Step | Hyp | Ref
| Expression |
1 | | elpwi 4117 |
. . . 4
⊢ (𝑐 ∈ 𝒫 𝐽 → 𝑐 ⊆ 𝐽) |
2 | | comppfsc.1 |
. . . . . . 7
⊢ 𝑋 = ∪
𝐽 |
3 | 2 | cmpcov 21002 |
. . . . . 6
⊢ ((𝐽 ∈ Comp ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) |
4 | | elfpw 8151 |
. . . . . . . 8
⊢ (𝑑 ∈ (𝒫 𝑐 ∩ Fin) ↔ (𝑑 ⊆ 𝑐 ∧ 𝑑 ∈ Fin)) |
5 | | finptfin 21131 |
. . . . . . . . . . 11
⊢ (𝑑 ∈ Fin → 𝑑 ∈ PtFin) |
6 | 5 | anim1i 590 |
. . . . . . . . . 10
⊢ ((𝑑 ∈ Fin ∧ (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → (𝑑 ∈ PtFin ∧ (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑))) |
7 | 6 | anassrs 678 |
. . . . . . . . 9
⊢ (((𝑑 ∈ Fin ∧ 𝑑 ⊆ 𝑐) ∧ 𝑋 = ∪ 𝑑) → (𝑑 ∈ PtFin ∧ (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑))) |
8 | 7 | ancom1s 843 |
. . . . . . . 8
⊢ (((𝑑 ⊆ 𝑐 ∧ 𝑑 ∈ Fin) ∧ 𝑋 = ∪ 𝑑) → (𝑑 ∈ PtFin ∧ (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑))) |
9 | 4, 8 | sylanb 488 |
. . . . . . 7
⊢ ((𝑑 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑋 = ∪
𝑑) → (𝑑 ∈ PtFin ∧ (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑))) |
10 | 9 | reximi2 2993 |
. . . . . 6
⊢
(∃𝑑 ∈
(𝒫 𝑐 ∩
Fin)𝑋 = ∪ 𝑑
→ ∃𝑑 ∈
PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) |
11 | 3, 10 | syl 17 |
. . . . 5
⊢ ((𝐽 ∈ Comp ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) |
12 | 11 | 3exp 1256 |
. . . 4
⊢ (𝐽 ∈ Comp → (𝑐 ⊆ 𝐽 → (𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)))) |
13 | 1, 12 | syl5 33 |
. . 3
⊢ (𝐽 ∈ Comp → (𝑐 ∈ 𝒫 𝐽 → (𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)))) |
14 | 13 | ralrimiv 2948 |
. 2
⊢ (𝐽 ∈ Comp →
∀𝑐 ∈ 𝒫
𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑))) |
15 | | elpwi 4117 |
. . . . . . 7
⊢ (𝑎 ∈ 𝒫 𝐽 → 𝑎 ⊆ 𝐽) |
16 | | 0elpw 4760 |
. . . . . . . . . . . 12
⊢ ∅
∈ 𝒫 𝑎 |
17 | | 0fin 8073 |
. . . . . . . . . . . 12
⊢ ∅
∈ Fin |
18 | | elin 3758 |
. . . . . . . . . . . 12
⊢ (∅
∈ (𝒫 𝑎 ∩
Fin) ↔ (∅ ∈ 𝒫 𝑎 ∧ ∅ ∈ Fin)) |
19 | 16, 17, 18 | mpbir2an 957 |
. . . . . . . . . . 11
⊢ ∅
∈ (𝒫 𝑎 ∩
Fin) |
20 | | unieq 4380 |
. . . . . . . . . . . . . 14
⊢ (𝑏 = ∅ → ∪ 𝑏 =
∪ ∅) |
21 | | uni0 4401 |
. . . . . . . . . . . . . 14
⊢ ∪ ∅ = ∅ |
22 | 20, 21 | syl6eq 2660 |
. . . . . . . . . . . . 13
⊢ (𝑏 = ∅ → ∪ 𝑏 =
∅) |
23 | 22 | eqeq2d 2620 |
. . . . . . . . . . . 12
⊢ (𝑏 = ∅ → (𝑋 = ∪
𝑏 ↔ 𝑋 = ∅)) |
24 | 23 | rspcev 3282 |
. . . . . . . . . . 11
⊢ ((∅
∈ (𝒫 𝑎 ∩
Fin) ∧ 𝑋 = ∅)
→ ∃𝑏 ∈
(𝒫 𝑎 ∩
Fin)𝑋 = ∪ 𝑏) |
25 | 19, 24 | mpan 702 |
. . . . . . . . . 10
⊢ (𝑋 = ∅ → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏) |
26 | 25 | a1d 25 |
. . . . . . . . 9
⊢ (𝑋 = ∅ → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏)) |
27 | 26 | a1i 11 |
. . . . . . . 8
⊢ ((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) → (𝑋 = ∅ → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏))) |
28 | | n0 3890 |
. . . . . . . . 9
⊢ (𝑋 ≠ ∅ ↔
∃𝑥 𝑥 ∈ 𝑋) |
29 | | simp2 1055 |
. . . . . . . . . . . . . 14
⊢ ((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) → 𝑋 = ∪ 𝑎) |
30 | 29 | eleq2d 2673 |
. . . . . . . . . . . . 13
⊢ ((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ ∪ 𝑎)) |
31 | 30 | biimpd 218 |
. . . . . . . . . . . 12
⊢ ((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) → (𝑥 ∈ 𝑋 → 𝑥 ∈ ∪ 𝑎)) |
32 | | eluni2 4376 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ∪ 𝑎
↔ ∃𝑠 ∈
𝑎 𝑥 ∈ 𝑠) |
33 | 31, 32 | syl6ib 240 |
. . . . . . . . . . 11
⊢ ((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) → (𝑥 ∈ 𝑋 → ∃𝑠 ∈ 𝑎 𝑥 ∈ 𝑠)) |
34 | | simpl3 1059 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑎 ⊆ 𝐽) |
35 | | simprl 790 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑠 ∈ 𝑎) |
36 | 34, 35 | sseldd 3569 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑠 ∈ 𝐽) |
37 | | elssuni 4403 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑠 ∈ 𝐽 → 𝑠 ⊆ ∪ 𝐽) |
38 | 37, 2 | syl6sseqr 3615 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑠 ∈ 𝐽 → 𝑠 ⊆ 𝑋) |
39 | 36, 38 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑠 ⊆ 𝑋) |
40 | 39 | ralrimivw 2950 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → ∀𝑝 ∈ 𝑎 𝑠 ⊆ 𝑋) |
41 | | iunss 4497 |
. . . . . . . . . . . . . . . . . 18
⊢ (∪ 𝑝 ∈ 𝑎 𝑠 ⊆ 𝑋 ↔ ∀𝑝 ∈ 𝑎 𝑠 ⊆ 𝑋) |
42 | 40, 41 | sylibr 223 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → ∪
𝑝 ∈ 𝑎 𝑠 ⊆ 𝑋) |
43 | | ssequn1 3745 |
. . . . . . . . . . . . . . . . 17
⊢ (∪ 𝑝 ∈ 𝑎 𝑠 ⊆ 𝑋 ↔ (∪
𝑝 ∈ 𝑎 𝑠 ∪ 𝑋) = 𝑋) |
44 | 42, 43 | sylib 207 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → (∪ 𝑝 ∈ 𝑎 𝑠 ∪ 𝑋) = 𝑋) |
45 | | simpl2 1058 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑋 = ∪ 𝑎) |
46 | | uniiun 4509 |
. . . . . . . . . . . . . . . . . 18
⊢ ∪ 𝑎 =
∪ 𝑝 ∈ 𝑎 𝑝 |
47 | 45, 46 | syl6eq 2660 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑋 = ∪ 𝑝 ∈ 𝑎 𝑝) |
48 | 47 | uneq2d 3729 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → (∪ 𝑝 ∈ 𝑎 𝑠 ∪ 𝑋) = (∪
𝑝 ∈ 𝑎 𝑠 ∪ ∪
𝑝 ∈ 𝑎 𝑝)) |
49 | 44, 48 | eqtr3d 2646 |
. . . . . . . . . . . . . . 15
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑋 = (∪
𝑝 ∈ 𝑎 𝑠 ∪ ∪
𝑝 ∈ 𝑎 𝑝)) |
50 | | iunun 4540 |
. . . . . . . . . . . . . . . 16
⊢ ∪ 𝑝 ∈ 𝑎 (𝑠 ∪ 𝑝) = (∪
𝑝 ∈ 𝑎 𝑠 ∪ ∪
𝑝 ∈ 𝑎 𝑝) |
51 | | vex 3176 |
. . . . . . . . . . . . . . . . . 18
⊢ 𝑠 ∈ V |
52 | | vex 3176 |
. . . . . . . . . . . . . . . . . 18
⊢ 𝑝 ∈ V |
53 | 51, 52 | unex 6854 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 ∪ 𝑝) ∈ V |
54 | 53 | dfiun3 5301 |
. . . . . . . . . . . . . . . 16
⊢ ∪ 𝑝 ∈ 𝑎 (𝑠 ∪ 𝑝) = ∪ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) |
55 | 50, 54 | eqtr3i 2634 |
. . . . . . . . . . . . . . 15
⊢ (∪ 𝑝 ∈ 𝑎 𝑠 ∪ ∪
𝑝 ∈ 𝑎 𝑝) = ∪ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) |
56 | 49, 55 | syl6eq 2660 |
. . . . . . . . . . . . . 14
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑋 = ∪ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝))) |
57 | | simpll1 1093 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ 𝑝 ∈ 𝑎) → 𝐽 ∈ Top) |
58 | 36 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ 𝑝 ∈ 𝑎) → 𝑠 ∈ 𝐽) |
59 | 34 | sselda 3568 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ 𝑝 ∈ 𝑎) → 𝑝 ∈ 𝐽) |
60 | | unopn 20533 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐽 ∈ Top ∧ 𝑠 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) → (𝑠 ∪ 𝑝) ∈ 𝐽) |
61 | 57, 58, 59, 60 | syl3anc 1318 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ 𝑝 ∈ 𝑎) → (𝑠 ∪ 𝑝) ∈ 𝐽) |
62 | | eqid 2610 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) = (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) |
63 | 61, 62 | fmptd 6292 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)):𝑎⟶𝐽) |
64 | | frn 5966 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)):𝑎⟶𝐽 → ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ⊆ 𝐽) |
65 | 63, 64 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ⊆ 𝐽) |
66 | | elpw2g 4754 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐽 ∈ Top → (ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∈ 𝒫 𝐽 ↔ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ⊆ 𝐽)) |
67 | 66 | 3ad2ant1 1075 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) → (ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∈ 𝒫 𝐽 ↔ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ⊆ 𝐽)) |
68 | 67 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → (ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∈ 𝒫 𝐽 ↔ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ⊆ 𝐽)) |
69 | 65, 68 | mpbird 246 |
. . . . . . . . . . . . . . 15
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∈ 𝒫 𝐽) |
70 | | unieq 4380 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑐 = ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) → ∪ 𝑐 = ∪
ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝))) |
71 | 70 | eqeq2d 2620 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑐 = ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) → (𝑋 = ∪ 𝑐 ↔ 𝑋 = ∪ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)))) |
72 | | sseq2 3590 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑐 = ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) → (𝑑 ⊆ 𝑐 ↔ 𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)))) |
73 | 72 | anbi1d 737 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑐 = ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) → ((𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑) ↔ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑))) |
74 | 73 | rexbidv 3034 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑐 = ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) → (∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑) ↔ ∃𝑑 ∈ PtFin (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑))) |
75 | 71, 74 | imbi12d 333 |
. . . . . . . . . . . . . . . 16
⊢ (𝑐 = ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) → ((𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) ↔ (𝑋 = ∪ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) → ∃𝑑 ∈ PtFin (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)))) |
76 | 75 | rspcv 3278 |
. . . . . . . . . . . . . . 15
⊢ (ran
(𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∈ 𝒫 𝐽 → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → (𝑋 = ∪ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) → ∃𝑑 ∈ PtFin (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)))) |
77 | 69, 76 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → (𝑋 = ∪ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) → ∃𝑑 ∈ PtFin (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)))) |
78 | 56, 77 | mpid 43 |
. . . . . . . . . . . . 13
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → ∃𝑑 ∈ PtFin (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑))) |
79 | | simprr 792 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑥 ∈ 𝑠) |
80 | | ssel2 3563 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑎 ⊆ 𝐽 ∧ 𝑠 ∈ 𝑎) → 𝑠 ∈ 𝐽) |
81 | 80 | 3ad2antl3 1218 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ 𝑠 ∈ 𝑎) → 𝑠 ∈ 𝐽) |
82 | 81 | adantrr 749 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑠 ∈ 𝐽) |
83 | | elunii 4377 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑥 ∈ 𝑠 ∧ 𝑠 ∈ 𝐽) → 𝑥 ∈ ∪ 𝐽) |
84 | 79, 82, 83 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑥 ∈ ∪ 𝐽) |
85 | 84, 2 | syl6eleqr 2699 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑥 ∈ 𝑋) |
86 | 85 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → 𝑥 ∈ 𝑋) |
87 | | simprr 792 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → 𝑋 = ∪ 𝑑) |
88 | 86, 87 | eleqtrd 2690 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → 𝑥 ∈ ∪ 𝑑) |
89 | | eqid 2610 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ∪ 𝑑 =
∪ 𝑑 |
90 | 89 | ptfinfin 21132 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑑 ∈ PtFin ∧ 𝑥 ∈ ∪ 𝑑)
→ {𝑧 ∈ 𝑑 ∣ 𝑥 ∈ 𝑧} ∈ Fin) |
91 | 90 | expcom 450 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈ ∪ 𝑑
→ (𝑑 ∈ PtFin
→ {𝑧 ∈ 𝑑 ∣ 𝑥 ∈ 𝑧} ∈ Fin)) |
92 | 88, 91 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → (𝑑 ∈ PtFin → {𝑧 ∈ 𝑑 ∣ 𝑥 ∈ 𝑧} ∈ Fin)) |
93 | | simprl 790 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → 𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝))) |
94 | | elun1 3742 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑥 ∈ 𝑠 → 𝑥 ∈ (𝑠 ∪ 𝑝)) |
95 | 94 | ad2antll 761 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑥 ∈ (𝑠 ∪ 𝑝)) |
96 | 95 | ralrimivw 2950 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → ∀𝑝 ∈ 𝑎 𝑥 ∈ (𝑠 ∪ 𝑝)) |
97 | 53 | rgenw 2908 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
∀𝑝 ∈
𝑎 (𝑠 ∪ 𝑝) ∈ V |
98 | | eleq2 2677 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑧 = (𝑠 ∪ 𝑝) → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ (𝑠 ∪ 𝑝))) |
99 | 62, 98 | ralrnmpt 6276 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(∀𝑝 ∈
𝑎 (𝑠 ∪ 𝑝) ∈ V → (∀𝑧 ∈ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝))𝑥 ∈ 𝑧 ↔ ∀𝑝 ∈ 𝑎 𝑥 ∈ (𝑠 ∪ 𝑝))) |
100 | 97, 99 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(∀𝑧 ∈
ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝))𝑥 ∈ 𝑧 ↔ ∀𝑝 ∈ 𝑎 𝑥 ∈ (𝑠 ∪ 𝑝)) |
101 | 96, 100 | sylibr 223 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → ∀𝑧 ∈ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝))𝑥 ∈ 𝑧) |
102 | 101 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → ∀𝑧 ∈ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝))𝑥 ∈ 𝑧) |
103 | | ssralv 3629 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) → (∀𝑧 ∈ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝))𝑥 ∈ 𝑧 → ∀𝑧 ∈ 𝑑 𝑥 ∈ 𝑧)) |
104 | 93, 102, 103 | sylc 63 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → ∀𝑧 ∈ 𝑑 𝑥 ∈ 𝑧) |
105 | | rabid2 3096 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑑 = {𝑧 ∈ 𝑑 ∣ 𝑥 ∈ 𝑧} ↔ ∀𝑧 ∈ 𝑑 𝑥 ∈ 𝑧) |
106 | 104, 105 | sylibr 223 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → 𝑑 = {𝑧 ∈ 𝑑 ∣ 𝑥 ∈ 𝑧}) |
107 | 106 | eleq1d 2672 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → (𝑑 ∈ Fin ↔ {𝑧 ∈ 𝑑 ∣ 𝑥 ∈ 𝑧} ∈ Fin)) |
108 | 107 | biimprd 237 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → ({𝑧 ∈ 𝑑 ∣ 𝑥 ∈ 𝑧} ∈ Fin → 𝑑 ∈ Fin)) |
109 | 62 | rnmpt 5292 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ran
(𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) = {𝑞 ∣ ∃𝑝 ∈ 𝑎 𝑞 = (𝑠 ∪ 𝑝)} |
110 | 93, 109 | syl6sseq 3614 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → 𝑑 ⊆ {𝑞 ∣ ∃𝑝 ∈ 𝑎 𝑞 = (𝑠 ∪ 𝑝)}) |
111 | | ssabral 3636 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑑 ⊆ {𝑞 ∣ ∃𝑝 ∈ 𝑎 𝑞 = (𝑠 ∪ 𝑝)} ↔ ∀𝑞 ∈ 𝑑 ∃𝑝 ∈ 𝑎 𝑞 = (𝑠 ∪ 𝑝)) |
112 | 110, 111 | sylib 207 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → ∀𝑞 ∈ 𝑑 ∃𝑝 ∈ 𝑎 𝑞 = (𝑠 ∪ 𝑝)) |
113 | | uneq2 3723 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑝 = (𝑓‘𝑞) → (𝑠 ∪ 𝑝) = (𝑠 ∪ (𝑓‘𝑞))) |
114 | 113 | eqeq2d 2620 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑝 = (𝑓‘𝑞) → (𝑞 = (𝑠 ∪ 𝑝) ↔ 𝑞 = (𝑠 ∪ (𝑓‘𝑞)))) |
115 | 114 | ac6sfi 8089 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑑 ∈ Fin ∧ ∀𝑞 ∈ 𝑑 ∃𝑝 ∈ 𝑎 𝑞 = (𝑠 ∪ 𝑝)) → ∃𝑓(𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞)))) |
116 | 115 | expcom 450 |
. . . . . . . . . . . . . . . . . . 19
⊢
(∀𝑞 ∈
𝑑 ∃𝑝 ∈ 𝑎 𝑞 = (𝑠 ∪ 𝑝) → (𝑑 ∈ Fin → ∃𝑓(𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) |
117 | 112, 116 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → (𝑑 ∈ Fin → ∃𝑓(𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) |
118 | | frn 5966 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑓:𝑑⟶𝑎 → ran 𝑓 ⊆ 𝑎) |
119 | 118 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))) → ran 𝑓 ⊆ 𝑎) |
120 | 119 | ad2antll 761 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((((𝐽 ∈ Top
∧ 𝑋 = ∪ 𝑎
∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → ran 𝑓 ⊆ 𝑎) |
121 | 35 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(((((𝐽 ∈ Top
∧ 𝑋 = ∪ 𝑎
∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → 𝑠 ∈ 𝑎) |
122 | 121 | snssd 4281 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((((𝐽 ∈ Top
∧ 𝑋 = ∪ 𝑎
∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → {𝑠} ⊆ 𝑎) |
123 | 120, 122 | unssd 3751 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝐽 ∈ Top
∧ 𝑋 = ∪ 𝑎
∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → (ran 𝑓 ∪ {𝑠}) ⊆ 𝑎) |
124 | | simprl 790 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(((((𝐽 ∈ Top
∧ 𝑋 = ∪ 𝑎
∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → 𝑑 ∈ Fin) |
125 | | simprrl 800 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
(((((𝐽 ∈ Top
∧ 𝑋 = ∪ 𝑎
∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → 𝑓:𝑑⟶𝑎) |
126 | | ffn 5958 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑓:𝑑⟶𝑎 → 𝑓 Fn 𝑑) |
127 | 125, 126 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
(((((𝐽 ∈ Top
∧ 𝑋 = ∪ 𝑎
∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → 𝑓 Fn 𝑑) |
128 | | dffn4 6034 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑓 Fn 𝑑 ↔ 𝑓:𝑑–onto→ran 𝑓) |
129 | 127, 128 | sylib 207 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(((((𝐽 ∈ Top
∧ 𝑋 = ∪ 𝑎
∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → 𝑓:𝑑–onto→ran 𝑓) |
130 | | fofi 8135 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑑 ∈ Fin ∧ 𝑓:𝑑–onto→ran 𝑓) → ran 𝑓 ∈ Fin) |
131 | 124, 129,
130 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((((𝐽 ∈ Top
∧ 𝑋 = ∪ 𝑎
∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → ran 𝑓 ∈ Fin) |
132 | | snfi 7923 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ {𝑠} ∈ Fin |
133 | | unfi 8112 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((ran
𝑓 ∈ Fin ∧ {𝑠} ∈ Fin) → (ran 𝑓 ∪ {𝑠}) ∈ Fin) |
134 | 131, 132,
133 | sylancl 693 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝐽 ∈ Top
∧ 𝑋 = ∪ 𝑎
∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → (ran 𝑓 ∪ {𝑠}) ∈ Fin) |
135 | | elfpw 8151 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((ran
𝑓 ∪ {𝑠}) ∈ (𝒫 𝑎 ∩ Fin) ↔ ((ran 𝑓 ∪ {𝑠}) ⊆ 𝑎 ∧ (ran 𝑓 ∪ {𝑠}) ∈ Fin)) |
136 | 123, 134,
135 | sylanbrc 695 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝐽 ∈ Top
∧ 𝑋 = ∪ 𝑎
∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → (ran 𝑓 ∪ {𝑠}) ∈ (𝒫 𝑎 ∩ Fin)) |
137 | | simplrr 797 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(((((𝐽 ∈ Top
∧ 𝑋 = ∪ 𝑎
∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → 𝑋 = ∪ 𝑑) |
138 | | uniiun 4509 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ∪ 𝑑 =
∪ 𝑞 ∈ 𝑑 𝑞 |
139 | | simprrr 801 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
(((((𝐽 ∈ Top
∧ 𝑋 = ∪ 𝑎
∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))) |
140 | | iuneq2 4473 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
(∀𝑞 ∈
𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞)) → ∪
𝑞 ∈ 𝑑 𝑞 = ∪ 𝑞 ∈ 𝑑 (𝑠 ∪ (𝑓‘𝑞))) |
141 | 139, 140 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
(((((𝐽 ∈ Top
∧ 𝑋 = ∪ 𝑎
∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → ∪ 𝑞 ∈ 𝑑 𝑞 = ∪ 𝑞 ∈ 𝑑 (𝑠 ∪ (𝑓‘𝑞))) |
142 | 138, 141 | syl5eq 2656 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(((((𝐽 ∈ Top
∧ 𝑋 = ∪ 𝑎
∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → ∪
𝑑 = ∪ 𝑞 ∈ 𝑑 (𝑠 ∪ (𝑓‘𝑞))) |
143 | 137, 142 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((((𝐽 ∈ Top
∧ 𝑋 = ∪ 𝑎
∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → 𝑋 = ∪ 𝑞 ∈ 𝑑 (𝑠 ∪ (𝑓‘𝑞))) |
144 | | ssun2 3739 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ {𝑠} ⊆ (ran 𝑓 ∪ {𝑠}) |
145 | | vsnid 4156 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ 𝑠 ∈ {𝑠} |
146 | 144, 145 | sselii 3565 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ 𝑠 ∈ (ran 𝑓 ∪ {𝑠}) |
147 | | elssuni 4403 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑠 ∈ (ran 𝑓 ∪ {𝑠}) → 𝑠 ⊆ ∪ (ran
𝑓 ∪ {𝑠})) |
148 | 146, 147 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ 𝑠 ⊆ ∪ (ran 𝑓 ∪ {𝑠}) |
149 | | fvssunirn 6127 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑓‘𝑞) ⊆ ∪ ran
𝑓 |
150 | | ssun1 3738 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ran 𝑓 ⊆ (ran 𝑓 ∪ {𝑠}) |
151 | 150 | unissi 4397 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ∪ ran 𝑓 ⊆ ∪ (ran
𝑓 ∪ {𝑠}) |
152 | 149, 151 | sstri 3577 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑓‘𝑞) ⊆ ∪ (ran
𝑓 ∪ {𝑠}) |
153 | 148, 152 | unssi 3750 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑠 ∪ (𝑓‘𝑞)) ⊆ ∪ (ran
𝑓 ∪ {𝑠}) |
154 | 153 | rgenw 2908 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
∀𝑞 ∈
𝑑 (𝑠 ∪ (𝑓‘𝑞)) ⊆ ∪ (ran
𝑓 ∪ {𝑠}) |
155 | | iunss 4497 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (∪ 𝑞 ∈ 𝑑 (𝑠 ∪ (𝑓‘𝑞)) ⊆ ∪ (ran
𝑓 ∪ {𝑠}) ↔ ∀𝑞 ∈ 𝑑 (𝑠 ∪ (𝑓‘𝑞)) ⊆ ∪ (ran
𝑓 ∪ {𝑠})) |
156 | 154, 155 | mpbir 220 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ∪ 𝑞 ∈ 𝑑 (𝑠 ∪ (𝑓‘𝑞)) ⊆ ∪ (ran
𝑓 ∪ {𝑠}) |
157 | 143, 156 | syl6eqss 3618 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝐽 ∈ Top
∧ 𝑋 = ∪ 𝑎
∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → 𝑋 ⊆ ∪ (ran
𝑓 ∪ {𝑠})) |
158 | 34 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
(((((𝐽 ∈ Top
∧ 𝑋 = ∪ 𝑎
∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → 𝑎 ⊆ 𝐽) |
159 | 120, 158 | sstrd 3578 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(((((𝐽 ∈ Top
∧ 𝑋 = ∪ 𝑎
∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → ran 𝑓 ⊆ 𝐽) |
160 | 36 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
(((((𝐽 ∈ Top
∧ 𝑋 = ∪ 𝑎
∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → 𝑠 ∈ 𝐽) |
161 | 160 | snssd 4281 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(((((𝐽 ∈ Top
∧ 𝑋 = ∪ 𝑎
∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → {𝑠} ⊆ 𝐽) |
162 | 159, 161 | unssd 3751 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((((𝐽 ∈ Top
∧ 𝑋 = ∪ 𝑎
∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → (ran 𝑓 ∪ {𝑠}) ⊆ 𝐽) |
163 | | uniss 4394 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((ran
𝑓 ∪ {𝑠}) ⊆ 𝐽 → ∪ (ran
𝑓 ∪ {𝑠}) ⊆ ∪ 𝐽) |
164 | 163, 2 | syl6sseqr 3615 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((ran
𝑓 ∪ {𝑠}) ⊆ 𝐽 → ∪ (ran
𝑓 ∪ {𝑠}) ⊆ 𝑋) |
165 | 162, 164 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝐽 ∈ Top
∧ 𝑋 = ∪ 𝑎
∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → ∪
(ran 𝑓 ∪ {𝑠}) ⊆ 𝑋) |
166 | 157, 165 | eqssd 3585 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝐽 ∈ Top
∧ 𝑋 = ∪ 𝑎
∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → 𝑋 = ∪ (ran 𝑓 ∪ {𝑠})) |
167 | | unieq 4380 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑏 = (ran 𝑓 ∪ {𝑠}) → ∪ 𝑏 = ∪
(ran 𝑓 ∪ {𝑠})) |
168 | 167 | eqeq2d 2620 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑏 = (ran 𝑓 ∪ {𝑠}) → (𝑋 = ∪ 𝑏 ↔ 𝑋 = ∪ (ran 𝑓 ∪ {𝑠}))) |
169 | 168 | rspcev 3282 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((ran
𝑓 ∪ {𝑠}) ∈ (𝒫 𝑎 ∩ Fin) ∧ 𝑋 = ∪ (ran 𝑓 ∪ {𝑠})) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏) |
170 | 136, 166,
169 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝐽 ∈ Top
∧ 𝑋 = ∪ 𝑎
∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏) |
171 | 170 | expr 641 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝐽 ∈ Top
∧ 𝑋 = ∪ 𝑎
∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ 𝑑 ∈ Fin) → ((𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏)) |
172 | 171 | exlimdv 1848 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝐽 ∈ Top
∧ 𝑋 = ∪ 𝑎
∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ 𝑑 ∈ Fin) → (∃𝑓(𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏)) |
173 | 172 | ex 449 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → (𝑑 ∈ Fin → (∃𝑓(𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏))) |
174 | 117, 173 | mpdd 42 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → (𝑑 ∈ Fin → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏)) |
175 | 92, 108, 174 | 3syld 58 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → (𝑑 ∈ PtFin → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏)) |
176 | 175 | ex 449 |
. . . . . . . . . . . . . . 15
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → ((𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑) → (𝑑 ∈ PtFin → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏))) |
177 | 176 | com23 84 |
. . . . . . . . . . . . . 14
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → (𝑑 ∈ PtFin → ((𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏))) |
178 | 177 | rexlimdv 3012 |
. . . . . . . . . . . . 13
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → (∃𝑑 ∈ PtFin (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏)) |
179 | 78, 178 | syld 46 |
. . . . . . . . . . . 12
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏)) |
180 | 179 | rexlimdvaa 3014 |
. . . . . . . . . . 11
⊢ ((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) → (∃𝑠 ∈ 𝑎 𝑥 ∈ 𝑠 → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏))) |
181 | 33, 180 | syld 46 |
. . . . . . . . . 10
⊢ ((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) → (𝑥 ∈ 𝑋 → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏))) |
182 | 181 | exlimdv 1848 |
. . . . . . . . 9
⊢ ((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) → (∃𝑥 𝑥 ∈ 𝑋 → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏))) |
183 | 28, 182 | syl5bi 231 |
. . . . . . . 8
⊢ ((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) → (𝑋 ≠ ∅ → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏))) |
184 | 27, 183 | pm2.61dne 2868 |
. . . . . . 7
⊢ ((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏)) |
185 | 15, 184 | syl3an3 1353 |
. . . . . 6
⊢ ((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ∈ 𝒫 𝐽) → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏)) |
186 | 185 | 3exp 1256 |
. . . . 5
⊢ (𝐽 ∈ Top → (𝑋 = ∪
𝑎 → (𝑎 ∈ 𝒫 𝐽 → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏)))) |
187 | 186 | com24 93 |
. . . 4
⊢ (𝐽 ∈ Top →
(∀𝑐 ∈ 𝒫
𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → (𝑎 ∈ 𝒫 𝐽 → (𝑋 = ∪ 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏)))) |
188 | 187 | ralrimdv 2951 |
. . 3
⊢ (𝐽 ∈ Top →
(∀𝑐 ∈ 𝒫
𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → ∀𝑎 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏))) |
189 | 2 | iscmp 21001 |
. . . 4
⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑎 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏))) |
190 | 189 | baibr 943 |
. . 3
⊢ (𝐽 ∈ Top →
(∀𝑎 ∈ 𝒫
𝐽(𝑋 = ∪ 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏) ↔ 𝐽 ∈ Comp)) |
191 | 188, 190 | sylibd 228 |
. 2
⊢ (𝐽 ∈ Top →
(∀𝑐 ∈ 𝒫
𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → 𝐽 ∈ Comp)) |
192 | 14, 191 | impbid2 215 |
1
⊢ (𝐽 ∈ Top → (𝐽 ∈ Comp ↔
∀𝑐 ∈ 𝒫
𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)))) |