Theorem dftermo3

Description: An alternate definition of df-termo depending on df-inito , without dummy variables. (Contributed by Zhi Wang, 29-Aug-2024)

	Ref	Expression
Assertion	dftermo3	⊢ TermO = ( InitO ∘ ( oppCat ↾ Cat ) )

Proof

Step	Hyp	Ref	Expression
1		fvres	⊢ ( 𝑐 ∈ Cat → ( ( oppCat ↾ Cat ) ‘ 𝑐 ) = ( oppCat ‘ 𝑐 ) )
2	1	fveq2d	⊢ ( 𝑐 ∈ Cat → ( InitO ‘ ( ( oppCat ↾ Cat ) ‘ 𝑐 ) ) = ( InitO ‘ ( oppCat ‘ 𝑐 ) ) )
3	2	mpteq2ia	⊢ ( 𝑐 ∈ Cat ↦ ( InitO ‘ ( ( oppCat ↾ Cat ) ‘ 𝑐 ) ) ) = ( 𝑐 ∈ Cat ↦ ( InitO ‘ ( oppCat ‘ 𝑐 ) ) )
4		initofn	⊢ InitO Fn Cat
5		dffn2	⊢ ( InitO Fn Cat ↔ InitO : Cat ⟶ V )
6	4 5	mpbi	⊢ InitO : Cat ⟶ V
7		oppccatf	⊢ ( oppCat ↾ Cat ) : Cat ⟶ Cat
8		fcompt	⊢ ( ( InitO : Cat ⟶ V ∧ ( oppCat ↾ Cat ) : Cat ⟶ Cat ) → ( InitO ∘ ( oppCat ↾ Cat ) ) = ( 𝑐 ∈ Cat ↦ ( InitO ‘ ( ( oppCat ↾ Cat ) ‘ 𝑐 ) ) ) )
9	6 7 8	mp2an	⊢ ( InitO ∘ ( oppCat ↾ Cat ) ) = ( 𝑐 ∈ Cat ↦ ( InitO ‘ ( ( oppCat ↾ Cat ) ‘ 𝑐 ) ) )
10		dftermo2	⊢ TermO = ( 𝑐 ∈ Cat ↦ ( InitO ‘ ( oppCat ‘ 𝑐 ) ) )
11	3 9 10	3eqtr4ri	⊢ TermO = ( InitO ∘ ( oppCat ↾ Cat ) )