This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem catcisoi

Description: A functor is an isomorphism of categories only if it is full and faithful, and is a bijection on the objects. Remark 3.28(2) in Adamek p. 34. (Contributed by Zhi Wang, 17-Nov-2025)

Ref Expression
Hypotheses catcisoi.c ⊢ 𝐶 = ( CatCat ‘ 𝑈 )
catcisoi.r ⊢ 𝑅 = ( Base ‘ 𝑋 )
catcisoi.s ⊢ 𝑆 = ( Base ‘ 𝑌 )
catcisoi.i ⊢ 𝐼 = ( Iso ‘ 𝐶 )
catcisoi.f ⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐼 𝑌 ) )
Assertion catcisoi ( 𝜑 → ( 𝐹 ∈ ( ( 𝑋 Full 𝑌 ) ∩ ( 𝑋 Faith 𝑌 ) ) ∧ ( 1st ‘ 𝐹 ) : 𝑅 –1-1-onto→ 𝑆 ) )

Proof

Step Hyp Ref Expression
1 catcisoi.c ⊢ 𝐶 = ( CatCat ‘ 𝑈 )
2 catcisoi.r ⊢ 𝑅 = ( Base ‘ 𝑋 )
3 catcisoi.s ⊢ 𝑆 = ( Base ‘ 𝑌 )
4 catcisoi.i ⊢ 𝐼 = ( Iso ‘ 𝐶 )
5 catcisoi.f ⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐼 𝑌 ) )
6 eqid ⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 )
7 4 5 6 isorcl2 ⊢ ( 𝜑 → ( 𝑋 ∈ ( Base ‘ 𝐶 ) ∧ 𝑌 ∈ ( Base ‘ 𝐶 ) ) )
8 7 simpld ⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ 𝐶 ) )
9 1 6 elbasfv ⊢ ( 𝑋 ∈ ( Base ‘ 𝐶 ) → 𝑈 ∈ V )
10 8 9 syl ⊢ ( 𝜑 → 𝑈 ∈ V )
11 7 simprd ⊢ ( 𝜑 → 𝑌 ∈ ( Base ‘ 𝐶 ) )
12 1 6 2 3 10 8 11 4 catciso ⊢ ( 𝜑 → ( 𝐹 ∈ ( 𝑋 𝐼 𝑌 ) ↔ ( 𝐹 ∈ ( ( 𝑋 Full 𝑌 ) ∩ ( 𝑋 Faith 𝑌 ) ) ∧ ( 1st ‘ 𝐹 ) : 𝑅 –1-1-onto→ 𝑆 ) ) )
13 5 12 mpbid ⊢ ( 𝜑 → ( 𝐹 ∈ ( ( 𝑋 Full 𝑌 ) ∩ ( 𝑋 Faith 𝑌 ) ) ∧ ( 1st ‘ 𝐹 ) : 𝑅 –1-1-onto→ 𝑆 ) )