Theorem oppchomf

Description: Hom-sets of the opposite category. (Contributed by Mario Carneiro, 17-Jan-2017)

	Ref	Expression
Hypotheses	oppcbas.1	⊢ 𝑂 = ( oppCat ‘ 𝐶 )
	oppchomf.h	⊢ 𝐻 = ( Homf ‘ 𝐶 )
Assertion	oppchomf	⊢ tpos 𝐻 = ( Homf ‘ 𝑂 )

Proof

Step	Hyp	Ref	Expression
1		oppcbas.1	⊢ 𝑂 = ( oppCat ‘ 𝐶 )
2		oppchomf.h	⊢ 𝐻 = ( Homf ‘ 𝐶 )
3		eqid	⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 )
4	3 1	oppchom	⊢ ( 𝑦 ( Hom ‘ 𝑂 ) 𝑥 ) = ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 )
5	4	a1i	⊢ ( ( 𝑦 ∈ ( Base ‘ 𝐶 ) ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) → ( 𝑦 ( Hom ‘ 𝑂 ) 𝑥 ) = ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) )
6	5	mpoeq3ia	⊢ ( 𝑦 ∈ ( Base ‘ 𝐶 ) , 𝑥 ∈ ( Base ‘ 𝐶 ) ↦ ( 𝑦 ( Hom ‘ 𝑂 ) 𝑥 ) ) = ( 𝑦 ∈ ( Base ‘ 𝐶 ) , 𝑥 ∈ ( Base ‘ 𝐶 ) ↦ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) )
7		eqid	⊢ ( Homf ‘ 𝑂 ) = ( Homf ‘ 𝑂 )
8		eqid	⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 )
9	1 8	oppcbas	⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝑂 )
10		eqid	⊢ ( Hom ‘ 𝑂 ) = ( Hom ‘ 𝑂 )
11	7 9 10	homffval	⊢ ( Homf ‘ 𝑂 ) = ( 𝑦 ∈ ( Base ‘ 𝐶 ) , 𝑥 ∈ ( Base ‘ 𝐶 ) ↦ ( 𝑦 ( Hom ‘ 𝑂 ) 𝑥 ) )
12	2 8 3	homffval	⊢ 𝐻 = ( 𝑥 ∈ ( Base ‘ 𝐶 ) , 𝑦 ∈ ( Base ‘ 𝐶 ) ↦ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) )
13	12	tposmpo	⊢ tpos 𝐻 = ( 𝑦 ∈ ( Base ‘ 𝐶 ) , 𝑥 ∈ ( Base ‘ 𝐶 ) ↦ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) )
14	6 11 13	3eqtr4ri	⊢ tpos 𝐻 = ( Homf ‘ 𝑂 )