| Step | Hyp | Ref
| Expression |
| 1 | | elpwi 4607 |
. . . 4
⊢ (𝑐 ∈ 𝒫 𝐽 → 𝑐 ⊆ 𝐽) |
| 2 | | comppfsc.1 |
. . . . . . 7
⊢ 𝑋 = ∪
𝐽 |
| 3 | 2 | cmpcov 23397 |
. . . . . 6
⊢ ((𝐽 ∈ Comp ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) |
| 4 | | elfpw 9394 |
. . . . . . . 8
⊢ (𝑑 ∈ (𝒫 𝑐 ∩ Fin) ↔ (𝑑 ⊆ 𝑐 ∧ 𝑑 ∈ Fin)) |
| 5 | | finptfin 23526 |
. . . . . . . . . . 11
⊢ (𝑑 ∈ Fin → 𝑑 ∈ PtFin) |
| 6 | 5 | anim1i 615 |
. . . . . . . . . 10
⊢ ((𝑑 ∈ Fin ∧ (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → (𝑑 ∈ PtFin ∧ (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑))) |
| 7 | 6 | anassrs 467 |
. . . . . . . . 9
⊢ (((𝑑 ∈ Fin ∧ 𝑑 ⊆ 𝑐) ∧ 𝑋 = ∪ 𝑑) → (𝑑 ∈ PtFin ∧ (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑))) |
| 8 | 7 | ancom1s 653 |
. . . . . . . 8
⊢ (((𝑑 ⊆ 𝑐 ∧ 𝑑 ∈ Fin) ∧ 𝑋 = ∪ 𝑑) → (𝑑 ∈ PtFin ∧ (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑))) |
| 9 | 4, 8 | sylanb 581 |
. . . . . . 7
⊢ ((𝑑 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑋 = ∪
𝑑) → (𝑑 ∈ PtFin ∧ (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑))) |
| 10 | 9 | reximi2 3079 |
. . . . . 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 1120 |
. . . 4
⊢ (𝐽 ∈ Comp → (𝑐 ⊆ 𝐽 → (𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)))) |
| 13 | 1, 12 | syl5 34 |
. . 3
⊢ (𝐽 ∈ Comp → (𝑐 ∈ 𝒫 𝐽 → (𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)))) |
| 14 | 13 | ralrimiv 3145 |
. 2
⊢ (𝐽 ∈ Comp →
∀𝑐 ∈ 𝒫
𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑))) |
| 15 | | elpwi 4607 |
. . . . . . 7
⊢ (𝑎 ∈ 𝒫 𝐽 → 𝑎 ⊆ 𝐽) |
| 16 | | 0elpw 5356 |
. . . . . . . . . . 11
⊢ ∅
∈ 𝒫 𝑎 |
| 17 | | 0fi 9082 |
. . . . . . . . . . 11
⊢ ∅
∈ Fin |
| 18 | 16, 17 | elini 4199 |
. . . . . . . . . 10
⊢ ∅
∈ (𝒫 𝑎 ∩
Fin) |
| 19 | | unieq 4918 |
. . . . . . . . . . . 12
⊢ (𝑏 = ∅ → ∪ 𝑏 =
∪ ∅) |
| 20 | | uni0 4935 |
. . . . . . . . . . . 12
⊢ ∪ ∅ = ∅ |
| 21 | 19, 20 | eqtrdi 2793 |
. . . . . . . . . . 11
⊢ (𝑏 = ∅ → ∪ 𝑏 =
∅) |
| 22 | 21 | rspceeqv 3645 |
. . . . . . . . . 10
⊢ ((∅
∈ (𝒫 𝑎 ∩
Fin) ∧ 𝑋 = ∅)
→ ∃𝑏 ∈
(𝒫 𝑎 ∩
Fin)𝑋 = ∪ 𝑏) |
| 23 | 18, 22 | mpan 690 |
. . . . . . . . 9
⊢ (𝑋 = ∅ → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏) |
| 24 | 23 | a1i13 27 |
. . . . . . . 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))) |
| 25 | | n0 4353 |
. . . . . . . . 9
⊢ (𝑋 ≠ ∅ ↔
∃𝑥 𝑥 ∈ 𝑋) |
| 26 | | simp2 1138 |
. . . . . . . . . . . . . 14
⊢ ((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) → 𝑋 = ∪ 𝑎) |
| 27 | 26 | eleq2d 2827 |
. . . . . . . . . . . . 13
⊢ ((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ ∪ 𝑎)) |
| 28 | 27 | biimpd 229 |
. . . . . . . . . . . 12
⊢ ((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) → (𝑥 ∈ 𝑋 → 𝑥 ∈ ∪ 𝑎)) |
| 29 | | eluni2 4911 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ∪ 𝑎
↔ ∃𝑠 ∈
𝑎 𝑥 ∈ 𝑠) |
| 30 | 28, 29 | imbitrdi 251 |
. . . . . . . . . . 11
⊢ ((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) → (𝑥 ∈ 𝑋 → ∃𝑠 ∈ 𝑎 𝑥 ∈ 𝑠)) |
| 31 | | simpl3 1194 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑎 ⊆ 𝐽) |
| 32 | | simprl 771 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑠 ∈ 𝑎) |
| 33 | 31, 32 | sseldd 3984 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑠 ∈ 𝐽) |
| 34 | | elssuni 4937 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑠 ∈ 𝐽 → 𝑠 ⊆ ∪ 𝐽) |
| 35 | 34, 2 | sseqtrrdi 4025 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑠 ∈ 𝐽 → 𝑠 ⊆ 𝑋) |
| 36 | 33, 35 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑠 ⊆ 𝑋) |
| 37 | 36 | ralrimivw 3150 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → ∀𝑝 ∈ 𝑎 𝑠 ⊆ 𝑋) |
| 38 | | iunss 5045 |
. . . . . . . . . . . . . . . . . 18
⊢ (∪ 𝑝 ∈ 𝑎 𝑠 ⊆ 𝑋 ↔ ∀𝑝 ∈ 𝑎 𝑠 ⊆ 𝑋) |
| 39 | 37, 38 | sylibr 234 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → ∪
𝑝 ∈ 𝑎 𝑠 ⊆ 𝑋) |
| 40 | | ssequn1 4186 |
. . . . . . . . . . . . . . . . 17
⊢ (∪ 𝑝 ∈ 𝑎 𝑠 ⊆ 𝑋 ↔ (∪
𝑝 ∈ 𝑎 𝑠 ∪ 𝑋) = 𝑋) |
| 41 | 39, 40 | sylib 218 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → (∪ 𝑝 ∈ 𝑎 𝑠 ∪ 𝑋) = 𝑋) |
| 42 | | simpl2 1193 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑋 = ∪ 𝑎) |
| 43 | | uniiun 5058 |
. . . . . . . . . . . . . . . . . 18
⊢ ∪ 𝑎 =
∪ 𝑝 ∈ 𝑎 𝑝 |
| 44 | 42, 43 | eqtrdi 2793 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑋 = ∪ 𝑝 ∈ 𝑎 𝑝) |
| 45 | 44 | uneq2d 4168 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → (∪ 𝑝 ∈ 𝑎 𝑠 ∪ 𝑋) = (∪
𝑝 ∈ 𝑎 𝑠 ∪ ∪
𝑝 ∈ 𝑎 𝑝)) |
| 46 | 41, 45 | eqtr3d 2779 |
. . . . . . . . . . . . . . 15
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑋 = (∪
𝑝 ∈ 𝑎 𝑠 ∪ ∪
𝑝 ∈ 𝑎 𝑝)) |
| 47 | | iunun 5093 |
. . . . . . . . . . . . . . . 16
⊢ ∪ 𝑝 ∈ 𝑎 (𝑠 ∪ 𝑝) = (∪
𝑝 ∈ 𝑎 𝑠 ∪ ∪
𝑝 ∈ 𝑎 𝑝) |
| 48 | | vex 3484 |
. . . . . . . . . . . . . . . . . 18
⊢ 𝑠 ∈ V |
| 49 | | vex 3484 |
. . . . . . . . . . . . . . . . . 18
⊢ 𝑝 ∈ V |
| 50 | 48, 49 | unex 7764 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 ∪ 𝑝) ∈ V |
| 51 | 50 | dfiun3 5980 |
. . . . . . . . . . . . . . . 16
⊢ ∪ 𝑝 ∈ 𝑎 (𝑠 ∪ 𝑝) = ∪ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) |
| 52 | 47, 51 | eqtr3i 2767 |
. . . . . . . . . . . . . . 15
⊢ (∪ 𝑝 ∈ 𝑎 𝑠 ∪ ∪
𝑝 ∈ 𝑎 𝑝) = ∪ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) |
| 53 | 46, 52 | eqtrdi 2793 |
. . . . . . . . . . . . . 14
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑋 = ∪ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝))) |
| 54 | | simpll1 1213 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ 𝑝 ∈ 𝑎) → 𝐽 ∈ Top) |
| 55 | 33 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ 𝑝 ∈ 𝑎) → 𝑠 ∈ 𝐽) |
| 56 | 31 | sselda 3983 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ 𝑝 ∈ 𝑎) → 𝑝 ∈ 𝐽) |
| 57 | | unopn 22909 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐽 ∈ Top ∧ 𝑠 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) → (𝑠 ∪ 𝑝) ∈ 𝐽) |
| 58 | 54, 55, 56, 57 | syl3anc 1373 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ 𝑝 ∈ 𝑎) → (𝑠 ∪ 𝑝) ∈ 𝐽) |
| 59 | 58 | fmpttd 7135 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)):𝑎⟶𝐽) |
| 60 | 59 | frnd 6744 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ⊆ 𝐽) |
| 61 | | elpw2g 5333 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐽 ∈ Top → (ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∈ 𝒫 𝐽 ↔ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ⊆ 𝐽)) |
| 62 | 61 | 3ad2ant1 1134 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) → (ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∈ 𝒫 𝐽 ↔ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ⊆ 𝐽)) |
| 63 | 62 | 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)) |
| 64 | 60, 63 | mpbird 257 |
. . . . . . . . . . . . . . 15
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∈ 𝒫 𝐽) |
| 65 | | unieq 4918 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑐 = ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) → ∪ 𝑐 = ∪
ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝))) |
| 66 | 65 | eqeq2d 2748 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑐 = ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) → (𝑋 = ∪ 𝑐 ↔ 𝑋 = ∪ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)))) |
| 67 | | sseq2 4010 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑐 = ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) → (𝑑 ⊆ 𝑐 ↔ 𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)))) |
| 68 | 67 | anbi1d 631 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑐 = ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) → ((𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑) ↔ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑))) |
| 69 | 68 | rexbidv 3179 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑐 = ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) → (∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑) ↔ ∃𝑑 ∈ PtFin (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑))) |
| 70 | 66, 69 | imbi12d 344 |
. . . . . . . . . . . . . . . 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)))) |
| 71 | 70 | rspcv 3618 |
. . . . . . . . . . . . . . 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)))) |
| 72 | 64, 71 | 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)))) |
| 73 | 53, 72 | mpid 44 |
. . . . . . . . . . . . 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))) |
| 74 | | simprr 773 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑥 ∈ 𝑠) |
| 75 | | ssel2 3978 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑎 ⊆ 𝐽 ∧ 𝑠 ∈ 𝑎) → 𝑠 ∈ 𝐽) |
| 76 | 75 | 3ad2antl3 1188 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ 𝑠 ∈ 𝑎) → 𝑠 ∈ 𝐽) |
| 77 | 76 | adantrr 717 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑠 ∈ 𝐽) |
| 78 | | elunii 4912 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑥 ∈ 𝑠 ∧ 𝑠 ∈ 𝐽) → 𝑥 ∈ ∪ 𝐽) |
| 79 | 74, 77, 78 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑥 ∈ ∪ 𝐽) |
| 80 | 79, 2 | eleqtrrdi 2852 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑥 ∈ 𝑋) |
| 81 | 80 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → 𝑥 ∈ 𝑋) |
| 82 | | simprr 773 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → 𝑋 = ∪ 𝑑) |
| 83 | 81, 82 | eleqtrd 2843 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → 𝑥 ∈ ∪ 𝑑) |
| 84 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ∪ 𝑑 =
∪ 𝑑 |
| 85 | 84 | ptfinfin 23527 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑑 ∈ PtFin ∧ 𝑥 ∈ ∪ 𝑑)
→ {𝑧 ∈ 𝑑 ∣ 𝑥 ∈ 𝑧} ∈ Fin) |
| 86 | 85 | expcom 413 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈ ∪ 𝑑
→ (𝑑 ∈ PtFin
→ {𝑧 ∈ 𝑑 ∣ 𝑥 ∈ 𝑧} ∈ Fin)) |
| 87 | 83, 86 | 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)) |
| 88 | | simprl 771 |
. . . . . . . . . . . . . . . . . . . . 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))) |
| 89 | | elun1 4182 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑥 ∈ 𝑠 → 𝑥 ∈ (𝑠 ∪ 𝑝)) |
| 90 | 89 | ad2antll 729 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑥 ∈ (𝑠 ∪ 𝑝)) |
| 91 | 90 | ralrimivw 3150 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → ∀𝑝 ∈ 𝑎 𝑥 ∈ (𝑠 ∪ 𝑝)) |
| 92 | 50 | rgenw 3065 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
∀𝑝 ∈
𝑎 (𝑠 ∪ 𝑝) ∈ V |
| 93 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) = (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) |
| 94 | | eleq2 2830 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑧 = (𝑠 ∪ 𝑝) → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ (𝑠 ∪ 𝑝))) |
| 95 | 93, 94 | ralrnmptw 7114 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(∀𝑝 ∈
𝑎 (𝑠 ∪ 𝑝) ∈ V → (∀𝑧 ∈ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝))𝑥 ∈ 𝑧 ↔ ∀𝑝 ∈ 𝑎 𝑥 ∈ (𝑠 ∪ 𝑝))) |
| 96 | 92, 95 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(∀𝑧 ∈
ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝))𝑥 ∈ 𝑧 ↔ ∀𝑝 ∈ 𝑎 𝑥 ∈ (𝑠 ∪ 𝑝)) |
| 97 | 91, 96 | sylibr 234 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → ∀𝑧 ∈ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝))𝑥 ∈ 𝑧) |
| 98 | 97 | 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) |
| 99 | | ssralv 4052 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) → (∀𝑧 ∈ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝))𝑥 ∈ 𝑧 → ∀𝑧 ∈ 𝑑 𝑥 ∈ 𝑧)) |
| 100 | 88, 98, 99 | sylc 65 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → ∀𝑧 ∈ 𝑑 𝑥 ∈ 𝑧) |
| 101 | | rabid2 3470 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑑 = {𝑧 ∈ 𝑑 ∣ 𝑥 ∈ 𝑧} ↔ ∀𝑧 ∈ 𝑑 𝑥 ∈ 𝑧) |
| 102 | 100, 101 | sylibr 234 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → 𝑑 = {𝑧 ∈ 𝑑 ∣ 𝑥 ∈ 𝑧}) |
| 103 | 102 | eleq1d 2826 |
. . . . . . . . . . . . . . . . . 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)) |
| 104 | 103 | biimprd 248 |
. . . . . . . . . . . . . . . . 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)) |
| 105 | 93 | rnmpt 5968 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ran
(𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) = {𝑞 ∣ ∃𝑝 ∈ 𝑎 𝑞 = (𝑠 ∪ 𝑝)} |
| 106 | 88, 105 | sseqtrdi 4024 |
. . . . . . . . . . . . . . . . . . . 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)}) |
| 107 | | ssabral 4065 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑑 ⊆ {𝑞 ∣ ∃𝑝 ∈ 𝑎 𝑞 = (𝑠 ∪ 𝑝)} ↔ ∀𝑞 ∈ 𝑑 ∃𝑝 ∈ 𝑎 𝑞 = (𝑠 ∪ 𝑝)) |
| 108 | 106, 107 | sylib 218 |
. . . . . . . . . . . . . . . . . . 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)) |
| 109 | | uneq2 4162 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑝 = (𝑓‘𝑞) → (𝑠 ∪ 𝑝) = (𝑠 ∪ (𝑓‘𝑞))) |
| 110 | 109 | eqeq2d 2748 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑝 = (𝑓‘𝑞) → (𝑞 = (𝑠 ∪ 𝑝) ↔ 𝑞 = (𝑠 ∪ (𝑓‘𝑞)))) |
| 111 | 110 | ac6sfi 9320 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑑 ∈ Fin ∧ ∀𝑞 ∈ 𝑑 ∃𝑝 ∈ 𝑎 𝑞 = (𝑠 ∪ 𝑝)) → ∃𝑓(𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞)))) |
| 112 | 111 | expcom 413 |
. . . . . . . . . . . . . . . . . . 19
⊢
(∀𝑞 ∈
𝑑 ∃𝑝 ∈ 𝑎 𝑞 = (𝑠 ∪ 𝑝) → (𝑑 ∈ Fin → ∃𝑓(𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) |
| 113 | 108, 112 | 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))))) |
| 114 | | frn 6743 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑓:𝑑⟶𝑎 → ran 𝑓 ⊆ 𝑎) |
| 115 | 114 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))) → ran 𝑓 ⊆ 𝑎) |
| 116 | 115 | ad2antll 729 |
. . . . . . . . . . . . . . . . . . . . . . . 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) |
| 117 | 32 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . . . . . . 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) |
| 118 | 117 | snssd 4809 |
. . . . . . . . . . . . . . . . . . . . . . . 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) |
| 119 | 116, 118 | unssd 4192 |
. . . . . . . . . . . . . . . . . . . . . . 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) |
| 120 | | simprl 771 |
. . . . . . . . . . . . . . . . . . . . . . . . 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) |
| 121 | | simprrl 781 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 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) |
| 122 | 121 | ffnd 6737 |
. . . . . . . . . . . . . . . . . . . . . . . . . 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) |
| 123 | | dffn4 6826 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑓 Fn 𝑑 ↔ 𝑓:𝑑–onto→ran 𝑓) |
| 124 | 122, 123 | sylib 218 |
. . . . . . . . . . . . . . . . . . . . . . . . 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) |
| 125 | | fofi 9351 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑑 ∈ Fin ∧ 𝑓:𝑑–onto→ran 𝑓) → ran 𝑓 ∈ Fin) |
| 126 | 120, 124,
125 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . . . . 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) |
| 127 | | snfi 9083 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ {𝑠} ∈ Fin |
| 128 | | unfi 9211 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((ran
𝑓 ∈ Fin ∧ {𝑠} ∈ Fin) → (ran 𝑓 ∪ {𝑠}) ∈ Fin) |
| 129 | 126, 127,
128 | sylancl 586 |
. . . . . . . . . . . . . . . . . . . . . . 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) |
| 130 | | elfpw 9394 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((ran
𝑓 ∪ {𝑠}) ∈ (𝒫 𝑎 ∩ Fin) ↔ ((ran 𝑓 ∪ {𝑠}) ⊆ 𝑎 ∧ (ran 𝑓 ∪ {𝑠}) ∈ Fin)) |
| 131 | 119, 129,
130 | sylanbrc 583 |
. . . . . . . . . . . . . . . . . . . . . 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)) |
| 132 | | simplrr 778 |
. . . . . . . . . . . . . . . . . . . . . . . . 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) |
| 133 | | uniiun 5058 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ∪ 𝑑 =
∪ 𝑞 ∈ 𝑑 𝑞 |
| 134 | | simprrr 782 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 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))) |
| 135 | | iuneq2 5011 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
(∀𝑞 ∈
𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞)) → ∪
𝑞 ∈ 𝑑 𝑞 = ∪ 𝑞 ∈ 𝑑 (𝑠 ∪ (𝑓‘𝑞))) |
| 136 | 134, 135 | 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))) |
| 137 | 133, 136 | eqtrid 2789 |
. . . . . . . . . . . . . . . . . . . . . . . . 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))) |
| 138 | 132, 137 | eqtrd 2777 |
. . . . . . . . . . . . . . . . . . . . . . . 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))) |
| 139 | | ssun2 4179 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ {𝑠} ⊆ (ran 𝑓 ∪ {𝑠}) |
| 140 | | vsnid 4663 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ 𝑠 ∈ {𝑠} |
| 141 | 139, 140 | sselii 3980 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ 𝑠 ∈ (ran 𝑓 ∪ {𝑠}) |
| 142 | | elssuni 4937 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑠 ∈ (ran 𝑓 ∪ {𝑠}) → 𝑠 ⊆ ∪ (ran
𝑓 ∪ {𝑠})) |
| 143 | 141, 142 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ 𝑠 ⊆ ∪ (ran 𝑓 ∪ {𝑠}) |
| 144 | | fvssunirn 6939 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑓‘𝑞) ⊆ ∪ ran
𝑓 |
| 145 | | ssun1 4178 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ran 𝑓 ⊆ (ran 𝑓 ∪ {𝑠}) |
| 146 | 145 | unissi 4916 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ∪ ran 𝑓 ⊆ ∪ (ran
𝑓 ∪ {𝑠}) |
| 147 | 144, 146 | sstri 3993 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑓‘𝑞) ⊆ ∪ (ran
𝑓 ∪ {𝑠}) |
| 148 | 143, 147 | unssi 4191 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑠 ∪ (𝑓‘𝑞)) ⊆ ∪ (ran
𝑓 ∪ {𝑠}) |
| 149 | 148 | rgenw 3065 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
∀𝑞 ∈
𝑑 (𝑠 ∪ (𝑓‘𝑞)) ⊆ ∪ (ran
𝑓 ∪ {𝑠}) |
| 150 | | iunss 5045 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (∪ 𝑞 ∈ 𝑑 (𝑠 ∪ (𝑓‘𝑞)) ⊆ ∪ (ran
𝑓 ∪ {𝑠}) ↔ ∀𝑞 ∈ 𝑑 (𝑠 ∪ (𝑓‘𝑞)) ⊆ ∪ (ran
𝑓 ∪ {𝑠})) |
| 151 | 149, 150 | mpbir 231 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ∪ 𝑞 ∈ 𝑑 (𝑠 ∪ (𝑓‘𝑞)) ⊆ ∪ (ran
𝑓 ∪ {𝑠}) |
| 152 | 138, 151 | eqsstrdi 4028 |
. . . . . . . . . . . . . . . . . . . . . . 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})) |
| 153 | 31 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . . . . . . . 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) |
| 154 | 116, 153 | sstrd 3994 |
. . . . . . . . . . . . . . . . . . . . . . . . 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) |
| 155 | 33 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . . . . . . . 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) |
| 156 | 155 | snssd 4809 |
. . . . . . . . . . . . . . . . . . . . . . . . 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) |
| 157 | 154, 156 | unssd 4192 |
. . . . . . . . . . . . . . . . . . . . . . . 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) |
| 158 | | uniss 4915 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((ran
𝑓 ∪ {𝑠}) ⊆ 𝐽 → ∪ (ran
𝑓 ∪ {𝑠}) ⊆ ∪ 𝐽) |
| 159 | 158, 2 | sseqtrrdi 4025 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((ran
𝑓 ∪ {𝑠}) ⊆ 𝐽 → ∪ (ran
𝑓 ∪ {𝑠}) ⊆ 𝑋) |
| 160 | 157, 159 | 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) |
| 161 | 152, 160 | eqssd 4001 |
. . . . . . . . . . . . . . . . . . . . . 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})) |
| 162 | | unieq 4918 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑏 = (ran 𝑓 ∪ {𝑠}) → ∪ 𝑏 = ∪
(ran 𝑓 ∪ {𝑠})) |
| 163 | 162 | rspceeqv 3645 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((ran
𝑓 ∪ {𝑠}) ∈ (𝒫 𝑎 ∩ Fin) ∧ 𝑋 = ∪ (ran 𝑓 ∪ {𝑠})) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏) |
| 164 | 131, 161,
163 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . 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) |
| 165 | 164 | expr 456 |
. . . . . . . . . . . . . . . . . . . 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)) |
| 166 | 165 | exlimdv 1933 |
. . . . . . . . . . . . . . . . . . 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)) |
| 167 | 166 | ex 412 |
. . . . . . . . . . . . . . . . . 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))) |
| 168 | 113, 167 | mpdd 43 |
. . . . . . . . . . . . . . . . 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)) |
| 169 | 87, 104, 168 | 3syld 60 |
. . . . . . . . . . . . . . . 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)) |
| 170 | 169 | ex 412 |
. . . . . . . . . . . . . . 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))) |
| 171 | 170 | com23 86 |
. . . . . . . . . . . . . 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))) |
| 172 | 171 | rexlimdv 3153 |
. . . . . . . . . . . . 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)) |
| 173 | 73, 172 | syld 47 |
. . . . . . . . . . . 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)) |
| 174 | 173 | rexlimdvaa 3156 |
. . . . . . . . . . 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))) |
| 175 | 30, 174 | syld 47 |
. . . . . . . . . 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))) |
| 176 | 175 | exlimdv 1933 |
. . . . . . . . 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))) |
| 177 | 25, 176 | biimtrid 242 |
. . . . . . . 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))) |
| 178 | 24, 177 | pm2.61dne 3028 |
. . . . . . 7
⊢ ((𝐽 ∈ Top ∧ 𝑋 = ∪
𝑎 ∧ 𝑎 ⊆ 𝐽) → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏)) |
| 179 | 15, 178 | syl3an3 1166 |
. . . . . 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)) |
| 180 | 179 | 3exp 1120 |
. . . . 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)))) |
| 181 | 180 | com24 95 |
. . . 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)))) |
| 182 | 181 | ralrimdv 3152 |
. . 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))) |
| 183 | 2 | iscmp 23396 |
. . . 4
⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑎 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏))) |
| 184 | 183 | baibr 536 |
. . 3
⊢ (𝐽 ∈ Top →
(∀𝑎 ∈ 𝒫
𝐽(𝑋 = ∪ 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏) ↔ 𝐽 ∈ Comp)) |
| 185 | 182, 184 | sylibd 239 |
. 2
⊢ (𝐽 ∈ Top →
(∀𝑐 ∈ 𝒫
𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → 𝐽 ∈ Comp)) |
| 186 | 14, 185 | impbid2 226 |
1
⊢ (𝐽 ∈ Top → (𝐽 ∈ Comp ↔
∀𝑐 ∈ 𝒫
𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)))) |