Theorem idfu1stf1o

Description: The identity functor/inclusion functor is bijective on objects. (Contributed by Zhi Wang, 16-Nov-2025)

	Ref	Expression
Hypotheses	idfu1stf1o.i	⊢ 𝐼 = ( idfunc ‘ 𝐶 )
	idfu1stf1o.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
Assertion	idfu1stf1o	⊢ ( 𝐶 ∈ Cat → ( 1st ‘ 𝐼 ) : 𝐵 –1-1-onto→ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		idfu1stf1o.i	⊢ 𝐼 = ( idfunc ‘ 𝐶 )
2		idfu1stf1o.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
3		f1oi	⊢ ( I ↾ 𝐵 ) : 𝐵 –1-1-onto→ 𝐵
4		id	⊢ ( 𝐶 ∈ Cat → 𝐶 ∈ Cat )
5	1 2 4	idfu1st	⊢ ( 𝐶 ∈ Cat → ( 1st ‘ 𝐼 ) = ( I ↾ 𝐵 ) )
6	5	f1oeq1d	⊢ ( 𝐶 ∈ Cat → ( ( 1st ‘ 𝐼 ) : 𝐵 –1-1-onto→ 𝐵 ↔ ( I ↾ 𝐵 ) : 𝐵 –1-1-onto→ 𝐵 ) )
7	3 6	mpbiri	⊢ ( 𝐶 ∈ Cat → ( 1st ‘ 𝐼 ) : 𝐵 –1-1-onto→ 𝐵 )