Step | Hyp | Ref
| Expression |
1 | | elfvex 6800 |
. 2
⊢ (𝐶 ∈ (ACS‘𝑋) → 𝑋 ∈ V) |
2 | | elfvex 6800 |
. . 3
⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ V) |
3 | 2 | adantr 481 |
. 2
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝐶 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))) → 𝑋 ∈ V) |
4 | | fveq2 6767 |
. . . . . 6
⊢ (𝑥 = 𝑋 → (Moore‘𝑥) = (Moore‘𝑋)) |
5 | | pweq 4550 |
. . . . . . . . 9
⊢ (𝑥 = 𝑋 → 𝒫 𝑥 = 𝒫 𝑋) |
6 | 5, 5 | feq23d 6588 |
. . . . . . . 8
⊢ (𝑥 = 𝑋 → (𝑓:𝒫 𝑥⟶𝒫 𝑥 ↔ 𝑓:𝒫 𝑋⟶𝒫 𝑋)) |
7 | 5 | raleqdv 3346 |
. . . . . . . 8
⊢ (𝑥 = 𝑋 → (∀𝑠 ∈ 𝒫 𝑥(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠) ↔ ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))) |
8 | 6, 7 | anbi12d 631 |
. . . . . . 7
⊢ (𝑥 = 𝑋 → ((𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠)) ↔ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠)))) |
9 | 8 | exbidv 1924 |
. . . . . 6
⊢ (𝑥 = 𝑋 → (∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠)) ↔ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠)))) |
10 | 4, 9 | rabeqbidv 3418 |
. . . . 5
⊢ (𝑥 = 𝑋 → {𝑐 ∈ (Moore‘𝑥) ∣ ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))} = {𝑐 ∈ (Moore‘𝑋) ∣ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))}) |
11 | | df-acs 17286 |
. . . . 5
⊢ ACS =
(𝑥 ∈ V ↦ {𝑐 ∈ (Moore‘𝑥) ∣ ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))}) |
12 | | fvex 6780 |
. . . . . 6
⊢
(Moore‘𝑋)
∈ V |
13 | 12 | rabex 5255 |
. . . . 5
⊢ {𝑐 ∈ (Moore‘𝑋) ∣ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))} ∈ V |
14 | 10, 11, 13 | fvmpt 6868 |
. . . 4
⊢ (𝑋 ∈ V →
(ACS‘𝑋) = {𝑐 ∈ (Moore‘𝑋) ∣ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))}) |
15 | 14 | eleq2d 2824 |
. . 3
⊢ (𝑋 ∈ V → (𝐶 ∈ (ACS‘𝑋) ↔ 𝐶 ∈ {𝑐 ∈ (Moore‘𝑋) ∣ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))})) |
16 | | eleq2 2827 |
. . . . . . . 8
⊢ (𝑐 = 𝐶 → (𝑠 ∈ 𝑐 ↔ 𝑠 ∈ 𝐶)) |
17 | 16 | bibi1d 344 |
. . . . . . 7
⊢ (𝑐 = 𝐶 → ((𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠) ↔ (𝑠 ∈ 𝐶 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))) |
18 | 17 | ralbidv 3108 |
. . . . . 6
⊢ (𝑐 = 𝐶 → (∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠) ↔ ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝐶 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))) |
19 | 18 | anbi2d 629 |
. . . . 5
⊢ (𝑐 = 𝐶 → ((𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠)) ↔ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝐶 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠)))) |
20 | 19 | exbidv 1924 |
. . . 4
⊢ (𝑐 = 𝐶 → (∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠)) ↔ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝐶 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠)))) |
21 | 20 | elrab 3624 |
. . 3
⊢ (𝐶 ∈ {𝑐 ∈ (Moore‘𝑋) ∣ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))} ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝐶 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠)))) |
22 | 15, 21 | bitrdi 287 |
. 2
⊢ (𝑋 ∈ V → (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝐶 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))))) |
23 | 1, 3, 22 | pm5.21nii 380 |
1
⊢ (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝐶 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠)))) |