Step | Hyp | Ref
| Expression |
1 | | tsmsfbas.a |
. 2
⊢ (𝜑 → 𝐴 ∈ 𝑊) |
2 | | elex 3185 |
. 2
⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ V) |
3 | | tsmsfbas.l |
. . 3
⊢ 𝐿 = ran 𝐹 |
4 | | ssrab2 3650 |
. . . . . . 7
⊢ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} ⊆ 𝑆 |
5 | | tsmsfbas.s |
. . . . . . . . . 10
⊢ 𝑆 = (𝒫 𝐴 ∩ Fin) |
6 | | pwexg 4776 |
. . . . . . . . . . 11
⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V) |
7 | | inex1g 4729 |
. . . . . . . . . . 11
⊢
(𝒫 𝐴 ∈
V → (𝒫 𝐴 ∩
Fin) ∈ V) |
8 | 6, 7 | syl 17 |
. . . . . . . . . 10
⊢ (𝐴 ∈ V → (𝒫
𝐴 ∩ Fin) ∈
V) |
9 | 5, 8 | syl5eqel 2692 |
. . . . . . . . 9
⊢ (𝐴 ∈ V → 𝑆 ∈ V) |
10 | 9 | adantr 480 |
. . . . . . . 8
⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) → 𝑆 ∈ V) |
11 | | elpw2g 4754 |
. . . . . . . 8
⊢ (𝑆 ∈ V → ({𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} ∈ 𝒫 𝑆 ↔ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} ⊆ 𝑆)) |
12 | 10, 11 | syl 17 |
. . . . . . 7
⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) → ({𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} ∈ 𝒫 𝑆 ↔ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} ⊆ 𝑆)) |
13 | 4, 12 | mpbiri 247 |
. . . . . 6
⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) → {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} ∈ 𝒫 𝑆) |
14 | | tsmsfbas.f |
. . . . . 6
⊢ 𝐹 = (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}) |
15 | 13, 14 | fmptd 6292 |
. . . . 5
⊢ (𝐴 ∈ V → 𝐹:𝑆⟶𝒫 𝑆) |
16 | | frn 5966 |
. . . . 5
⊢ (𝐹:𝑆⟶𝒫 𝑆 → ran 𝐹 ⊆ 𝒫 𝑆) |
17 | 15, 16 | syl 17 |
. . . 4
⊢ (𝐴 ∈ V → ran 𝐹 ⊆ 𝒫 𝑆) |
18 | | 0ss 3924 |
. . . . . . . . . 10
⊢ ∅
⊆ 𝐴 |
19 | | 0fin 8073 |
. . . . . . . . . 10
⊢ ∅
∈ Fin |
20 | | elfpw 8151 |
. . . . . . . . . 10
⊢ (∅
∈ (𝒫 𝐴 ∩
Fin) ↔ (∅ ⊆ 𝐴 ∧ ∅ ∈ Fin)) |
21 | 18, 19, 20 | mpbir2an 957 |
. . . . . . . . 9
⊢ ∅
∈ (𝒫 𝐴 ∩
Fin) |
22 | 21, 5 | eleqtrri 2687 |
. . . . . . . 8
⊢ ∅
∈ 𝑆 |
23 | | 0ss 3924 |
. . . . . . . . 9
⊢ ∅
⊆ 𝑦 |
24 | 23 | rgenw 2908 |
. . . . . . . 8
⊢
∀𝑦 ∈
𝑆 ∅ ⊆ 𝑦 |
25 | | rabid2 3096 |
. . . . . . . . . 10
⊢ (𝑆 = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} ↔ ∀𝑦 ∈ 𝑆 𝑧 ⊆ 𝑦) |
26 | | sseq1 3589 |
. . . . . . . . . . 11
⊢ (𝑧 = ∅ → (𝑧 ⊆ 𝑦 ↔ ∅ ⊆ 𝑦)) |
27 | 26 | ralbidv 2969 |
. . . . . . . . . 10
⊢ (𝑧 = ∅ → (∀𝑦 ∈ 𝑆 𝑧 ⊆ 𝑦 ↔ ∀𝑦 ∈ 𝑆 ∅ ⊆ 𝑦)) |
28 | 25, 27 | syl5bb 271 |
. . . . . . . . 9
⊢ (𝑧 = ∅ → (𝑆 = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} ↔ ∀𝑦 ∈ 𝑆 ∅ ⊆ 𝑦)) |
29 | 28 | rspcev 3282 |
. . . . . . . 8
⊢ ((∅
∈ 𝑆 ∧
∀𝑦 ∈ 𝑆 ∅ ⊆ 𝑦) → ∃𝑧 ∈ 𝑆 𝑆 = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}) |
30 | 22, 24, 29 | mp2an 704 |
. . . . . . 7
⊢
∃𝑧 ∈
𝑆 𝑆 = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} |
31 | 14 | elrnmpt 5293 |
. . . . . . . 8
⊢ (𝑆 ∈ V → (𝑆 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝑆 𝑆 = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦})) |
32 | 9, 31 | syl 17 |
. . . . . . 7
⊢ (𝐴 ∈ V → (𝑆 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝑆 𝑆 = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦})) |
33 | 30, 32 | mpbiri 247 |
. . . . . 6
⊢ (𝐴 ∈ V → 𝑆 ∈ ran 𝐹) |
34 | | ne0i 3880 |
. . . . . 6
⊢ (𝑆 ∈ ran 𝐹 → ran 𝐹 ≠ ∅) |
35 | 33, 34 | syl 17 |
. . . . 5
⊢ (𝐴 ∈ V → ran 𝐹 ≠ ∅) |
36 | | simpr 476 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ 𝑆) |
37 | | ssid 3587 |
. . . . . . . . . . . 12
⊢ 𝑧 ⊆ 𝑧 |
38 | | sseq2 3590 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑧 → (𝑧 ⊆ 𝑦 ↔ 𝑧 ⊆ 𝑧)) |
39 | 38 | rspcev 3282 |
. . . . . . . . . . . 12
⊢ ((𝑧 ∈ 𝑆 ∧ 𝑧 ⊆ 𝑧) → ∃𝑦 ∈ 𝑆 𝑧 ⊆ 𝑦) |
40 | 36, 37, 39 | sylancl 693 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) → ∃𝑦 ∈ 𝑆 𝑧 ⊆ 𝑦) |
41 | | rabn0 3912 |
. . . . . . . . . . 11
⊢ ({𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} ≠ ∅ ↔ ∃𝑦 ∈ 𝑆 𝑧 ⊆ 𝑦) |
42 | 40, 41 | sylibr 223 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) → {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} ≠ ∅) |
43 | 42 | necomd 2837 |
. . . . . . . . 9
⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) → ∅ ≠ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}) |
44 | 43 | neneqd 2787 |
. . . . . . . 8
⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ 𝑆) → ¬ ∅ = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}) |
45 | 44 | nrexdv 2984 |
. . . . . . 7
⊢ (𝐴 ∈ V → ¬
∃𝑧 ∈ 𝑆 ∅ = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}) |
46 | | 0ex 4718 |
. . . . . . . 8
⊢ ∅
∈ V |
47 | 14 | elrnmpt 5293 |
. . . . . . . 8
⊢ (∅
∈ V → (∅ ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝑆 ∅ = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦})) |
48 | 46, 47 | ax-mp 5 |
. . . . . . 7
⊢ (∅
∈ ran 𝐹 ↔
∃𝑧 ∈ 𝑆 ∅ = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}) |
49 | 45, 48 | sylnibr 318 |
. . . . . 6
⊢ (𝐴 ∈ V → ¬ ∅
∈ ran 𝐹) |
50 | | df-nel 2783 |
. . . . . 6
⊢ (∅
∉ ran 𝐹 ↔ ¬
∅ ∈ ran 𝐹) |
51 | 49, 50 | sylibr 223 |
. . . . 5
⊢ (𝐴 ∈ V → ∅ ∉
ran 𝐹) |
52 | | elfpw 8151 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑢 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑢 ⊆ 𝐴 ∧ 𝑢 ∈ Fin)) |
53 | 52 | simplbi 475 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑢 ∈ (𝒫 𝐴 ∩ Fin) → 𝑢 ⊆ 𝐴) |
54 | 53, 5 | eleq2s 2706 |
. . . . . . . . . . . . . . . 16
⊢ (𝑢 ∈ 𝑆 → 𝑢 ⊆ 𝐴) |
55 | | elfpw 8151 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑣 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑣 ⊆ 𝐴 ∧ 𝑣 ∈ Fin)) |
56 | 55 | simplbi 475 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑣 ∈ (𝒫 𝐴 ∩ Fin) → 𝑣 ⊆ 𝐴) |
57 | 56, 5 | eleq2s 2706 |
. . . . . . . . . . . . . . . 16
⊢ (𝑣 ∈ 𝑆 → 𝑣 ⊆ 𝐴) |
58 | 54, 57 | anim12i 588 |
. . . . . . . . . . . . . . 15
⊢ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → (𝑢 ⊆ 𝐴 ∧ 𝑣 ⊆ 𝐴)) |
59 | | unss 3749 |
. . . . . . . . . . . . . . 15
⊢ ((𝑢 ⊆ 𝐴 ∧ 𝑣 ⊆ 𝐴) ↔ (𝑢 ∪ 𝑣) ⊆ 𝐴) |
60 | 58, 59 | sylib 207 |
. . . . . . . . . . . . . 14
⊢ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → (𝑢 ∪ 𝑣) ⊆ 𝐴) |
61 | 52 | simprbi 479 |
. . . . . . . . . . . . . . . 16
⊢ (𝑢 ∈ (𝒫 𝐴 ∩ Fin) → 𝑢 ∈ Fin) |
62 | 61, 5 | eleq2s 2706 |
. . . . . . . . . . . . . . 15
⊢ (𝑢 ∈ 𝑆 → 𝑢 ∈ Fin) |
63 | 55 | simprbi 479 |
. . . . . . . . . . . . . . . 16
⊢ (𝑣 ∈ (𝒫 𝐴 ∩ Fin) → 𝑣 ∈ Fin) |
64 | 63, 5 | eleq2s 2706 |
. . . . . . . . . . . . . . 15
⊢ (𝑣 ∈ 𝑆 → 𝑣 ∈ Fin) |
65 | | unfi 8112 |
. . . . . . . . . . . . . . 15
⊢ ((𝑢 ∈ Fin ∧ 𝑣 ∈ Fin) → (𝑢 ∪ 𝑣) ∈ Fin) |
66 | 62, 64, 65 | syl2an 493 |
. . . . . . . . . . . . . 14
⊢ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → (𝑢 ∪ 𝑣) ∈ Fin) |
67 | | elfpw 8151 |
. . . . . . . . . . . . . 14
⊢ ((𝑢 ∪ 𝑣) ∈ (𝒫 𝐴 ∩ Fin) ↔ ((𝑢 ∪ 𝑣) ⊆ 𝐴 ∧ (𝑢 ∪ 𝑣) ∈ Fin)) |
68 | 60, 66, 67 | sylanbrc 695 |
. . . . . . . . . . . . 13
⊢ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → (𝑢 ∪ 𝑣) ∈ (𝒫 𝐴 ∩ Fin)) |
69 | 68 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → (𝑢 ∪ 𝑣) ∈ (𝒫 𝐴 ∩ Fin)) |
70 | 69, 5 | syl6eleqr 2699 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → (𝑢 ∪ 𝑣) ∈ 𝑆) |
71 | | eqidd 2611 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}) |
72 | | sseq1 3589 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = (𝑢 ∪ 𝑣) → (𝑎 ⊆ 𝑦 ↔ (𝑢 ∪ 𝑣) ⊆ 𝑦)) |
73 | 72 | rabbidv 3164 |
. . . . . . . . . . . . 13
⊢ (𝑎 = (𝑢 ∪ 𝑣) → {𝑦 ∈ 𝑆 ∣ 𝑎 ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}) |
74 | 73 | eqeq2d 2620 |
. . . . . . . . . . . 12
⊢ (𝑎 = (𝑢 ∪ 𝑣) → ({𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑎 ⊆ 𝑦} ↔ {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦})) |
75 | 74 | rspcev 3282 |
. . . . . . . . . . 11
⊢ (((𝑢 ∪ 𝑣) ∈ 𝑆 ∧ {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}) → ∃𝑎 ∈ 𝑆 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑎 ⊆ 𝑦}) |
76 | 70, 71, 75 | syl2anc 691 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → ∃𝑎 ∈ 𝑆 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑎 ⊆ 𝑦}) |
77 | 9 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → 𝑆 ∈ V) |
78 | | rabexg 4739 |
. . . . . . . . . . . 12
⊢ (𝑆 ∈ V → {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} ∈ V) |
79 | 77, 78 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} ∈ V) |
80 | | sseq1 3589 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 = 𝑎 → (𝑧 ⊆ 𝑦 ↔ 𝑎 ⊆ 𝑦)) |
81 | 80 | rabbidv 3164 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = 𝑎 → {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑎 ⊆ 𝑦}) |
82 | 81 | cbvmptv 4678 |
. . . . . . . . . . . . 13
⊢ (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}) = (𝑎 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑎 ⊆ 𝑦}) |
83 | 14, 82 | eqtri 2632 |
. . . . . . . . . . . 12
⊢ 𝐹 = (𝑎 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑎 ⊆ 𝑦}) |
84 | 83 | elrnmpt 5293 |
. . . . . . . . . . 11
⊢ ({𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} ∈ V → ({𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} ∈ ran 𝐹 ↔ ∃𝑎 ∈ 𝑆 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑎 ⊆ 𝑦})) |
85 | 79, 84 | syl 17 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → ({𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} ∈ ran 𝐹 ↔ ∃𝑎 ∈ 𝑆 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑎 ⊆ 𝑦})) |
86 | 76, 85 | mpbird 246 |
. . . . . . . . 9
⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} ∈ ran 𝐹) |
87 | | pwidg 4121 |
. . . . . . . . . 10
⊢ ({𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} ∈ V → {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} ∈ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}) |
88 | 79, 87 | syl 17 |
. . . . . . . . 9
⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} ∈ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}) |
89 | | inelcm 3984 |
. . . . . . . . 9
⊢ (({𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} ∈ ran 𝐹 ∧ {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} ∈ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}) → (ran 𝐹 ∩ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}) ≠ ∅) |
90 | 86, 88, 89 | syl2anc 691 |
. . . . . . . 8
⊢ ((𝐴 ∈ V ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → (ran 𝐹 ∩ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}) ≠ ∅) |
91 | 90 | ralrimivva 2954 |
. . . . . . 7
⊢ (𝐴 ∈ V → ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑆 (ran 𝐹 ∩ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}) ≠ ∅) |
92 | | rabexg 4739 |
. . . . . . . . . 10
⊢ (𝑆 ∈ V → {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} ∈ V) |
93 | 9, 92 | syl 17 |
. . . . . . . . 9
⊢ (𝐴 ∈ V → {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} ∈ V) |
94 | 93 | ralrimivw 2950 |
. . . . . . . 8
⊢ (𝐴 ∈ V → ∀𝑢 ∈ 𝑆 {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} ∈ V) |
95 | | sseq1 3589 |
. . . . . . . . . . . 12
⊢ (𝑧 = 𝑢 → (𝑧 ⊆ 𝑦 ↔ 𝑢 ⊆ 𝑦)) |
96 | 95 | rabbidv 3164 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝑢 → {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦}) |
97 | 96 | cbvmptv 4678 |
. . . . . . . . . 10
⊢ (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}) = (𝑢 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦}) |
98 | 14, 97 | eqtri 2632 |
. . . . . . . . 9
⊢ 𝐹 = (𝑢 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦}) |
99 | | ineq1 3769 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} → (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦}) = ({𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})) |
100 | | inrab 3858 |
. . . . . . . . . . . . . . 15
⊢ ({𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦}) = {𝑦 ∈ 𝑆 ∣ (𝑢 ⊆ 𝑦 ∧ 𝑣 ⊆ 𝑦)} |
101 | | unss 3749 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑢 ⊆ 𝑦 ∧ 𝑣 ⊆ 𝑦) ↔ (𝑢 ∪ 𝑣) ⊆ 𝑦) |
102 | 101 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ 𝑆 → ((𝑢 ⊆ 𝑦 ∧ 𝑣 ⊆ 𝑦) ↔ (𝑢 ∪ 𝑣) ⊆ 𝑦)) |
103 | 102 | rabbiia 3161 |
. . . . . . . . . . . . . . 15
⊢ {𝑦 ∈ 𝑆 ∣ (𝑢 ⊆ 𝑦 ∧ 𝑣 ⊆ 𝑦)} = {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} |
104 | 100, 103 | eqtri 2632 |
. . . . . . . . . . . . . 14
⊢ ({𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦}) = {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦} |
105 | 99, 104 | syl6eq 2660 |
. . . . . . . . . . . . 13
⊢ (𝑎 = {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} → (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦}) = {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}) |
106 | 105 | pweqd 4113 |
. . . . . . . . . . . 12
⊢ (𝑎 = {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} → 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦}) = 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}) |
107 | 106 | ineq2d 3776 |
. . . . . . . . . . 11
⊢ (𝑎 = {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} → (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})) = (ran 𝐹 ∩ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦})) |
108 | 107 | neeq1d 2841 |
. . . . . . . . . 10
⊢ (𝑎 = {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} → ((ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})) ≠ ∅ ↔ (ran 𝐹 ∩ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}) ≠ ∅)) |
109 | 108 | ralbidv 2969 |
. . . . . . . . 9
⊢ (𝑎 = {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} → (∀𝑣 ∈ 𝑆 (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})) ≠ ∅ ↔ ∀𝑣 ∈ 𝑆 (ran 𝐹 ∩ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}) ≠ ∅)) |
110 | 98, 109 | ralrnmpt 6276 |
. . . . . . . 8
⊢
(∀𝑢 ∈
𝑆 {𝑦 ∈ 𝑆 ∣ 𝑢 ⊆ 𝑦} ∈ V → (∀𝑎 ∈ ran 𝐹∀𝑣 ∈ 𝑆 (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})) ≠ ∅ ↔ ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑆 (ran 𝐹 ∩ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}) ≠ ∅)) |
111 | 94, 110 | syl 17 |
. . . . . . 7
⊢ (𝐴 ∈ V → (∀𝑎 ∈ ran 𝐹∀𝑣 ∈ 𝑆 (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})) ≠ ∅ ↔ ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑆 (ran 𝐹 ∩ 𝒫 {𝑦 ∈ 𝑆 ∣ (𝑢 ∪ 𝑣) ⊆ 𝑦}) ≠ ∅)) |
112 | 91, 111 | mpbird 246 |
. . . . . 6
⊢ (𝐴 ∈ V → ∀𝑎 ∈ ran 𝐹∀𝑣 ∈ 𝑆 (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})) ≠ ∅) |
113 | | rabexg 4739 |
. . . . . . . . . 10
⊢ (𝑆 ∈ V → {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦} ∈ V) |
114 | 9, 113 | syl 17 |
. . . . . . . . 9
⊢ (𝐴 ∈ V → {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦} ∈ V) |
115 | 114 | ralrimivw 2950 |
. . . . . . . 8
⊢ (𝐴 ∈ V → ∀𝑣 ∈ 𝑆 {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦} ∈ V) |
116 | | sseq1 3589 |
. . . . . . . . . . . 12
⊢ (𝑧 = 𝑣 → (𝑧 ⊆ 𝑦 ↔ 𝑣 ⊆ 𝑦)) |
117 | 116 | rabbidv 3164 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝑣 → {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦}) |
118 | 117 | cbvmptv 4678 |
. . . . . . . . . 10
⊢ (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}) = (𝑣 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦}) |
119 | 14, 118 | eqtri 2632 |
. . . . . . . . 9
⊢ 𝐹 = (𝑣 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦}) |
120 | | ineq2 3770 |
. . . . . . . . . . . 12
⊢ (𝑏 = {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦} → (𝑎 ∩ 𝑏) = (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})) |
121 | 120 | pweqd 4113 |
. . . . . . . . . . 11
⊢ (𝑏 = {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦} → 𝒫 (𝑎 ∩ 𝑏) = 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})) |
122 | 121 | ineq2d 3776 |
. . . . . . . . . 10
⊢ (𝑏 = {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦} → (ran 𝐹 ∩ 𝒫 (𝑎 ∩ 𝑏)) = (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦}))) |
123 | 122 | neeq1d 2841 |
. . . . . . . . 9
⊢ (𝑏 = {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦} → ((ran 𝐹 ∩ 𝒫 (𝑎 ∩ 𝑏)) ≠ ∅ ↔ (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})) ≠ ∅)) |
124 | 119, 123 | ralrnmpt 6276 |
. . . . . . . 8
⊢
(∀𝑣 ∈
𝑆 {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦} ∈ V → (∀𝑏 ∈ ran 𝐹(ran 𝐹 ∩ 𝒫 (𝑎 ∩ 𝑏)) ≠ ∅ ↔ ∀𝑣 ∈ 𝑆 (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})) ≠ ∅)) |
125 | 115, 124 | syl 17 |
. . . . . . 7
⊢ (𝐴 ∈ V → (∀𝑏 ∈ ran 𝐹(ran 𝐹 ∩ 𝒫 (𝑎 ∩ 𝑏)) ≠ ∅ ↔ ∀𝑣 ∈ 𝑆 (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})) ≠ ∅)) |
126 | 125 | ralbidv 2969 |
. . . . . 6
⊢ (𝐴 ∈ V → (∀𝑎 ∈ ran 𝐹∀𝑏 ∈ ran 𝐹(ran 𝐹 ∩ 𝒫 (𝑎 ∩ 𝑏)) ≠ ∅ ↔ ∀𝑎 ∈ ran 𝐹∀𝑣 ∈ 𝑆 (ran 𝐹 ∩ 𝒫 (𝑎 ∩ {𝑦 ∈ 𝑆 ∣ 𝑣 ⊆ 𝑦})) ≠ ∅)) |
127 | 112, 126 | mpbird 246 |
. . . . 5
⊢ (𝐴 ∈ V → ∀𝑎 ∈ ran 𝐹∀𝑏 ∈ ran 𝐹(ran 𝐹 ∩ 𝒫 (𝑎 ∩ 𝑏)) ≠ ∅) |
128 | 35, 51, 127 | 3jca 1235 |
. . . 4
⊢ (𝐴 ∈ V → (ran 𝐹 ≠ ∅ ∧ ∅
∉ ran 𝐹 ∧
∀𝑎 ∈ ran 𝐹∀𝑏 ∈ ran 𝐹(ran 𝐹 ∩ 𝒫 (𝑎 ∩ 𝑏)) ≠ ∅)) |
129 | | isfbas 21443 |
. . . . 5
⊢ (𝑆 ∈ V → (ran 𝐹 ∈ (fBas‘𝑆) ↔ (ran 𝐹 ⊆ 𝒫 𝑆 ∧ (ran 𝐹 ≠ ∅ ∧ ∅ ∉ ran
𝐹 ∧ ∀𝑎 ∈ ran 𝐹∀𝑏 ∈ ran 𝐹(ran 𝐹 ∩ 𝒫 (𝑎 ∩ 𝑏)) ≠ ∅)))) |
130 | 9, 129 | syl 17 |
. . . 4
⊢ (𝐴 ∈ V → (ran 𝐹 ∈ (fBas‘𝑆) ↔ (ran 𝐹 ⊆ 𝒫 𝑆 ∧ (ran 𝐹 ≠ ∅ ∧ ∅ ∉ ran
𝐹 ∧ ∀𝑎 ∈ ran 𝐹∀𝑏 ∈ ran 𝐹(ran 𝐹 ∩ 𝒫 (𝑎 ∩ 𝑏)) ≠ ∅)))) |
131 | 17, 128, 130 | mpbir2and 959 |
. . 3
⊢ (𝐴 ∈ V → ran 𝐹 ∈ (fBas‘𝑆)) |
132 | 3, 131 | syl5eqel 2692 |
. 2
⊢ (𝐴 ∈ V → 𝐿 ∈ (fBas‘𝑆)) |
133 | 1, 2, 132 | 3syl 18 |
1
⊢ (𝜑 → 𝐿 ∈ (fBas‘𝑆)) |