Proof of Theorem lpvtx
| Step | Hyp | Ref
| Expression |
| 1 | | simp1 1137 |
. . . 4
⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → 𝐺 ∈ UHGraph) |
| 2 | | lpvtx.i |
. . . . . . 7
⊢ 𝐼 = (iEdg‘𝐺) |
| 3 | 2 | uhgrfun 29083 |
. . . . . 6
⊢ (𝐺 ∈ UHGraph → Fun 𝐼) |
| 4 | 3 | funfnd 6597 |
. . . . 5
⊢ (𝐺 ∈ UHGraph → 𝐼 Fn dom 𝐼) |
| 5 | 4 | 3ad2ant1 1134 |
. . . 4
⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → 𝐼 Fn dom 𝐼) |
| 6 | | simp2 1138 |
. . . 4
⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → 𝐽 ∈ dom 𝐼) |
| 7 | 2 | uhgrn0 29084 |
. . . 4
⊢ ((𝐺 ∈ UHGraph ∧ 𝐼 Fn dom 𝐼 ∧ 𝐽 ∈ dom 𝐼) → (𝐼‘𝐽) ≠ ∅) |
| 8 | 1, 5, 6, 7 | syl3anc 1373 |
. . 3
⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → (𝐼‘𝐽) ≠ ∅) |
| 9 | | neeq1 3003 |
. . . . 5
⊢ ((𝐼‘𝐽) = {𝐴} → ((𝐼‘𝐽) ≠ ∅ ↔ {𝐴} ≠ ∅)) |
| 10 | 9 | biimpd 229 |
. . . 4
⊢ ((𝐼‘𝐽) = {𝐴} → ((𝐼‘𝐽) ≠ ∅ → {𝐴} ≠ ∅)) |
| 11 | 10 | 3ad2ant3 1136 |
. . 3
⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → ((𝐼‘𝐽) ≠ ∅ → {𝐴} ≠ ∅)) |
| 12 | 8, 11 | mpd 15 |
. 2
⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → {𝐴} ≠ ∅) |
| 13 | | eqid 2737 |
. . . . . 6
⊢
(Vtx‘𝐺) =
(Vtx‘𝐺) |
| 14 | 13, 2 | uhgrss 29081 |
. . . . 5
⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼) → (𝐼‘𝐽) ⊆ (Vtx‘𝐺)) |
| 15 | 14 | 3adant3 1133 |
. . . 4
⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → (𝐼‘𝐽) ⊆ (Vtx‘𝐺)) |
| 16 | | sseq1 4009 |
. . . . 5
⊢ ((𝐼‘𝐽) = {𝐴} → ((𝐼‘𝐽) ⊆ (Vtx‘𝐺) ↔ {𝐴} ⊆ (Vtx‘𝐺))) |
| 17 | 16 | 3ad2ant3 1136 |
. . . 4
⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → ((𝐼‘𝐽) ⊆ (Vtx‘𝐺) ↔ {𝐴} ⊆ (Vtx‘𝐺))) |
| 18 | 15, 17 | mpbid 232 |
. . 3
⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → {𝐴} ⊆ (Vtx‘𝐺)) |
| 19 | | snnzb 4718 |
. . . 4
⊢ (𝐴 ∈ V ↔ {𝐴} ≠ ∅) |
| 20 | | snssg 4783 |
. . . 4
⊢ (𝐴 ∈ V → (𝐴 ∈ (Vtx‘𝐺) ↔ {𝐴} ⊆ (Vtx‘𝐺))) |
| 21 | 19, 20 | sylbir 235 |
. . 3
⊢ ({𝐴} ≠ ∅ → (𝐴 ∈ (Vtx‘𝐺) ↔ {𝐴} ⊆ (Vtx‘𝐺))) |
| 22 | 18, 21 | syl5ibrcom 247 |
. 2
⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → ({𝐴} ≠ ∅ → 𝐴 ∈ (Vtx‘𝐺))) |
| 23 | 12, 22 | mpd 15 |
1
⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → 𝐴 ∈ (Vtx‘𝐺)) |