Theorem dfinito3

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

	Ref	Expression
Assertion	dfinito3	⊢ InitO = ( TermO ∘ ( oppCat ↾ Cat ) )

Proof

Step	Hyp	Ref	Expression
1		fvres	⊢ ( 𝑐 ∈ Cat → ( ( oppCat ↾ Cat ) ‘ 𝑐 ) = ( oppCat ‘ 𝑐 ) )
2	1	fveq2d	⊢ ( 𝑐 ∈ Cat → ( TermO ‘ ( ( oppCat ↾ Cat ) ‘ 𝑐 ) ) = ( TermO ‘ ( oppCat ‘ 𝑐 ) ) )
3	2	mpteq2ia	⊢ ( 𝑐 ∈ Cat ↦ ( TermO ‘ ( ( oppCat ↾ Cat ) ‘ 𝑐 ) ) ) = ( 𝑐 ∈ Cat ↦ ( TermO ‘ ( oppCat ‘ 𝑐 ) ) )
4		termofn	⊢ TermO Fn Cat
5		dffn2	⊢ ( TermO Fn Cat ↔ TermO : Cat ⟶ V )
6	4 5	mpbi	⊢ TermO : Cat ⟶ V
7		oppccatf	⊢ ( oppCat ↾ Cat ) : Cat ⟶ Cat
8		fcompt	⊢ ( ( TermO : Cat ⟶ V ∧ ( oppCat ↾ Cat ) : Cat ⟶ Cat ) → ( TermO ∘ ( oppCat ↾ Cat ) ) = ( 𝑐 ∈ Cat ↦ ( TermO ‘ ( ( oppCat ↾ Cat ) ‘ 𝑐 ) ) ) )
9	6 7 8	mp2an	⊢ ( TermO ∘ ( oppCat ↾ Cat ) ) = ( 𝑐 ∈ Cat ↦ ( TermO ‘ ( ( oppCat ↾ Cat ) ‘ 𝑐 ) ) )
10		dfinito2	⊢ InitO = ( 𝑐 ∈ Cat ↦ ( TermO ‘ ( oppCat ‘ 𝑐 ) ) )
11	3 9 10	3eqtr4ri	⊢ InitO = ( TermO ∘ ( oppCat ↾ Cat ) )