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

Metamath Proof Explorer

Theorem homarel

Description: An arrow is an ordered pair. (Contributed by Mario Carneiro, 11-Jan-2017)

Ref Expression
Hypothesis homahom.h ⊢ 𝐻 = ( Homa ‘ 𝐶 )
Assertion homarel Rel ( 𝑋 𝐻 𝑌 )

Proof

Step Hyp Ref Expression
1 homahom.h ⊢ 𝐻 = ( Homa ‘ 𝐶 )
2 xpss ⊢ ( ( ( Base ‘ 𝐶 ) × ( Base ‘ 𝐶 ) ) × V ) ⊆ ( V × V )
3 eqid ⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 )
4 1 homarcl ⊢ ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) → 𝐶 ∈ Cat )
5 1 3 4 homaf ⊢ ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) → 𝐻 : ( ( Base ‘ 𝐶 ) × ( Base ‘ 𝐶 ) ) ⟶ 𝒫 ( ( ( Base ‘ 𝐶 ) × ( Base ‘ 𝐶 ) ) × V ) )
6 1 3 homarcl2 ⊢ ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) → ( 𝑋 ∈ ( Base ‘ 𝐶 ) ∧ 𝑌 ∈ ( Base ‘ 𝐶 ) ) )
7 6 simpld ⊢ ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) → 𝑋 ∈ ( Base ‘ 𝐶 ) )
8 6 simprd ⊢ ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) → 𝑌 ∈ ( Base ‘ 𝐶 ) )
9 5 7 8 fovcdmd ⊢ ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) → ( 𝑋 𝐻 𝑌 ) ∈ 𝒫 ( ( ( Base ‘ 𝐶 ) × ( Base ‘ 𝐶 ) ) × V ) )
10 elelpwi ⊢ ( ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ∧ ( 𝑋 𝐻 𝑌 ) ∈ 𝒫 ( ( ( Base ‘ 𝐶 ) × ( Base ‘ 𝐶 ) ) × V ) ) → 𝑓 ∈ ( ( ( Base ‘ 𝐶 ) × ( Base ‘ 𝐶 ) ) × V ) )
11 9 10 mpdan ⊢ ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) → 𝑓 ∈ ( ( ( Base ‘ 𝐶 ) × ( Base ‘ 𝐶 ) ) × V ) )
12 2 11 sselid ⊢ ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) → 𝑓 ∈ ( V × V ) )
13 12 ssriv ⊢ ( 𝑋 𝐻 𝑌 ) ⊆ ( V × V )
14 df-rel ⊢ ( Rel ( 𝑋 𝐻 𝑌 ) ↔ ( 𝑋 𝐻 𝑌 ) ⊆ ( V × V ) )
15 13 14 mpbir ⊢ Rel ( 𝑋 𝐻 𝑌 )