| Step | Hyp | Ref
| Expression |
| 1 | | vex 3484 |
. . . . . . 7
⊢ 𝑓 ∈ V |
| 2 | 1 | dmex 7931 |
. . . . . 6
⊢ dom 𝑓 ∈ V |
| 3 | 2 | a1i 11 |
. . . . 5
⊢
((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → dom 𝑓 ∈ V) |
| 4 | | simpr 484 |
. . . . 5
⊢
((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → 𝑓:dom 𝑓⟶Comp) |
| 5 | | fvex 6919 |
. . . . . . . 8
⊢
(∏t‘𝑓) ∈ V |
| 6 | 5 | uniex 7761 |
. . . . . . 7
⊢ ∪ (∏t‘𝑓) ∈ V |
| 7 | | acufl 23925 |
. . . . . . . 8
⊢
(CHOICE → UFL = V) |
| 8 | 7 | adantr 480 |
. . . . . . 7
⊢
((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → UFL =
V) |
| 9 | 6, 8 | eleqtrrid 2848 |
. . . . . 6
⊢
((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → ∪ (∏t‘𝑓) ∈ UFL) |
| 10 | | simpl 482 |
. . . . . . . 8
⊢
((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) →
CHOICE) |
| 11 | | dfac10 10178 |
. . . . . . . 8
⊢
(CHOICE ↔ dom card = V) |
| 12 | 10, 11 | sylib 218 |
. . . . . . 7
⊢
((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → dom card =
V) |
| 13 | 6, 12 | eleqtrrid 2848 |
. . . . . 6
⊢
((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → ∪ (∏t‘𝑓) ∈ dom card) |
| 14 | 9, 13 | elind 4200 |
. . . . 5
⊢
((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → ∪ (∏t‘𝑓) ∈ (UFL ∩ dom
card)) |
| 15 | | eqid 2737 |
. . . . . 6
⊢
(∏t‘𝑓) = (∏t‘𝑓) |
| 16 | | eqid 2737 |
. . . . . 6
⊢ ∪ (∏t‘𝑓) = ∪
(∏t‘𝑓) |
| 17 | 15, 16 | ptcmpg 24065 |
. . . . 5
⊢ ((dom
𝑓 ∈ V ∧ 𝑓:dom 𝑓⟶Comp ∧ ∪ (∏t‘𝑓) ∈ (UFL ∩ dom card)) →
(∏t‘𝑓) ∈ Comp) |
| 18 | 3, 4, 14, 17 | syl3anc 1373 |
. . . 4
⊢
((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) →
(∏t‘𝑓) ∈ Comp) |
| 19 | 18 | ex 412 |
. . 3
⊢
(CHOICE → (𝑓:dom 𝑓⟶Comp →
(∏t‘𝑓) ∈ Comp)) |
| 20 | 19 | alrimiv 1927 |
. 2
⊢
(CHOICE → ∀𝑓(𝑓:dom 𝑓⟶Comp →
(∏t‘𝑓) ∈ Comp)) |
| 21 | | fvex 6919 |
. . . . . . . . 9
⊢ (𝑔‘𝑦) ∈ V |
| 22 | | kelac2lem 43076 |
. . . . . . . . 9
⊢ ((𝑔‘𝑦) ∈ V → (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}}) ∈ Comp) |
| 23 | 21, 22 | mp1i 13 |
. . . . . . . 8
⊢ (((Fun
𝑔 ∧ ∅ ∉ ran
𝑔) ∧ 𝑦 ∈ dom 𝑔) → (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}}) ∈ Comp) |
| 24 | 23 | fmpttd 7135 |
. . . . . . 7
⊢ ((Fun
𝑔 ∧ ∅ ∉ ran
𝑔) → (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}})):dom 𝑔⟶Comp) |
| 25 | 24 | ffdmd 6766 |
. . . . . 6
⊢ ((Fun
𝑔 ∧ ∅ ∉ ran
𝑔) → (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}})):dom (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}}))⟶Comp) |
| 26 | | vex 3484 |
. . . . . . . . 9
⊢ 𝑔 ∈ V |
| 27 | 26 | dmex 7931 |
. . . . . . . 8
⊢ dom 𝑔 ∈ V |
| 28 | 27 | mptex 7243 |
. . . . . . 7
⊢ (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}})) ∈ V |
| 29 | | id 22 |
. . . . . . . . 9
⊢ (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}})) → 𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}}))) |
| 30 | | dmeq 5914 |
. . . . . . . . 9
⊢ (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}})) → dom 𝑓 = dom (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}}))) |
| 31 | 29, 30 | feq12d 6724 |
. . . . . . . 8
⊢ (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}})) → (𝑓:dom 𝑓⟶Comp ↔ (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}})):dom (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}}))⟶Comp)) |
| 32 | | fveq2 6906 |
. . . . . . . . 9
⊢ (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}})) → (∏t‘𝑓) =
(∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}})))) |
| 33 | 32 | eleq1d 2826 |
. . . . . . . 8
⊢ (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}})) → ((∏t‘𝑓) ∈ Comp ↔
(∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}}))) ∈ Comp)) |
| 34 | 31, 33 | imbi12d 344 |
. . . . . . 7
⊢ (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}})) → ((𝑓:dom 𝑓⟶Comp →
(∏t‘𝑓) ∈ Comp) ↔ ((𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}})):dom (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}}))⟶Comp →
(∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}}))) ∈ Comp))) |
| 35 | 28, 34 | spcv 3605 |
. . . . . 6
⊢
(∀𝑓(𝑓:dom 𝑓⟶Comp →
(∏t‘𝑓) ∈ Comp) → ((𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}})):dom (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}}))⟶Comp →
(∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}}))) ∈ Comp)) |
| 36 | 25, 35 | syl5com 31 |
. . . . 5
⊢ ((Fun
𝑔 ∧ ∅ ∉ ran
𝑔) → (∀𝑓(𝑓:dom 𝑓⟶Comp →
(∏t‘𝑓) ∈ Comp) →
(∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}}))) ∈ Comp)) |
| 37 | | fvex 6919 |
. . . . . . . 8
⊢ (𝑔‘𝑥) ∈ V |
| 38 | 37 | a1i 11 |
. . . . . . 7
⊢ ((((Fun
𝑔 ∧ ∅ ∉ ran
𝑔) ∧
(∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}}))) ∈ Comp) ∧ 𝑥 ∈ dom 𝑔) → (𝑔‘𝑥) ∈ V) |
| 39 | | df-nel 3047 |
. . . . . . . . . . 11
⊢ (∅
∉ ran 𝑔 ↔ ¬
∅ ∈ ran 𝑔) |
| 40 | 39 | biimpi 216 |
. . . . . . . . . 10
⊢ (∅
∉ ran 𝑔 → ¬
∅ ∈ ran 𝑔) |
| 41 | 40 | ad2antlr 727 |
. . . . . . . . 9
⊢ (((Fun
𝑔 ∧ ∅ ∉ ran
𝑔) ∧ 𝑥 ∈ dom 𝑔) → ¬ ∅ ∈ ran 𝑔) |
| 42 | | fvelrn 7096 |
. . . . . . . . . . . 12
⊢ ((Fun
𝑔 ∧ 𝑥 ∈ dom 𝑔) → (𝑔‘𝑥) ∈ ran 𝑔) |
| 43 | 42 | adantlr 715 |
. . . . . . . . . . 11
⊢ (((Fun
𝑔 ∧ ∅ ∉ ran
𝑔) ∧ 𝑥 ∈ dom 𝑔) → (𝑔‘𝑥) ∈ ran 𝑔) |
| 44 | | eleq1 2829 |
. . . . . . . . . . 11
⊢ ((𝑔‘𝑥) = ∅ → ((𝑔‘𝑥) ∈ ran 𝑔 ↔ ∅ ∈ ran 𝑔)) |
| 45 | 43, 44 | syl5ibcom 245 |
. . . . . . . . . 10
⊢ (((Fun
𝑔 ∧ ∅ ∉ ran
𝑔) ∧ 𝑥 ∈ dom 𝑔) → ((𝑔‘𝑥) = ∅ → ∅ ∈ ran 𝑔)) |
| 46 | 45 | necon3bd 2954 |
. . . . . . . . 9
⊢ (((Fun
𝑔 ∧ ∅ ∉ ran
𝑔) ∧ 𝑥 ∈ dom 𝑔) → (¬ ∅ ∈ ran 𝑔 → (𝑔‘𝑥) ≠ ∅)) |
| 47 | 41, 46 | mpd 15 |
. . . . . . . 8
⊢ (((Fun
𝑔 ∧ ∅ ∉ ran
𝑔) ∧ 𝑥 ∈ dom 𝑔) → (𝑔‘𝑥) ≠ ∅) |
| 48 | 47 | adantlr 715 |
. . . . . . 7
⊢ ((((Fun
𝑔 ∧ ∅ ∉ ran
𝑔) ∧
(∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}}))) ∈ Comp) ∧ 𝑥 ∈ dom 𝑔) → (𝑔‘𝑥) ≠ ∅) |
| 49 | | fveq2 6906 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑥 → (𝑔‘𝑦) = (𝑔‘𝑥)) |
| 50 | 49 | unieqd 4920 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 = 𝑥 → ∪ (𝑔‘𝑦) = ∪ (𝑔‘𝑥)) |
| 51 | 50 | pweqd 4617 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = 𝑥 → 𝒫 ∪ (𝑔‘𝑦) = 𝒫 ∪
(𝑔‘𝑥)) |
| 52 | 51 | sneqd 4638 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑥 → {𝒫 ∪ (𝑔‘𝑦)} = {𝒫 ∪
(𝑔‘𝑥)}) |
| 53 | 49, 52 | preq12d 4741 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑥 → {(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}} = {(𝑔‘𝑥), {𝒫 ∪
(𝑔‘𝑥)}}) |
| 54 | 53 | fveq2d 6910 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑥 → (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}}) = (topGen‘{(𝑔‘𝑥), {𝒫 ∪
(𝑔‘𝑥)}})) |
| 55 | 54 | cbvmptv 5255 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}})) = (𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑥), {𝒫 ∪
(𝑔‘𝑥)}})) |
| 56 | 55 | fveq2i 6909 |
. . . . . . . . . 10
⊢
(∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}}))) = (∏t‘(𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑥), {𝒫 ∪
(𝑔‘𝑥)}}))) |
| 57 | 56 | eleq1i 2832 |
. . . . . . . . 9
⊢
((∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}}))) ∈ Comp ↔
(∏t‘(𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑥), {𝒫 ∪
(𝑔‘𝑥)}}))) ∈ Comp) |
| 58 | 57 | biimpi 216 |
. . . . . . . 8
⊢
((∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}}))) ∈ Comp →
(∏t‘(𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑥), {𝒫 ∪
(𝑔‘𝑥)}}))) ∈ Comp) |
| 59 | 58 | adantl 481 |
. . . . . . 7
⊢ (((Fun
𝑔 ∧ ∅ ∉ ran
𝑔) ∧
(∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}}))) ∈ Comp) →
(∏t‘(𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑥), {𝒫 ∪
(𝑔‘𝑥)}}))) ∈ Comp) |
| 60 | 38, 48, 59 | kelac2 43077 |
. . . . . 6
⊢ (((Fun
𝑔 ∧ ∅ ∉ ran
𝑔) ∧
(∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}}))) ∈ Comp) → X𝑥 ∈
dom 𝑔(𝑔‘𝑥) ≠ ∅) |
| 61 | 60 | ex 412 |
. . . . 5
⊢ ((Fun
𝑔 ∧ ∅ ∉ ran
𝑔) →
((∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}}))) ∈ Comp → X𝑥 ∈
dom 𝑔(𝑔‘𝑥) ≠ ∅)) |
| 62 | 36, 61 | syldc 48 |
. . . 4
⊢
(∀𝑓(𝑓:dom 𝑓⟶Comp →
(∏t‘𝑓) ∈ Comp) → ((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → X𝑥 ∈
dom 𝑔(𝑔‘𝑥) ≠ ∅)) |
| 63 | 62 | alrimiv 1927 |
. . 3
⊢
(∀𝑓(𝑓:dom 𝑓⟶Comp →
(∏t‘𝑓) ∈ Comp) → ∀𝑔((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → X𝑥 ∈ dom 𝑔(𝑔‘𝑥) ≠ ∅)) |
| 64 | | dfac9 10177 |
. . 3
⊢
(CHOICE ↔ ∀𝑔((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → X𝑥 ∈ dom 𝑔(𝑔‘𝑥) ≠ ∅)) |
| 65 | 63, 64 | sylibr 234 |
. 2
⊢
(∀𝑓(𝑓:dom 𝑓⟶Comp →
(∏t‘𝑓) ∈ Comp) →
CHOICE) |
| 66 | 20, 65 | impbii 209 |
1
⊢
(CHOICE ↔ ∀𝑓(𝑓:dom 𝑓⟶Comp →
(∏t‘𝑓) ∈ Comp)) |