| Step | Hyp | Ref
| Expression |
| 1 | | elex 3501 |
. . 3
⊢ (𝐺 ∈ 𝑊 → 𝐺 ∈ V) |
| 2 | 1 | adantr 480 |
. 2
⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → 𝐺 ∈ V) |
| 3 | | isisubgr.v |
. . . . . 6
⊢ 𝑉 = (Vtx‘𝐺) |
| 4 | 3 | fvexi 6920 |
. . . . 5
⊢ 𝑉 ∈ V |
| 5 | 4 | a1i 11 |
. . . 4
⊢ (𝑆 ⊆ 𝑉 → 𝑉 ∈ V) |
| 6 | | id 22 |
. . . 4
⊢ (𝑆 ⊆ 𝑉 → 𝑆 ⊆ 𝑉) |
| 7 | 5, 6 | sselpwd 5328 |
. . 3
⊢ (𝑆 ⊆ 𝑉 → 𝑆 ∈ 𝒫 𝑉) |
| 8 | 7 | adantl 481 |
. 2
⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → 𝑆 ∈ 𝒫 𝑉) |
| 9 | | opex 5469 |
. . 3
⊢
〈𝑆, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆})〉 ∈ V |
| 10 | 9 | a1i 11 |
. 2
⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → 〈𝑆, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆})〉 ∈ V) |
| 11 | | simpr 484 |
. . . 4
⊢ ((𝑔 = 𝐺 ∧ 𝑣 = 𝑆) → 𝑣 = 𝑆) |
| 12 | | fvexd 6921 |
. . . . 5
⊢ ((𝑔 = 𝐺 ∧ 𝑣 = 𝑆) → (iEdg‘𝑔) ∈ V) |
| 13 | | fveq2 6906 |
. . . . . . . . . 10
⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺)) |
| 14 | | isisubgr.e |
. . . . . . . . . 10
⊢ 𝐸 = (iEdg‘𝐺) |
| 15 | 13, 14 | eqtr4di 2795 |
. . . . . . . . 9
⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = 𝐸) |
| 16 | 15 | eqeq2d 2748 |
. . . . . . . 8
⊢ (𝑔 = 𝐺 → (𝑒 = (iEdg‘𝑔) ↔ 𝑒 = 𝐸)) |
| 17 | 16 | adantr 480 |
. . . . . . 7
⊢ ((𝑔 = 𝐺 ∧ 𝑣 = 𝑆) → (𝑒 = (iEdg‘𝑔) ↔ 𝑒 = 𝐸)) |
| 18 | | simpr 484 |
. . . . . . . . . 10
⊢ ((𝑣 = 𝑆 ∧ 𝑒 = 𝐸) → 𝑒 = 𝐸) |
| 19 | | dmeq 5914 |
. . . . . . . . . . . 12
⊢ (𝑒 = 𝐸 → dom 𝑒 = dom 𝐸) |
| 20 | 19 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝑣 = 𝑆 ∧ 𝑒 = 𝐸) → dom 𝑒 = dom 𝐸) |
| 21 | | fveq1 6905 |
. . . . . . . . . . . . 13
⊢ (𝑒 = 𝐸 → (𝑒‘𝑥) = (𝐸‘𝑥)) |
| 22 | 21 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝑣 = 𝑆 ∧ 𝑒 = 𝐸) → (𝑒‘𝑥) = (𝐸‘𝑥)) |
| 23 | | simpl 482 |
. . . . . . . . . . . 12
⊢ ((𝑣 = 𝑆 ∧ 𝑒 = 𝐸) → 𝑣 = 𝑆) |
| 24 | 22, 23 | sseq12d 4017 |
. . . . . . . . . . 11
⊢ ((𝑣 = 𝑆 ∧ 𝑒 = 𝐸) → ((𝑒‘𝑥) ⊆ 𝑣 ↔ (𝐸‘𝑥) ⊆ 𝑆)) |
| 25 | 20, 24 | rabeqbidv 3455 |
. . . . . . . . . 10
⊢ ((𝑣 = 𝑆 ∧ 𝑒 = 𝐸) → {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) ⊆ 𝑣} = {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆}) |
| 26 | 18, 25 | reseq12d 5998 |
. . . . . . . . 9
⊢ ((𝑣 = 𝑆 ∧ 𝑒 = 𝐸) → (𝑒 ↾ {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) ⊆ 𝑣}) = (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆})) |
| 27 | 26 | ex 412 |
. . . . . . . 8
⊢ (𝑣 = 𝑆 → (𝑒 = 𝐸 → (𝑒 ↾ {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) ⊆ 𝑣}) = (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆}))) |
| 28 | 27 | adantl 481 |
. . . . . . 7
⊢ ((𝑔 = 𝐺 ∧ 𝑣 = 𝑆) → (𝑒 = 𝐸 → (𝑒 ↾ {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) ⊆ 𝑣}) = (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆}))) |
| 29 | 17, 28 | sylbid 240 |
. . . . . 6
⊢ ((𝑔 = 𝐺 ∧ 𝑣 = 𝑆) → (𝑒 = (iEdg‘𝑔) → (𝑒 ↾ {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) ⊆ 𝑣}) = (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆}))) |
| 30 | 29 | imp 406 |
. . . . 5
⊢ (((𝑔 = 𝐺 ∧ 𝑣 = 𝑆) ∧ 𝑒 = (iEdg‘𝑔)) → (𝑒 ↾ {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) ⊆ 𝑣}) = (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆})) |
| 31 | 12, 30 | csbied 3935 |
. . . 4
⊢ ((𝑔 = 𝐺 ∧ 𝑣 = 𝑆) → ⦋(iEdg‘𝑔) / 𝑒⦌(𝑒 ↾ {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) ⊆ 𝑣}) = (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆})) |
| 32 | 11, 31 | opeq12d 4881 |
. . 3
⊢ ((𝑔 = 𝐺 ∧ 𝑣 = 𝑆) → 〈𝑣, ⦋(iEdg‘𝑔) / 𝑒⦌(𝑒 ↾ {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) ⊆ 𝑣})〉 = 〈𝑆, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆})〉) |
| 33 | | fveq2 6906 |
. . . . 5
⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺)) |
| 34 | 33, 3 | eqtr4di 2795 |
. . . 4
⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉) |
| 35 | 34 | pweqd 4617 |
. . 3
⊢ (𝑔 = 𝐺 → 𝒫 (Vtx‘𝑔) = 𝒫 𝑉) |
| 36 | | df-isubgr 47847 |
. . 3
⊢ ISubGr =
(𝑔 ∈ V, 𝑣 ∈ 𝒫
(Vtx‘𝑔) ↦
〈𝑣,
⦋(iEdg‘𝑔) / 𝑒⦌(𝑒 ↾ {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) ⊆ 𝑣})〉) |
| 37 | 32, 35, 36 | ovmpox 7586 |
. 2
⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ 𝒫 𝑉 ∧ 〈𝑆, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆})〉 ∈ V) → (𝐺 ISubGr 𝑆) = 〈𝑆, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆})〉) |
| 38 | 2, 8, 10, 37 | syl3anc 1373 |
1
⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) = 〈𝑆, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆})〉) |