 8 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ ((oppCat‘(𝐷 FuncCat 𝐶)) Func (𝑃 FuncCat 𝑂)))
20	12, 19	cofucl 18113	. . . . . . 7 ⊢ (𝜑 → (⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿)) ∈ (𝑂 Func (𝑃 FuncCat 𝑂)))
21	20	func1st2nd 50521	. . . . . 6 ⊢ (𝜑 → (1st ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿)))(𝑂 Func (𝑃 FuncCat 𝑂))(2nd ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿))))
22	3, 5, 21	funcf1 18091	. . . . 5 ⊢ (𝜑 → (1st ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿))):(Base‘𝐶)⟶(𝑃 Func 𝑂))
23	22	ffnd 6740	. . . 4 ⊢ (𝜑 → (1st ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿))) Fn (Base‘𝐶))
24		eqid 2763	. . . . . . . 8 ⊢ (𝑂Δfunc𝑃) = (𝑂Δfunc𝑃)
25	1	oppccat 17946	. . . . . . . . 9 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
26	8, 25	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑂 ∈ Cat)
27	13	oppccat 17946	. . . . . . . . 9 ⊢ (𝐷 ∈ Cat → 𝑃 ∈ Cat)
28	9, 27	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ Cat)
29	24, 26, 28, 4	diagcl 18465	. . . . . . 7 ⊢ (𝜑 → (𝑂Δfunc𝑃) ∈ (𝑂 Func (𝑃 FuncCat 𝑂)))
30	29	func1st2nd 50521	. . . . . 6 ⊢ (𝜑 → (1st ‘(𝑂Δfunc𝑃))(𝑂 Func (𝑃 FuncCat 𝑂))(2nd ‘(𝑂Δfunc𝑃)))
31	3, 5, 30	funcf1 18091	. . . . 5 ⊢ (𝜑 → (1st ‘(𝑂Δfunc𝑃)):(Base‘𝐶)⟶(𝑃 Func 𝑂))
32	31	ffnd 6740	. . . 4 ⊢ (𝜑 → (1st ‘(𝑂Δfunc𝑃)) Fn (Base‘𝐶))
33	12	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ( oppFunc ‘𝐿) ∈ (𝑂 Func (oppCat‘(𝐷 FuncCat 𝐶))))
34	19	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ⟨𝐹, 𝐺⟩ ∈ ((oppCat‘(𝐷 FuncCat 𝐶)) Func (𝑃 FuncCat 𝑂)))
35		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
36	3, 33, 34, 35	cofu1 18109	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿)))‘𝑥) = ((1st ‘⟨𝐹, 𝐺⟩)‘((1st ‘( oppFunc ‘𝐿))‘𝑥)))
37	17	func1st 50522	. . . . . . 7 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
38	11	oppf1 50586	. . . . . . . 8 ⊢ (𝜑 → (1st ‘( oppFunc ‘𝐿)) = (1st ‘𝐿))
39	38	fveq1d 6918	. . . . . . 7 ⊢ (𝜑 → ((1st ‘( oppFunc ‘𝐿))‘𝑥) = ((1st ‘𝐿)‘𝑥))
40	37, 39	fveq12d 6923	. . . . . 6 ⊢ (𝜑 → ((1st ‘⟨𝐹, 𝐺⟩)‘((1st ‘( oppFunc ‘𝐿))‘𝑥)) = (𝐹‘((1st ‘𝐿)‘𝑥)))
41	40	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘⟨𝐹, 𝐺⟩)‘((1st ‘( oppFunc ‘𝐿))‘𝑥)) = (𝐹‘((1st ‘𝐿)‘𝑥)))
42	8	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
43	9	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
44	15	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐹 = ( oppFunc ↾ (𝐷 Func 𝐶)))
45	1, 13, 7, 42, 43, 44, 2, 35	oppfdiag1 50880	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝐹‘((1st ‘𝐿)‘𝑥)) = ((1st ‘(𝑂Δfunc𝑃))‘𝑥))
46	36, 41, 45	3eqtrd 2802	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿)))‘𝑥) = ((1st ‘(𝑂Δfunc𝑃))‘𝑥))
47	23, 32, 46	eqfnfvd 7063	. . 3 ⊢ (𝜑 → (1st ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿))) = (1st ‘(𝑂Δfunc𝑃)))
48	3, 21	funcfn2 18094	. . . 4 ⊢ (𝜑 → (2nd ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿))) Fn ((Base‘𝐶) × (Base‘𝐶)))
49	3, 30	funcfn2 18094	. . . 4 ⊢ (𝜑 → (2nd ‘(𝑂Δfunc𝑃)) Fn ((Base‘𝐶) × (Base‘𝐶)))
50		eqid 2763	. . . . . . . . 9 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
51	50, 1	oppchom 17939	. . . . . . . 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 2763	. . . . . . . 8 ⊢ (Hom ‘𝑂) = (Hom ‘𝑂)
54		eqid 2763	. . . . . . . . 9 ⊢ (𝑃 Nat 𝑂) = (𝑃 Nat 𝑂)
55	4, 54	fuchom 18189	. . . . . . . 8 ⊢ (𝑃 Nat 𝑂) = (Hom ‘(𝑃 FuncCat 𝑂))
56	21	adantr 486	. . . . . . . 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 783	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
58		simprr 785	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
59	3, 53, 55, 56, 57, 58	funcf2 18093	. . . . . . 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 6725	. . . . . 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 6740	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿)))𝑦) Fn (𝑦(Hom ‘𝐶)𝑥))
62	30	adantr 486	. . . . . . . 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 18093	. . . . . . 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 6725	. . . . . 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 6740	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘(𝑂Δfunc𝑃))𝑦) Fn (𝑦(Hom ‘𝐶)𝑥))
66	12	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → ( oppFunc ‘𝐿) ∈ (𝑂 Func (oppCat‘(𝐷 FuncCat 𝐶))))
67	19	ad2antrr 739	. . . . . . . 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 486	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → 𝑥 ∈ (Base‘𝐶))
69	58	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → 𝑦 ∈ (Base‘𝐶))
70		simpr 490	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥))
71	70, 51	eleqtrrdi 2875	. . . . . . . 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 18111	. . . . . . 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 50523	. . . . . . . . . 10 ⊢ (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
74	38	fveq1d 6918	. . . . . . . . . 10 ⊢ (𝜑 → ((1st ‘( oppFunc ‘𝐿))‘𝑦) = ((1st ‘𝐿)‘𝑦))
75	73, 39, 74	oveq123d 7472	. . . . . . . . 9 ⊢ (𝜑 → (((1st ‘( oppFunc ‘𝐿))‘𝑥)(2nd ‘⟨𝐹, 𝐺⟩)((1st ‘( oppFunc ‘𝐿))‘𝑦)) = (((1st ‘𝐿)‘𝑥)𝐺((1st ‘𝐿)‘𝑦)))
76	11	oppf2 50587	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥(2nd ‘( oppFunc ‘𝐿))𝑦) = (𝑦(2nd ‘𝐿)𝑥))
77	76	fveq1d 6918	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥(2nd ‘( oppFunc ‘𝐿))𝑦)‘𝑓) = ((𝑦(2nd ‘𝐿)𝑥)‘𝑓))
78	75, 77	fveq12d 6923	. . . . . . . 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 739	. . . . . . 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 739	. . . . . . . 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 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → 𝐶 ∈ Cat)
82	9	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → 𝐷 ∈ Cat)
83		eqid 2763	. . . . . . . . 9 ⊢ ((1st ‘𝐿)‘𝑥) = ((1st ‘𝐿)‘𝑥)
84	7, 81, 82, 2, 68, 83	diag1cl 18466	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → ((1st ‘𝐿)‘𝑥) ∈ (𝐷 Func 𝐶))
85		eqid 2763	. . . . . . . . 9 ⊢ ((1st ‘𝐿)‘𝑦) = ((1st ‘𝐿)‘𝑦)
86	7, 81, 82, 2, 69, 85	diag1cl 18466	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → ((1st ‘𝐿)‘𝑦) ∈ (𝐷 Func 𝐶))
87		eqid 2763	. . . . . . . . 9 ⊢ (Base‘𝐷) = (Base‘𝐷)
88	7, 2, 87, 50, 81, 82, 69, 68, 70	diag2 18469	. . . . . . . 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 18470	. . . . . . . 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 50872	. . . . . . 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 2802	. . . . . 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 17942	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝑃)
93	81, 25	syl 18	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) → 𝑂 ∈ Cat)
94	82, 27	syl 18	. . . . . . 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 18469	. . . . . 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 2801	. . . . 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 7063	. . . 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 50308	. . 3 ⊢ (𝜑 → (2nd ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿))) = (2nd ‘(𝑂Δfunc𝑃)))
99	47, 98	opeq12d 4853	. 2 ⊢ (𝜑 → ⟨(1st ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿))), (2nd ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿)))⟩ = ⟨(1st ‘(𝑂Δfunc𝑃)), (2nd ‘(𝑂Δfunc𝑃))⟩)
100		relfunc 18087	. . 3 ⊢ Rel (𝑂 Func (𝑃 FuncCat 𝑂))
101		1st2nd 8084	. . 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 599	. 2 ⊢ (𝜑 → (⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿)) = ⟨(1st ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿))), (2nd ‘(⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿)))⟩)
103		1st2nd 8084	. . 3 ⊢ ((Rel (𝑂 Func (𝑃 FuncCat 𝑂)) ∧ (𝑂Δfunc𝑃) ∈ (𝑂 Func (𝑃 FuncCat 𝑂))) → (𝑂Δfunc𝑃) = ⟨(1st ‘(𝑂Δfunc𝑃)), (2nd ‘(𝑂Δfunc𝑃))⟩)
104	100, 29, 103	sylancr 599	. 2 ⊢ (𝜑 → (𝑂Δfunc𝑃) = ⟨(1st ‘(𝑂Δfunc𝑃)), (2nd ‘(𝑂Δfunc𝑃))⟩)
105	99, 102, 104	3eqtr4d 2808	1 ⊢ (𝜑 → (⟨𝐹, 𝐺⟩ ∘func ( oppFunc ‘𝐿)) = (𝑂Δfunc𝑃))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {csn 4588  ⟨cop 4840   class class class wbr 5108   I cid 5552   × cxp 5656   ↾ cres 5660  Rel wrel 5663  ‘cfv 6566  (class class class)co 7451   ∈ cmpo 7453  1^st c1st 8033  2^nd c2nd 8034  Basecbs 17431  Hom chom 17483  Catccat 17888  oppCatcoppc 17935   Func cfunc 18079   ∘_func ccofu 18081   Nat cnat 18169   FuncCat cfuc 18170  Δ_funccdiag 18436   oppFunc coppf 50569

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-tpos 8281  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-er 8755  df-map 8887  df-ixp 8964  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-2 12445  df-3 12446  df-4 12447  df-5 12448  df-6 12449  df-7 12450  df-8 12451  df-9 12452  df-n0 12647  df-z 12734  df-dec 12855  df-uz 13006  df-fz 13680  df-struct 17369  df-sets 17386  df-slot 17404  df-ndx 17416  df-base 17432  df-hom 17497  df-cco 17498  df-cat 17892  df-cid 17893  df-homf 17894  df-comf 17895  df-oppc 17936  df-sect 17972  df-inv 17973  df-iso 17974  df-func 18083  df-idfu 18084  df-cofu 18085  df-full 18131  df-fth 18132  df-nat 18171  df-fuc 18172  df-catc 18324  df-xpc 18396  df-1stf 18397  df-curf 18438  df-diag 18440  df-oppf 50570

This theorem is used by:  lmddu  51181