Proof of Theorem isf34lem5
| Step | Hyp | Ref
| Expression |
| 1 | | imassrn 6089 |
. . . . . . 7
⊢ (𝐹 “ 𝑋) ⊆ ran 𝐹 |
| 2 | | compss.a |
. . . . . . . . . 10
⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) |
| 3 | 2 | isf34lem2 10413 |
. . . . . . . . 9
⊢ (𝐴 ∈ 𝑉 → 𝐹:𝒫 𝐴⟶𝒫 𝐴) |
| 4 | 3 | adantr 480 |
. . . . . . . 8
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → 𝐹:𝒫 𝐴⟶𝒫 𝐴) |
| 5 | 4 | frnd 6744 |
. . . . . . 7
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → ran 𝐹 ⊆ 𝒫 𝐴) |
| 6 | 1, 5 | sstrid 3995 |
. . . . . 6
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹 “ 𝑋) ⊆ 𝒫 𝐴) |
| 7 | | simprl 771 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → 𝑋 ⊆ 𝒫 𝐴) |
| 8 | 4 | fdmd 6746 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → dom 𝐹 = 𝒫 𝐴) |
| 9 | 7, 8 | sseqtrrd 4021 |
. . . . . . . . 9
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → 𝑋 ⊆ dom 𝐹) |
| 10 | | sseqin2 4223 |
. . . . . . . . 9
⊢ (𝑋 ⊆ dom 𝐹 ↔ (dom 𝐹 ∩ 𝑋) = 𝑋) |
| 11 | 9, 10 | sylib 218 |
. . . . . . . 8
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (dom 𝐹 ∩ 𝑋) = 𝑋) |
| 12 | | simprr 773 |
. . . . . . . 8
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → 𝑋 ≠ ∅) |
| 13 | 11, 12 | eqnetrd 3008 |
. . . . . . 7
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (dom 𝐹 ∩ 𝑋) ≠ ∅) |
| 14 | | imadisj 6098 |
. . . . . . . 8
⊢ ((𝐹 “ 𝑋) = ∅ ↔ (dom 𝐹 ∩ 𝑋) = ∅) |
| 15 | 14 | necon3bii 2993 |
. . . . . . 7
⊢ ((𝐹 “ 𝑋) ≠ ∅ ↔ (dom 𝐹 ∩ 𝑋) ≠ ∅) |
| 16 | 13, 15 | sylibr 234 |
. . . . . 6
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹 “ 𝑋) ≠ ∅) |
| 17 | 6, 16 | jca 511 |
. . . . 5
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → ((𝐹 “ 𝑋) ⊆ 𝒫 𝐴 ∧ (𝐹 “ 𝑋) ≠ ∅)) |
| 18 | 2 | isf34lem4 10417 |
. . . . 5
⊢ ((𝐴 ∈ 𝑉 ∧ ((𝐹 “ 𝑋) ⊆ 𝒫 𝐴 ∧ (𝐹 “ 𝑋) ≠ ∅)) → (𝐹‘∪ (𝐹 “ 𝑋)) = ∩ (𝐹 “ (𝐹 “ 𝑋))) |
| 19 | 17, 18 | syldan 591 |
. . . 4
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹‘∪ (𝐹 “ 𝑋)) = ∩ (𝐹 “ (𝐹 “ 𝑋))) |
| 20 | 2 | isf34lem3 10415 |
. . . . . 6
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ⊆ 𝒫 𝐴) → (𝐹 “ (𝐹 “ 𝑋)) = 𝑋) |
| 21 | 20 | adantrr 717 |
. . . . 5
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹 “ (𝐹 “ 𝑋)) = 𝑋) |
| 22 | 21 | inteqd 4951 |
. . . 4
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → ∩ (𝐹
“ (𝐹 “ 𝑋)) = ∩ 𝑋) |
| 23 | 19, 22 | eqtrd 2777 |
. . 3
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹‘∪ (𝐹 “ 𝑋)) = ∩ 𝑋) |
| 24 | 23 | fveq2d 6910 |
. 2
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹‘(𝐹‘∪ (𝐹 “ 𝑋))) = (𝐹‘∩ 𝑋)) |
| 25 | 2 | compsscnv 10411 |
. . . 4
⊢ ◡𝐹 = 𝐹 |
| 26 | 25 | fveq1i 6907 |
. . 3
⊢ (◡𝐹‘(𝐹‘∪ (𝐹 “ 𝑋))) = (𝐹‘(𝐹‘∪ (𝐹 “ 𝑋))) |
| 27 | 2 | compssiso 10414 |
. . . . 5
⊢ (𝐴 ∈ 𝑉 → 𝐹 Isom [⊊] , ◡ [⊊] (𝒫 𝐴, 𝒫 𝐴)) |
| 28 | | isof1o 7343 |
. . . . 5
⊢ (𝐹 Isom [⊊] , ◡ [⊊] (𝒫 𝐴, 𝒫 𝐴) → 𝐹:𝒫 𝐴–1-1-onto→𝒫 𝐴) |
| 29 | 27, 28 | syl 17 |
. . . 4
⊢ (𝐴 ∈ 𝑉 → 𝐹:𝒫 𝐴–1-1-onto→𝒫 𝐴) |
| 30 | | sspwuni 5100 |
. . . . . 6
⊢ ((𝐹 “ 𝑋) ⊆ 𝒫 𝐴 ↔ ∪ (𝐹 “ 𝑋) ⊆ 𝐴) |
| 31 | 6, 30 | sylib 218 |
. . . . 5
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → ∪ (𝐹
“ 𝑋) ⊆ 𝐴) |
| 32 | | elpw2g 5333 |
. . . . . 6
⊢ (𝐴 ∈ 𝑉 → (∪ (𝐹 “ 𝑋) ∈ 𝒫 𝐴 ↔ ∪ (𝐹 “ 𝑋) ⊆ 𝐴)) |
| 33 | 32 | adantr 480 |
. . . . 5
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (∪ (𝐹
“ 𝑋) ∈ 𝒫
𝐴 ↔ ∪ (𝐹
“ 𝑋) ⊆ 𝐴)) |
| 34 | 31, 33 | mpbird 257 |
. . . 4
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → ∪ (𝐹
“ 𝑋) ∈ 𝒫
𝐴) |
| 35 | | f1ocnvfv1 7296 |
. . . 4
⊢ ((𝐹:𝒫 𝐴–1-1-onto→𝒫 𝐴 ∧ ∪ (𝐹 “ 𝑋) ∈ 𝒫 𝐴) → (◡𝐹‘(𝐹‘∪ (𝐹 “ 𝑋))) = ∪ (𝐹 “ 𝑋)) |
| 36 | 29, 34, 35 | syl2an2r 685 |
. . 3
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (◡𝐹‘(𝐹‘∪ (𝐹 “ 𝑋))) = ∪ (𝐹 “ 𝑋)) |
| 37 | 26, 36 | eqtr3id 2791 |
. 2
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹‘(𝐹‘∪ (𝐹 “ 𝑋))) = ∪ (𝐹 “ 𝑋)) |
| 38 | 24, 37 | eqtr3d 2779 |
1
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅)) → (𝐹‘∩ 𝑋) = ∪
(𝐹 “ 𝑋)) |