Theorem idemb

Description: The inclusion functor is an embedding. Remark 4.4(1) in Adamek p. 49. (Contributed by Zhi Wang, 16-Nov-2025)

	Ref	Expression
Hypothesis	idfth.i	⊢ 𝐼 = ( idfunc ‘ 𝐶 )
Assertion	idemb	⊢ ( 𝐼 ∈ ( 𝐷 Func 𝐸 ) → ( 𝐼 ∈ ( 𝐷 Faith 𝐸 ) ∧ Fun ◡ ( 1st ‘ 𝐼 ) ) )

Proof

Step	Hyp	Ref	Expression
1		idfth.i	⊢ 𝐼 = ( idfunc ‘ 𝐶 )
2	1	idfth	⊢ ( 𝐼 ∈ ( 𝐷 Func 𝐸 ) → 𝐼 ∈ ( 𝐷 Faith 𝐸 ) )
3	1	eleq1i	⊢ ( 𝐼 ∈ ( 𝐷 Func 𝐸 ) ↔ ( idfunc ‘ 𝐶 ) ∈ ( 𝐷 Func 𝐸 ) )
4		idfurcl	⊢ ( ( idfunc ‘ 𝐶 ) ∈ ( 𝐷 Func 𝐸 ) → 𝐶 ∈ Cat )
5	3 4	sylbi	⊢ ( 𝐼 ∈ ( 𝐷 Func 𝐸 ) → 𝐶 ∈ Cat )
6		eqid	⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 )
7	1 6	idfu1stf1o	⊢ ( 𝐶 ∈ Cat → ( 1st ‘ 𝐼 ) : ( Base ‘ 𝐶 ) –1-1-onto→ ( Base ‘ 𝐶 ) )
8		dff1o4	⊢ ( ( 1st ‘ 𝐼 ) : ( Base ‘ 𝐶 ) –1-1-onto→ ( Base ‘ 𝐶 ) ↔ ( ( 1st ‘ 𝐼 ) Fn ( Base ‘ 𝐶 ) ∧ ◡ ( 1st ‘ 𝐼 ) Fn ( Base ‘ 𝐶 ) ) )
9	8	simprbi	⊢ ( ( 1st ‘ 𝐼 ) : ( Base ‘ 𝐶 ) –1-1-onto→ ( Base ‘ 𝐶 ) → ◡ ( 1st ‘ 𝐼 ) Fn ( Base ‘ 𝐶 ) )
10	5 7 9	3syl	⊢ ( 𝐼 ∈ ( 𝐷 Func 𝐸 ) → ◡ ( 1st ‘ 𝐼 ) Fn ( Base ‘ 𝐶 ) )
11	10	fnfund	⊢ ( 𝐼 ∈ ( 𝐷 Func 𝐸 ) → Fun ◡ ( 1st ‘ 𝐼 ) )
12	2 11	jca	⊢ ( 𝐼 ∈ ( 𝐷 Func 𝐸 ) → ( 𝐼 ∈ ( 𝐷 Faith 𝐸 ) ∧ Fun ◡ ( 1st ‘ 𝐼 ) ) )