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Metamath Proof Explorer

Theorem thincmod

Description: At most one morphism in each hom-set (deduction form). (Contributed by Zhi Wang, 21-Sep-2024)

Ref Expression
Hypotheses thincmo.c ⊢ ( 𝜑 → 𝐶 ∈ ThinCat )
thincmo.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
thincmo.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
thincn0eu.b ⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐶 ) )
thincn0eu.h ⊢ ( 𝜑 → 𝐻 = ( Hom ‘ 𝐶 ) )
Assertion thincmod ( 𝜑 → ∃* 𝑓 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) )

Proof

Step Hyp Ref Expression
1 thincmo.c ⊢ ( 𝜑 → 𝐶 ∈ ThinCat )
2 thincmo.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
3 thincmo.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
4 thincn0eu.b ⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐶 ) )
5 thincn0eu.h ⊢ ( 𝜑 → 𝐻 = ( Hom ‘ 𝐶 ) )
6 2 4 eleqtrd ⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ 𝐶 ) )
7 3 4 eleqtrd ⊢ ( 𝜑 → 𝑌 ∈ ( Base ‘ 𝐶 ) )
8 eqid ⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 )
9 eqid ⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 )
10 1 6 7 8 9 thincmo ⊢ ( 𝜑 → ∃* 𝑓 𝑓 ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) )
11 5 oveqd ⊢ ( 𝜑 → ( 𝑋 𝐻 𝑌 ) = ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) )
12 11 eleq2d ⊢ ( 𝜑 → ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ↔ 𝑓 ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) ) )
13 12 mobidv ⊢ ( 𝜑 → ( ∃* 𝑓 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ↔ ∃* 𝑓 𝑓 ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) ) )
14 10 13 mpbird ⊢ ( 𝜑 → ∃* 𝑓 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) )