| Step | Hyp | Ref
| Expression |
| 1 | | df-3an 1089 |
. 2
⊢ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp ∧ 𝐹 ∈ {𝑐 ∈ (𝑅 GrpHom 𝑆) ∣ 𝑐:𝐵–1-1-onto→𝐶}) ↔ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ 𝐹 ∈ {𝑐 ∈ (𝑅 GrpHom 𝑆) ∣ 𝑐:𝐵–1-1-onto→𝐶})) |
| 2 | | df-gim 19277 |
. . 3
⊢ GrpIso =
(𝑎 ∈ Grp, 𝑏 ∈ Grp ↦ {𝑐 ∈ (𝑎 GrpHom 𝑏) ∣ 𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏)}) |
| 3 | | ovex 7464 |
. . . 4
⊢ (𝑎 GrpHom 𝑏) ∈ V |
| 4 | 3 | rabex 5339 |
. . 3
⊢ {𝑐 ∈ (𝑎 GrpHom 𝑏) ∣ 𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏)} ∈ V |
| 5 | | oveq12 7440 |
. . . 4
⊢ ((𝑎 = 𝑅 ∧ 𝑏 = 𝑆) → (𝑎 GrpHom 𝑏) = (𝑅 GrpHom 𝑆)) |
| 6 | | fveq2 6906 |
. . . . . 6
⊢ (𝑎 = 𝑅 → (Base‘𝑎) = (Base‘𝑅)) |
| 7 | | isgim.b |
. . . . . 6
⊢ 𝐵 = (Base‘𝑅) |
| 8 | 6, 7 | eqtr4di 2795 |
. . . . 5
⊢ (𝑎 = 𝑅 → (Base‘𝑎) = 𝐵) |
| 9 | | fveq2 6906 |
. . . . . 6
⊢ (𝑏 = 𝑆 → (Base‘𝑏) = (Base‘𝑆)) |
| 10 | | isgim.c |
. . . . . 6
⊢ 𝐶 = (Base‘𝑆) |
| 11 | 9, 10 | eqtr4di 2795 |
. . . . 5
⊢ (𝑏 = 𝑆 → (Base‘𝑏) = 𝐶) |
| 12 | | f1oeq23 6839 |
. . . . 5
⊢
(((Base‘𝑎) =
𝐵 ∧ (Base‘𝑏) = 𝐶) → (𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏) ↔ 𝑐:𝐵–1-1-onto→𝐶)) |
| 13 | 8, 11, 12 | syl2an 596 |
. . . 4
⊢ ((𝑎 = 𝑅 ∧ 𝑏 = 𝑆) → (𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏) ↔ 𝑐:𝐵–1-1-onto→𝐶)) |
| 14 | 5, 13 | rabeqbidv 3455 |
. . 3
⊢ ((𝑎 = 𝑅 ∧ 𝑏 = 𝑆) → {𝑐 ∈ (𝑎 GrpHom 𝑏) ∣ 𝑐:(Base‘𝑎)–1-1-onto→(Base‘𝑏)} = {𝑐 ∈ (𝑅 GrpHom 𝑆) ∣ 𝑐:𝐵–1-1-onto→𝐶}) |
| 15 | 2, 4, 14 | elovmpo 7678 |
. 2
⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝑅 ∈ Grp ∧ 𝑆 ∈ Grp ∧ 𝐹 ∈ {𝑐 ∈ (𝑅 GrpHom 𝑆) ∣ 𝑐:𝐵–1-1-onto→𝐶})) |
| 16 | | ghmgrp1 19236 |
. . . . . 6
⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑅 ∈ Grp) |
| 17 | | ghmgrp2 19237 |
. . . . . 6
⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑆 ∈ Grp) |
| 18 | 16, 17 | jca 511 |
. . . . 5
⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝑅 ∈ Grp ∧ 𝑆 ∈ Grp)) |
| 19 | 18 | adantr 480 |
. . . 4
⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (𝑅 ∈ Grp ∧ 𝑆 ∈ Grp)) |
| 20 | 19 | pm4.71ri 560 |
. . 3
⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ↔ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶))) |
| 21 | | f1oeq1 6836 |
. . . . 5
⊢ (𝑐 = 𝐹 → (𝑐:𝐵–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐶)) |
| 22 | 21 | elrab 3692 |
. . . 4
⊢ (𝐹 ∈ {𝑐 ∈ (𝑅 GrpHom 𝑆) ∣ 𝑐:𝐵–1-1-onto→𝐶} ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶)) |
| 23 | 22 | anbi2i 623 |
. . 3
⊢ (((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ 𝐹 ∈ {𝑐 ∈ (𝑅 GrpHom 𝑆) ∣ 𝑐:𝐵–1-1-onto→𝐶}) ↔ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶))) |
| 24 | 20, 23 | bitr4i 278 |
. 2
⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) ↔ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ 𝐹 ∈ {𝑐 ∈ (𝑅 GrpHom 𝑆) ∣ 𝑐:𝐵–1-1-onto→𝐶})) |
| 25 | 1, 15, 24 | 3bitr4i 303 |
1
⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶)) |