Theorem oppcval

Description: Value of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017)

	Ref	Expression
Hypotheses	oppcval.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	oppcval.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
	oppcval.x	⊢ · = ( comp ‘ 𝐶 )
	oppcval.o	⊢ 𝑂 = ( oppCat ‘ 𝐶 )
Assertion	oppcval	⊢ ( 𝐶 ∈ 𝑉 → 𝑂 = ( ( 𝐶 sSet ⟨ ( Hom ‘ ndx ) , tpos 𝐻 ⟩ ) sSet ⟨ ( comp ‘ ndx ) , ( 𝑢 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ tpos ( ⟨ 𝑧 , ( 2nd ‘ 𝑢 ) ⟩ · ( 1st ‘ 𝑢 ) ) ) ⟩ ) )

Proof

Step	Hyp	Ref	Expression
1		oppcval.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
2		oppcval.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
3		oppcval.x	⊢ · = ( comp ‘ 𝐶 )
4		oppcval.o	⊢ 𝑂 = ( oppCat ‘ 𝐶 )
5		elex	⊢ ( 𝐶 ∈ 𝑉 → 𝐶 ∈ V )
6		id	⊢ ( 𝑐 = 𝐶 → 𝑐 = 𝐶 )
7		fveq2	⊢ ( 𝑐 = 𝐶 → ( Hom ‘ 𝑐 ) = ( Hom ‘ 𝐶 ) )
8	7 2	eqtr4di	⊢ ( 𝑐 = 𝐶 → ( Hom ‘ 𝑐 ) = 𝐻 )
9	8	tposeqd	⊢ ( 𝑐 = 𝐶 → tpos ( Hom ‘ 𝑐 ) = tpos 𝐻 )
10	9	opeq2d	⊢ ( 𝑐 = 𝐶 → ⟨ ( Hom ‘ ndx ) , tpos ( Hom ‘ 𝑐 ) ⟩ = ⟨ ( Hom ‘ ndx ) , tpos 𝐻 ⟩ )
11	6 10	oveq12d	⊢ ( 𝑐 = 𝐶 → ( 𝑐 sSet ⟨ ( Hom ‘ ndx ) , tpos ( Hom ‘ 𝑐 ) ⟩ ) = ( 𝐶 sSet ⟨ ( Hom ‘ ndx ) , tpos 𝐻 ⟩ ) )
12		fveq2	⊢ ( 𝑐 = 𝐶 → ( Base ‘ 𝑐 ) = ( Base ‘ 𝐶 ) )
13	12 1	eqtr4di	⊢ ( 𝑐 = 𝐶 → ( Base ‘ 𝑐 ) = 𝐵 )
14	13	sqxpeqd	⊢ ( 𝑐 = 𝐶 → ( ( Base ‘ 𝑐 ) × ( Base ‘ 𝑐 ) ) = ( 𝐵 × 𝐵 ) )
15		fveq2	⊢ ( 𝑐 = 𝐶 → ( comp ‘ 𝑐 ) = ( comp ‘ 𝐶 ) )
16	15 3	eqtr4di	⊢ ( 𝑐 = 𝐶 → ( comp ‘ 𝑐 ) = · )
17	16	oveqd	⊢ ( 𝑐 = 𝐶 → ( ⟨ 𝑧 , ( 2nd ‘ 𝑢 ) ⟩ ( comp ‘ 𝑐 ) ( 1st ‘ 𝑢 ) ) = ( ⟨ 𝑧 , ( 2nd ‘ 𝑢 ) ⟩ · ( 1st ‘ 𝑢 ) ) )
18	17	tposeqd	⊢ ( 𝑐 = 𝐶 → tpos ( ⟨ 𝑧 , ( 2nd ‘ 𝑢 ) ⟩ ( comp ‘ 𝑐 ) ( 1st ‘ 𝑢 ) ) = tpos ( ⟨ 𝑧 , ( 2nd ‘ 𝑢 ) ⟩ · ( 1st ‘ 𝑢 ) ) )
19	14 13 18	mpoeq123dv	⊢ ( 𝑐 = 𝐶 → ( 𝑢 ∈ ( ( Base ‘ 𝑐 ) × ( Base ‘ 𝑐 ) ) , 𝑧 ∈ ( Base ‘ 𝑐 ) ↦ tpos ( ⟨ 𝑧 , ( 2nd ‘ 𝑢 ) ⟩ ( comp ‘ 𝑐 ) ( 1st ‘ 𝑢 ) ) ) = ( 𝑢 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ tpos ( ⟨ 𝑧 , ( 2nd ‘ 𝑢 ) ⟩ · ( 1st ‘ 𝑢 ) ) ) )
20	19	opeq2d	⊢ ( 𝑐 = 𝐶 → ⟨ ( comp ‘ ndx ) , ( 𝑢 ∈ ( ( Base ‘ 𝑐 ) × ( Base ‘ 𝑐 ) ) , 𝑧 ∈ ( Base ‘ 𝑐 ) ↦ tpos ( ⟨ 𝑧 , ( 2nd ‘ 𝑢 ) ⟩ ( comp ‘ 𝑐 ) ( 1st ‘ 𝑢 ) ) ) ⟩ = ⟨ ( comp ‘ ndx ) , ( 𝑢 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ tpos ( ⟨ 𝑧 , ( 2nd ‘ 𝑢 ) ⟩ · ( 1st ‘ 𝑢 ) ) ) ⟩ )
21	11 20	oveq12d	⊢ ( 𝑐 = 𝐶 → ( ( 𝑐 sSet ⟨ ( Hom ‘ ndx ) , tpos ( Hom ‘ 𝑐 ) ⟩ ) sSet ⟨ ( comp ‘ ndx ) , ( 𝑢 ∈ ( ( Base ‘ 𝑐 ) × ( Base ‘ 𝑐 ) ) , 𝑧 ∈ ( Base ‘ 𝑐 ) ↦ tpos ( ⟨ 𝑧 , ( 2nd ‘ 𝑢 ) ⟩ ( comp ‘ 𝑐 ) ( 1st ‘ 𝑢 ) ) ) ⟩ ) = ( ( 𝐶 sSet ⟨ ( Hom ‘ ndx ) , tpos 𝐻 ⟩ ) sSet ⟨ ( comp ‘ ndx ) , ( 𝑢 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ tpos ( ⟨ 𝑧 , ( 2nd ‘ 𝑢 ) ⟩ · ( 1st ‘ 𝑢 ) ) ) ⟩ ) )
22		df-oppc	⊢ oppCat = ( 𝑐 ∈ V ↦ ( ( 𝑐 sSet ⟨ ( Hom ‘ ndx ) , tpos ( Hom ‘ 𝑐 ) ⟩ ) sSet ⟨ ( comp ‘ ndx ) , ( 𝑢 ∈ ( ( Base ‘ 𝑐 ) × ( Base ‘ 𝑐 ) ) , 𝑧 ∈ ( Base ‘ 𝑐 ) ↦ tpos ( ⟨ 𝑧 , ( 2nd ‘ 𝑢 ) ⟩ ( comp ‘ 𝑐 ) ( 1st ‘ 𝑢 ) ) ) ⟩ ) )
23		ovex	⊢ ( ( 𝐶 sSet ⟨ ( Hom ‘ ndx ) , tpos 𝐻 ⟩ ) sSet ⟨ ( comp ‘ ndx ) , ( 𝑢 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ tpos ( ⟨ 𝑧 , ( 2nd ‘ 𝑢 ) ⟩ · ( 1st ‘ 𝑢 ) ) ) ⟩ ) ∈ V
24	21 22 23	fvmpt	⊢ ( 𝐶 ∈ V → ( oppCat ‘ 𝐶 ) = ( ( 𝐶 sSet ⟨ ( Hom ‘ ndx ) , tpos 𝐻 ⟩ ) sSet ⟨ ( comp ‘ ndx ) , ( 𝑢 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ tpos ( ⟨ 𝑧 , ( 2nd ‘ 𝑢 ) ⟩ · ( 1st ‘ 𝑢 ) ) ) ⟩ ) )
25	5 24	syl	⊢ ( 𝐶 ∈ 𝑉 → ( oppCat ‘ 𝐶 ) = ( ( 𝐶 sSet ⟨ ( Hom ‘ ndx ) , tpos 𝐻 ⟩ ) sSet ⟨ ( comp ‘ ndx ) , ( 𝑢 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ tpos ( ⟨ 𝑧 , ( 2nd ‘ 𝑢 ) ⟩ · ( 1st ‘ 𝑢 ) ) ) ⟩ ) )
26	4 25	eqtrid	⊢ ( 𝐶 ∈ 𝑉 → 𝑂 = ( ( 𝐶 sSet ⟨ ( Hom ‘ ndx ) , tpos 𝐻 ⟩ ) sSet ⟨ ( comp ‘ ndx ) , ( 𝑢 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ tpos ( ⟨ 𝑧 , ( 2nd ‘ 𝑢 ) ⟩ · ( 1st ‘ 𝑢 ) ) ) ⟩ ) )