Theorem cic

Description: Objects X and Y in a category are isomorphic provided that there is an isomorphism f : X --> Y , see definition 3.15 of Adamek p. 29. (Contributed by AV, 4-Apr-2020)

	Ref	Expression
Hypotheses	cic.i	⊢ 𝐼 = ( Iso ‘ 𝐶 )
	cic.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	cic.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
	cic.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	cic.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
Assertion	cic	⊢ ( 𝜑 → ( 𝑋 ( ≃𝑐 ‘ 𝐶 ) 𝑌 ↔ ∃ 𝑓 𝑓 ∈ ( 𝑋 𝐼 𝑌 ) ) )

Proof

Step	Hyp	Ref	Expression
1		cic.i	⊢ 𝐼 = ( Iso ‘ 𝐶 )
2		cic.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
3		cic.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
4		cic.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5		cic.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
6	1 2 3 4 5	brcic	⊢ ( 𝜑 → ( 𝑋 ( ≃𝑐 ‘ 𝐶 ) 𝑌 ↔ ( 𝑋 𝐼 𝑌 ) ≠ ∅ ) )
7		n0	⊢ ( ( 𝑋 𝐼 𝑌 ) ≠ ∅ ↔ ∃ 𝑓 𝑓 ∈ ( 𝑋 𝐼 𝑌 ) )
8	6 7	bitrdi	⊢ ( 𝜑 → ( 𝑋 ( ≃𝑐 ‘ 𝐶 ) 𝑌 ↔ ∃ 𝑓 𝑓 ∈ ( 𝑋 𝐼 𝑌 ) ) )