Theorem thinccic

Description: In a thin category, two objects are isomorphic iff there are morphisms between them in both directions. (Contributed by Zhi Wang, 25-Sep-2024)

	Ref	Expression
Hypotheses	thincsect.c	⊢ ( 𝜑 → 𝐶 ∈ ThinCat )
	thincsect.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	thincsect.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	thincsect.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
	thinciso.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
Assertion	thinccic	⊢ ( 𝜑 → ( 𝑋 ( ≃𝑐 ‘ 𝐶 ) 𝑌 ↔ ( ( 𝑋 𝐻 𝑌 ) ≠ ∅ ∧ ( 𝑌 𝐻 𝑋 ) ≠ ∅ ) ) )

Proof

Step	Hyp	Ref	Expression
1		thincsect.c	⊢ ( 𝜑 → 𝐶 ∈ ThinCat )
2		thincsect.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
3		thincsect.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
4		thincsect.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
5		thinciso.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
6		eqid	⊢ ( Iso ‘ 𝐶 ) = ( Iso ‘ 𝐶 )
7	1	thinccd	⊢ ( 𝜑 → 𝐶 ∈ Cat )
8	2 5 6 7 3 4	isohom	⊢ ( 𝜑 → ( 𝑋 ( Iso ‘ 𝐶 ) 𝑌 ) ⊆ ( 𝑋 𝐻 𝑌 ) )
9	8	sselda	⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑋 ( Iso ‘ 𝐶 ) 𝑌 ) ) → 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) )
10	1	adantr	⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ) → 𝐶 ∈ ThinCat )
11	3	adantr	⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ) → 𝑋 ∈ 𝐵 )
12	4	adantr	⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ) → 𝑌 ∈ 𝐵 )
13		simpr	⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ) → 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) )
14	10 2 11 12 5 6 13	thinciso	⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ) → ( 𝑓 ∈ ( 𝑋 ( Iso ‘ 𝐶 ) 𝑌 ) ↔ ( 𝑌 𝐻 𝑋 ) ≠ ∅ ) )
15	9 14	biadanid	⊢ ( 𝜑 → ( 𝑓 ∈ ( 𝑋 ( Iso ‘ 𝐶 ) 𝑌 ) ↔ ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ∧ ( 𝑌 𝐻 𝑋 ) ≠ ∅ ) ) )
16	15	exbidv	⊢ ( 𝜑 → ( ∃ 𝑓 𝑓 ∈ ( 𝑋 ( Iso ‘ 𝐶 ) 𝑌 ) ↔ ∃ 𝑓 ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ∧ ( 𝑌 𝐻 𝑋 ) ≠ ∅ ) ) )
17	6 2 7 3 4	cic	⊢ ( 𝜑 → ( 𝑋 ( ≃𝑐 ‘ 𝐶 ) 𝑌 ↔ ∃ 𝑓 𝑓 ∈ ( 𝑋 ( Iso ‘ 𝐶 ) 𝑌 ) ) )
18		n0	⊢ ( ( 𝑋 𝐻 𝑌 ) ≠ ∅ ↔ ∃ 𝑓 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) )
19	18	anbi1i	⊢ ( ( ( 𝑋 𝐻 𝑌 ) ≠ ∅ ∧ ( 𝑌 𝐻 𝑋 ) ≠ ∅ ) ↔ ( ∃ 𝑓 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ∧ ( 𝑌 𝐻 𝑋 ) ≠ ∅ ) )
20		19.41v	⊢ ( ∃ 𝑓 ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ∧ ( 𝑌 𝐻 𝑋 ) ≠ ∅ ) ↔ ( ∃ 𝑓 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ∧ ( 𝑌 𝐻 𝑋 ) ≠ ∅ ) )
21	19 20	bitr4i	⊢ ( ( ( 𝑋 𝐻 𝑌 ) ≠ ∅ ∧ ( 𝑌 𝐻 𝑋 ) ≠ ∅ ) ↔ ∃ 𝑓 ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ∧ ( 𝑌 𝐻 𝑋 ) ≠ ∅ ) )
22	21	a1i	⊢ ( 𝜑 → ( ( ( 𝑋 𝐻 𝑌 ) ≠ ∅ ∧ ( 𝑌 𝐻 𝑋 ) ≠ ∅ ) ↔ ∃ 𝑓 ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ∧ ( 𝑌 𝐻 𝑋 ) ≠ ∅ ) ) )
23	16 17 22	3bitr4d	⊢ ( 𝜑 → ( 𝑋 ( ≃𝑐 ‘ 𝐶 ) 𝑌 ↔ ( ( 𝑋 𝐻 𝑌 ) ≠ ∅ ∧ ( 𝑌 𝐻 𝑋 ) ≠ ∅ ) ) )