Description: The set of vertices of an extensible structure with (at least) two slots. (Contributed by AV, 12-Oct-2020) (Revised by AV, 7-Jun-2021) (Revised by AV, 12-Nov-2021)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | funvtxdmge2val | ⊢ ( ( Fun ( 𝐺 ∖ { ∅ } ) ∧ 2 ≤ ( ♯ ‘ dom 𝐺 ) ) → ( Vtx ‘ 𝐺 ) = ( Base ‘ 𝐺 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | vtxval | ⊢ ( Vtx ‘ 𝐺 ) = if ( 𝐺 ∈ ( V × V ) , ( 1st ‘ 𝐺 ) , ( Base ‘ 𝐺 ) ) | |
| 2 | fundmge2nop0 | ⊢ ( ( Fun ( 𝐺 ∖ { ∅ } ) ∧ 2 ≤ ( ♯ ‘ dom 𝐺 ) ) → ¬ 𝐺 ∈ ( V × V ) ) | |
| 3 | 2 | iffalsed | ⊢ ( ( Fun ( 𝐺 ∖ { ∅ } ) ∧ 2 ≤ ( ♯ ‘ dom 𝐺 ) ) → if ( 𝐺 ∈ ( V × V ) , ( 1st ‘ 𝐺 ) , ( Base ‘ 𝐺 ) ) = ( Base ‘ 𝐺 ) ) |
| 4 | 1 3 | syl5eq | ⊢ ( ( Fun ( 𝐺 ∖ { ∅ } ) ∧ 2 ≤ ( ♯ ‘ dom 𝐺 ) ) → ( Vtx ‘ 𝐺 ) = ( Base ‘ 𝐺 ) ) |