Step | Hyp | Ref
| Expression |
1 | | df-subg 18394 |
. . 3
⊢ SubGrp =
(𝑤 ∈ Grp ↦
{𝑠 ∈ 𝒫
(Base‘𝑤) ∣
(𝑤 ↾s
𝑠) ∈
Grp}) |
2 | 1 | mptrcl 6784 |
. 2
⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) |
3 | | simp1 1137 |
. 2
⊢ ((𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp) → 𝐺 ∈ Grp) |
4 | | fveq2 6674 |
. . . . . . . . . 10
⊢ (𝑤 = 𝐺 → (Base‘𝑤) = (Base‘𝐺)) |
5 | | issubg.b |
. . . . . . . . . 10
⊢ 𝐵 = (Base‘𝐺) |
6 | 4, 5 | eqtr4di 2791 |
. . . . . . . . 9
⊢ (𝑤 = 𝐺 → (Base‘𝑤) = 𝐵) |
7 | 6 | pweqd 4507 |
. . . . . . . 8
⊢ (𝑤 = 𝐺 → 𝒫 (Base‘𝑤) = 𝒫 𝐵) |
8 | | oveq1 7177 |
. . . . . . . . 9
⊢ (𝑤 = 𝐺 → (𝑤 ↾s 𝑠) = (𝐺 ↾s 𝑠)) |
9 | 8 | eleq1d 2817 |
. . . . . . . 8
⊢ (𝑤 = 𝐺 → ((𝑤 ↾s 𝑠) ∈ Grp ↔ (𝐺 ↾s 𝑠) ∈ Grp)) |
10 | 7, 9 | rabeqbidv 3387 |
. . . . . . 7
⊢ (𝑤 = 𝐺 → {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp} = {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺 ↾s 𝑠) ∈ Grp}) |
11 | 5 | fvexi 6688 |
. . . . . . . . 9
⊢ 𝐵 ∈ V |
12 | 11 | pwex 5247 |
. . . . . . . 8
⊢ 𝒫
𝐵 ∈ V |
13 | 12 | rabex 5200 |
. . . . . . 7
⊢ {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺 ↾s 𝑠) ∈ Grp} ∈ V |
14 | 10, 1, 13 | fvmpt 6775 |
. . . . . 6
⊢ (𝐺 ∈ Grp →
(SubGrp‘𝐺) = {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺 ↾s 𝑠) ∈ Grp}) |
15 | 14 | eleq2d 2818 |
. . . . 5
⊢ (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ 𝑆 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺 ↾s 𝑠) ∈ Grp})) |
16 | | oveq2 7178 |
. . . . . . . 8
⊢ (𝑠 = 𝑆 → (𝐺 ↾s 𝑠) = (𝐺 ↾s 𝑆)) |
17 | 16 | eleq1d 2817 |
. . . . . . 7
⊢ (𝑠 = 𝑆 → ((𝐺 ↾s 𝑠) ∈ Grp ↔ (𝐺 ↾s 𝑆) ∈ Grp)) |
18 | 17 | elrab 3588 |
. . . . . 6
⊢ (𝑆 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺 ↾s 𝑠) ∈ Grp} ↔ (𝑆 ∈ 𝒫 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp)) |
19 | 11 | elpw2 5213 |
. . . . . . 7
⊢ (𝑆 ∈ 𝒫 𝐵 ↔ 𝑆 ⊆ 𝐵) |
20 | 19 | anbi1i 627 |
. . . . . 6
⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp) ↔ (𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp)) |
21 | 18, 20 | bitri 278 |
. . . . 5
⊢ (𝑆 ∈ {𝑠 ∈ 𝒫 𝐵 ∣ (𝐺 ↾s 𝑠) ∈ Grp} ↔ (𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp)) |
22 | 15, 21 | bitrdi 290 |
. . . 4
⊢ (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp))) |
23 | | ibar 532 |
. . . 4
⊢ (𝐺 ∈ Grp → ((𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp) ↔ (𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp)))) |
24 | 22, 23 | bitrd 282 |
. . 3
⊢ (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp)))) |
25 | | 3anass 1096 |
. . 3
⊢ ((𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp) ↔ (𝐺 ∈ Grp ∧ (𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp))) |
26 | 24, 25 | bitr4di 292 |
. 2
⊢ (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp))) |
27 | 2, 3, 26 | pm5.21nii 383 |
1
⊢ (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp)) |