Step | Hyp | Ref
| Expression |
1 | | vex 3203 |
. . . . . . 7
⊢ 𝑓 ∈ V |
2 | 1 | dmex 7099 |
. . . . . 6
⊢ dom 𝑓 ∈ V |
3 | 2 | a1i 11 |
. . . . 5
⊢
((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → dom 𝑓 ∈ V) |
4 | | simpr 477 |
. . . . 5
⊢
((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → 𝑓:dom 𝑓⟶Comp) |
5 | | fvex 6201 |
. . . . . . . 8
⊢
(∏t‘𝑓) ∈ V |
6 | 5 | uniex 6953 |
. . . . . . 7
⊢ ∪ (∏t‘𝑓) ∈ V |
7 | | acufl 21721 |
. . . . . . . 8
⊢
(CHOICE → UFL = V) |
8 | 7 | adantr 481 |
. . . . . . 7
⊢
((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → UFL =
V) |
9 | 6, 8 | syl5eleqr 2708 |
. . . . . 6
⊢
((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → ∪ (∏t‘𝑓) ∈ UFL) |
10 | | simpl 473 |
. . . . . . . 8
⊢
((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) →
CHOICE) |
11 | | dfac10 8959 |
. . . . . . . 8
⊢
(CHOICE ↔ dom card = V) |
12 | 10, 11 | sylib 208 |
. . . . . . 7
⊢
((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → dom card =
V) |
13 | 6, 12 | syl5eleqr 2708 |
. . . . . 6
⊢
((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → ∪ (∏t‘𝑓) ∈ dom card) |
14 | 9, 13 | elind 3798 |
. . . . 5
⊢
((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → ∪ (∏t‘𝑓) ∈ (UFL ∩ dom
card)) |
15 | | eqid 2622 |
. . . . . 6
⊢
(∏t‘𝑓) = (∏t‘𝑓) |
16 | | eqid 2622 |
. . . . . 6
⊢ ∪ (∏t‘𝑓) = ∪
(∏t‘𝑓) |
17 | 15, 16 | ptcmpg 21861 |
. . . . 5
⊢ ((dom
𝑓 ∈ V ∧ 𝑓:dom 𝑓⟶Comp ∧ ∪ (∏t‘𝑓) ∈ (UFL ∩ dom card)) →
(∏t‘𝑓) ∈ Comp) |
18 | 3, 4, 14, 17 | syl3anc 1326 |
. . . 4
⊢
((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) →
(∏t‘𝑓) ∈ Comp) |
19 | 18 | ex 450 |
. . 3
⊢
(CHOICE → (𝑓:dom 𝑓⟶Comp →
(∏t‘𝑓) ∈ Comp)) |
20 | 19 | alrimiv 1855 |
. 2
⊢
(CHOICE → ∀𝑓(𝑓:dom 𝑓⟶Comp →
(∏t‘𝑓) ∈ Comp)) |
21 | | fvex 6201 |
. . . . . . . . . 10
⊢ (𝑔‘𝑦) ∈ V |
22 | | kelac2lem 37634 |
. . . . . . . . . 10
⊢ ((𝑔‘𝑦) ∈ V → (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}}) ∈ Comp) |
23 | 21, 22 | mp1i 13 |
. . . . . . . . 9
⊢ (((Fun
𝑔 ∧ ∅ ∉ ran
𝑔) ∧ 𝑦 ∈ dom 𝑔) → (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}}) ∈ Comp) |
24 | | eqid 2622 |
. . . . . . . . 9
⊢ (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}})) = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}})) |
25 | 23, 24 | fmptd 6385 |
. . . . . . . 8
⊢ ((Fun
𝑔 ∧ ∅ ∉ ran
𝑔) → (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}})):dom 𝑔⟶Comp) |
26 | | ffdm 6062 |
. . . . . . . 8
⊢ ((𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}})):dom 𝑔⟶Comp → ((𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}})):dom (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}}))⟶Comp ∧ dom (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}})) ⊆ dom 𝑔)) |
27 | 25, 26 | syl 17 |
. . . . . . 7
⊢ ((Fun
𝑔 ∧ ∅ ∉ ran
𝑔) → ((𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}})):dom (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}}))⟶Comp ∧ dom (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}})) ⊆ dom 𝑔)) |
28 | 27 | simpld 475 |
. . . . . 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) |
29 | | vex 3203 |
. . . . . . . . 9
⊢ 𝑔 ∈ V |
30 | 29 | dmex 7099 |
. . . . . . . 8
⊢ dom 𝑔 ∈ V |
31 | 30 | mptex 6486 |
. . . . . . 7
⊢ (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}})) ∈ V |
32 | | id 22 |
. . . . . . . . 9
⊢ (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}})) → 𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}}))) |
33 | | dmeq 5324 |
. . . . . . . . 9
⊢ (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}})) → dom 𝑓 = dom (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}}))) |
34 | 32, 33 | feq12d 6033 |
. . . . . . . 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)) |
35 | | fveq2 6191 |
. . . . . . . . 9
⊢ (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}})) → (∏t‘𝑓) =
(∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}})))) |
36 | 35 | eleq1d 2686 |
. . . . . . . 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)) |
37 | 34, 36 | imbi12d 334 |
. . . . . . 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))) |
38 | 31, 37 | spcv 3299 |
. . . . . 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)) |
39 | 28, 38 | 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)) |
40 | | fvex 6201 |
. . . . . . . 8
⊢ (𝑔‘𝑥) ∈ V |
41 | 40 | a1i 11 |
. . . . . . 7
⊢ ((((Fun
𝑔 ∧ ∅ ∉ ran
𝑔) ∧
(∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}}))) ∈ Comp) ∧ 𝑥 ∈ dom 𝑔) → (𝑔‘𝑥) ∈ V) |
42 | | df-nel 2898 |
. . . . . . . . . . 11
⊢ (∅
∉ ran 𝑔 ↔ ¬
∅ ∈ ran 𝑔) |
43 | 42 | biimpi 206 |
. . . . . . . . . 10
⊢ (∅
∉ ran 𝑔 → ¬
∅ ∈ ran 𝑔) |
44 | 43 | ad2antlr 763 |
. . . . . . . . 9
⊢ (((Fun
𝑔 ∧ ∅ ∉ ran
𝑔) ∧ 𝑥 ∈ dom 𝑔) → ¬ ∅ ∈ ran 𝑔) |
45 | | fvelrn 6352 |
. . . . . . . . . . . 12
⊢ ((Fun
𝑔 ∧ 𝑥 ∈ dom 𝑔) → (𝑔‘𝑥) ∈ ran 𝑔) |
46 | 45 | adantlr 751 |
. . . . . . . . . . 11
⊢ (((Fun
𝑔 ∧ ∅ ∉ ran
𝑔) ∧ 𝑥 ∈ dom 𝑔) → (𝑔‘𝑥) ∈ ran 𝑔) |
47 | | eleq1 2689 |
. . . . . . . . . . 11
⊢ ((𝑔‘𝑥) = ∅ → ((𝑔‘𝑥) ∈ ran 𝑔 ↔ ∅ ∈ ran 𝑔)) |
48 | 46, 47 | syl5ibcom 235 |
. . . . . . . . . 10
⊢ (((Fun
𝑔 ∧ ∅ ∉ ran
𝑔) ∧ 𝑥 ∈ dom 𝑔) → ((𝑔‘𝑥) = ∅ → ∅ ∈ ran 𝑔)) |
49 | 48 | necon3bd 2808 |
. . . . . . . . 9
⊢ (((Fun
𝑔 ∧ ∅ ∉ ran
𝑔) ∧ 𝑥 ∈ dom 𝑔) → (¬ ∅ ∈ ran 𝑔 → (𝑔‘𝑥) ≠ ∅)) |
50 | 44, 49 | mpd 15 |
. . . . . . . 8
⊢ (((Fun
𝑔 ∧ ∅ ∉ ran
𝑔) ∧ 𝑥 ∈ dom 𝑔) → (𝑔‘𝑥) ≠ ∅) |
51 | 50 | adantlr 751 |
. . . . . . 7
⊢ ((((Fun
𝑔 ∧ ∅ ∉ ran
𝑔) ∧
(∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}}))) ∈ Comp) ∧ 𝑥 ∈ dom 𝑔) → (𝑔‘𝑥) ≠ ∅) |
52 | | fveq2 6191 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑥 → (𝑔‘𝑦) = (𝑔‘𝑥)) |
53 | 52 | unieqd 4446 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 = 𝑥 → ∪ (𝑔‘𝑦) = ∪ (𝑔‘𝑥)) |
54 | 53 | pweqd 4163 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = 𝑥 → 𝒫 ∪ (𝑔‘𝑦) = 𝒫 ∪
(𝑔‘𝑥)) |
55 | 54 | sneqd 4189 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑥 → {𝒫 ∪ (𝑔‘𝑦)} = {𝒫 ∪
(𝑔‘𝑥)}) |
56 | 52, 55 | preq12d 4276 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑥 → {(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}} = {(𝑔‘𝑥), {𝒫 ∪
(𝑔‘𝑥)}}) |
57 | 56 | fveq2d 6195 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑥 → (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}}) = (topGen‘{(𝑔‘𝑥), {𝒫 ∪
(𝑔‘𝑥)}})) |
58 | 57 | cbvmptv 4750 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}})) = (𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑥), {𝒫 ∪
(𝑔‘𝑥)}})) |
59 | 58 | fveq2i 6194 |
. . . . . . . . . 10
⊢
(∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}}))) = (∏t‘(𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑥), {𝒫 ∪
(𝑔‘𝑥)}}))) |
60 | 59 | eleq1i 2692 |
. . . . . . . . 9
⊢
((∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}}))) ∈ Comp ↔
(∏t‘(𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑥), {𝒫 ∪
(𝑔‘𝑥)}}))) ∈ Comp) |
61 | 60 | biimpi 206 |
. . . . . . . 8
⊢
((∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}}))) ∈ Comp →
(∏t‘(𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑥), {𝒫 ∪
(𝑔‘𝑥)}}))) ∈ Comp) |
62 | 61 | adantl 482 |
. . . . . . 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) |
63 | 41, 51, 62 | kelac2 37635 |
. . . . . 6
⊢ (((Fun
𝑔 ∧ ∅ ∉ ran
𝑔) ∧
(∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}}))) ∈ Comp) → X𝑥 ∈
dom 𝑔(𝑔‘𝑥) ≠ ∅) |
64 | 63 | ex 450 |
. . . . 5
⊢ ((Fun
𝑔 ∧ ∅ ∉ ran
𝑔) →
((∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪
(𝑔‘𝑦)}}))) ∈ Comp → X𝑥 ∈
dom 𝑔(𝑔‘𝑥) ≠ ∅)) |
65 | 39, 64 | syldc 48 |
. . . 4
⊢
(∀𝑓(𝑓:dom 𝑓⟶Comp →
(∏t‘𝑓) ∈ Comp) → ((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → X𝑥 ∈
dom 𝑔(𝑔‘𝑥) ≠ ∅)) |
66 | 65 | alrimiv 1855 |
. . 3
⊢
(∀𝑓(𝑓:dom 𝑓⟶Comp →
(∏t‘𝑓) ∈ Comp) → ∀𝑔((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → X𝑥 ∈ dom 𝑔(𝑔‘𝑥) ≠ ∅)) |
67 | | dfac9 8958 |
. . 3
⊢
(CHOICE ↔ ∀𝑔((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → X𝑥 ∈ dom 𝑔(𝑔‘𝑥) ≠ ∅)) |
68 | 66, 67 | sylibr 224 |
. 2
⊢
(∀𝑓(𝑓:dom 𝑓⟶Comp →
(∏t‘𝑓) ∈ Comp) →
CHOICE) |
69 | 20, 68 | impbii 199 |
1
⊢
(CHOICE ↔ ∀𝑓(𝑓:dom 𝑓⟶Comp →
(∏t‘𝑓) ∈ Comp)) |