| Step | Hyp | Ref
| Expression |
| 1 | | fin23lem.e |
. . 3
⊢ 𝑍 = if(𝑃 ∈ Fin, (𝑡 ∘ 𝑅), ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran
𝑈)) ∘ 𝑄)) |
| 2 | | eqif 4567 |
. . 3
⊢ (𝑍 = if(𝑃 ∈ Fin, (𝑡 ∘ 𝑅), ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran
𝑈)) ∘ 𝑄)) ↔ ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡 ∘ 𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran
𝑈)) ∘ 𝑄)))) |
| 3 | 1, 2 | mpbi 230 |
. 2
⊢ ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡 ∘ 𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran
𝑈)) ∘ 𝑄))) |
| 4 | | difss 4136 |
. . . . . . . . 9
⊢ (ω
∖ 𝑃) ⊆
ω |
| 5 | | ominf 9294 |
. . . . . . . . . 10
⊢ ¬
ω ∈ Fin |
| 6 | | fin23lem.b |
. . . . . . . . . . . . . 14
⊢ 𝑃 = {𝑣 ∈ ω ∣ ∩ ran 𝑈 ⊆ (𝑡‘𝑣)} |
| 7 | 6 | ssrab3 4082 |
. . . . . . . . . . . . 13
⊢ 𝑃 ⊆
ω |
| 8 | | undif 4482 |
. . . . . . . . . . . . 13
⊢ (𝑃 ⊆ ω ↔ (𝑃 ∪ (ω ∖ 𝑃)) = ω) |
| 9 | 7, 8 | mpbi 230 |
. . . . . . . . . . . 12
⊢ (𝑃 ∪ (ω ∖ 𝑃)) = ω |
| 10 | | unfi 9211 |
. . . . . . . . . . . 12
⊢ ((𝑃 ∈ Fin ∧ (ω
∖ 𝑃) ∈ Fin)
→ (𝑃 ∪ (ω
∖ 𝑃)) ∈
Fin) |
| 11 | 9, 10 | eqeltrrid 2846 |
. . . . . . . . . . 11
⊢ ((𝑃 ∈ Fin ∧ (ω
∖ 𝑃) ∈ Fin)
→ ω ∈ Fin) |
| 12 | 11 | ex 412 |
. . . . . . . . . 10
⊢ (𝑃 ∈ Fin → ((ω
∖ 𝑃) ∈ Fin
→ ω ∈ Fin)) |
| 13 | 5, 12 | mtoi 199 |
. . . . . . . . 9
⊢ (𝑃 ∈ Fin → ¬
(ω ∖ 𝑃) ∈
Fin) |
| 14 | | fin23lem.d |
. . . . . . . . . 10
⊢ 𝑅 = (𝑤 ∈ ω ↦ (℩𝑥 ∈ (ω ∖ 𝑃)(𝑥 ∩ (ω ∖ 𝑃)) ≈ 𝑤)) |
| 15 | 14 | fin23lem22 10367 |
. . . . . . . . 9
⊢
(((ω ∖ 𝑃) ⊆ ω ∧ ¬ (ω
∖ 𝑃) ∈ Fin)
→ 𝑅:ω–1-1-onto→(ω ∖ 𝑃)) |
| 16 | 4, 13, 15 | sylancr 587 |
. . . . . . . 8
⊢ (𝑃 ∈ Fin → 𝑅:ω–1-1-onto→(ω ∖ 𝑃)) |
| 17 | 16 | adantl 481 |
. . . . . . 7
⊢ ((𝑡:ω–1-1→V ∧ 𝑃 ∈ Fin) → 𝑅:ω–1-1-onto→(ω ∖ 𝑃)) |
| 18 | | f1of1 6847 |
. . . . . . 7
⊢ (𝑅:ω–1-1-onto→(ω ∖ 𝑃) → 𝑅:ω–1-1→(ω ∖ 𝑃)) |
| 19 | | f1ss 6809 |
. . . . . . . 8
⊢ ((𝑅:ω–1-1→(ω ∖ 𝑃) ∧ (ω ∖ 𝑃) ⊆ ω) → 𝑅:ω–1-1→ω) |
| 20 | 4, 19 | mpan2 691 |
. . . . . . 7
⊢ (𝑅:ω–1-1→(ω ∖ 𝑃) → 𝑅:ω–1-1→ω) |
| 21 | 17, 18, 20 | 3syl 18 |
. . . . . 6
⊢ ((𝑡:ω–1-1→V ∧ 𝑃 ∈ Fin) → 𝑅:ω–1-1→ω) |
| 22 | | f1co 6815 |
. . . . . 6
⊢ ((𝑡:ω–1-1→V ∧ 𝑅:ω–1-1→ω) → (𝑡 ∘ 𝑅):ω–1-1→V) |
| 23 | 21, 22 | syldan 591 |
. . . . 5
⊢ ((𝑡:ω–1-1→V ∧ 𝑃 ∈ Fin) → (𝑡 ∘ 𝑅):ω–1-1→V) |
| 24 | | f1eq1 6799 |
. . . . 5
⊢ (𝑍 = (𝑡 ∘ 𝑅) → (𝑍:ω–1-1→V ↔ (𝑡 ∘ 𝑅):ω–1-1→V)) |
| 25 | 23, 24 | syl5ibrcom 247 |
. . . 4
⊢ ((𝑡:ω–1-1→V ∧ 𝑃 ∈ Fin) → (𝑍 = (𝑡 ∘ 𝑅) → 𝑍:ω–1-1→V)) |
| 26 | 25 | impr 454 |
. . 3
⊢ ((𝑡:ω–1-1→V ∧ (𝑃 ∈ Fin ∧ 𝑍 = (𝑡 ∘ 𝑅))) → 𝑍:ω–1-1→V) |
| 27 | | fvex 6919 |
. . . . . . . . . . 11
⊢ (𝑡‘𝑧) ∈ V |
| 28 | 27 | difexi 5330 |
. . . . . . . . . 10
⊢ ((𝑡‘𝑧) ∖ ∩ ran
𝑈) ∈
V |
| 29 | 28 | rgenw 3065 |
. . . . . . . . 9
⊢
∀𝑧 ∈
𝑃 ((𝑡‘𝑧) ∖ ∩ ran
𝑈) ∈
V |
| 30 | | eqid 2737 |
. . . . . . . . . 10
⊢ (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran
𝑈)) = (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran
𝑈)) |
| 31 | 30 | fmpt 7130 |
. . . . . . . . 9
⊢
(∀𝑧 ∈
𝑃 ((𝑡‘𝑧) ∖ ∩ ran
𝑈) ∈ V ↔ (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran
𝑈)):𝑃⟶V) |
| 32 | 29, 31 | mpbi 230 |
. . . . . . . 8
⊢ (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran
𝑈)):𝑃⟶V |
| 33 | 32 | a1i 11 |
. . . . . . 7
⊢ (𝑡:ω–1-1→V → (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran
𝑈)):𝑃⟶V) |
| 34 | | fveq2 6906 |
. . . . . . . . . . . . 13
⊢ (𝑧 = 𝑎 → (𝑡‘𝑧) = (𝑡‘𝑎)) |
| 35 | 34 | difeq1d 4125 |
. . . . . . . . . . . 12
⊢ (𝑧 = 𝑎 → ((𝑡‘𝑧) ∖ ∩ ran
𝑈) = ((𝑡‘𝑎) ∖ ∩ ran
𝑈)) |
| 36 | | fvex 6919 |
. . . . . . . . . . . . 13
⊢ (𝑡‘𝑎) ∈ V |
| 37 | 36 | difexi 5330 |
. . . . . . . . . . . 12
⊢ ((𝑡‘𝑎) ∖ ∩ ran
𝑈) ∈
V |
| 38 | 35, 30, 37 | fvmpt 7016 |
. . . . . . . . . . 11
⊢ (𝑎 ∈ 𝑃 → ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran
𝑈))‘𝑎) = ((𝑡‘𝑎) ∖ ∩ ran
𝑈)) |
| 39 | 38 | ad2antrl 728 |
. . . . . . . . . 10
⊢ ((𝑡:ω–1-1→V ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran
𝑈))‘𝑎) = ((𝑡‘𝑎) ∖ ∩ ran
𝑈)) |
| 40 | | fveq2 6906 |
. . . . . . . . . . . . 13
⊢ (𝑧 = 𝑏 → (𝑡‘𝑧) = (𝑡‘𝑏)) |
| 41 | 40 | difeq1d 4125 |
. . . . . . . . . . . 12
⊢ (𝑧 = 𝑏 → ((𝑡‘𝑧) ∖ ∩ ran
𝑈) = ((𝑡‘𝑏) ∖ ∩ ran
𝑈)) |
| 42 | | fvex 6919 |
. . . . . . . . . . . . 13
⊢ (𝑡‘𝑏) ∈ V |
| 43 | 42 | difexi 5330 |
. . . . . . . . . . . 12
⊢ ((𝑡‘𝑏) ∖ ∩ ran
𝑈) ∈
V |
| 44 | 41, 30, 43 | fvmpt 7016 |
. . . . . . . . . . 11
⊢ (𝑏 ∈ 𝑃 → ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran
𝑈))‘𝑏) = ((𝑡‘𝑏) ∖ ∩ ran
𝑈)) |
| 45 | 44 | ad2antll 729 |
. . . . . . . . . 10
⊢ ((𝑡:ω–1-1→V ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran
𝑈))‘𝑏) = ((𝑡‘𝑏) ∖ ∩ ran
𝑈)) |
| 46 | 39, 45 | eqeq12d 2753 |
. . . . . . . . 9
⊢ ((𝑡:ω–1-1→V ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran
𝑈))‘𝑎) = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran
𝑈))‘𝑏) ↔ ((𝑡‘𝑎) ∖ ∩ ran
𝑈) = ((𝑡‘𝑏) ∖ ∩ ran
𝑈))) |
| 47 | | uneq2 4162 |
. . . . . . . . . . 11
⊢ (((𝑡‘𝑎) ∖ ∩ ran
𝑈) = ((𝑡‘𝑏) ∖ ∩ ran
𝑈) → (∩ ran 𝑈 ∪ ((𝑡‘𝑎) ∖ ∩ ran
𝑈)) = (∩ ran 𝑈 ∪ ((𝑡‘𝑏) ∖ ∩ ran
𝑈))) |
| 48 | | fveq2 6906 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑣 = 𝑎 → (𝑡‘𝑣) = (𝑡‘𝑎)) |
| 49 | 48 | sseq2d 4016 |
. . . . . . . . . . . . . . . 16
⊢ (𝑣 = 𝑎 → (∩ ran
𝑈 ⊆ (𝑡‘𝑣) ↔ ∩ ran
𝑈 ⊆ (𝑡‘𝑎))) |
| 50 | 49, 6 | elrab2 3695 |
. . . . . . . . . . . . . . 15
⊢ (𝑎 ∈ 𝑃 ↔ (𝑎 ∈ ω ∧ ∩ ran 𝑈 ⊆ (𝑡‘𝑎))) |
| 51 | 50 | simprbi 496 |
. . . . . . . . . . . . . 14
⊢ (𝑎 ∈ 𝑃 → ∩ ran
𝑈 ⊆ (𝑡‘𝑎)) |
| 52 | 51 | ad2antrl 728 |
. . . . . . . . . . . . 13
⊢ ((𝑡:ω–1-1→V ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ∩ ran
𝑈 ⊆ (𝑡‘𝑎)) |
| 53 | | undif 4482 |
. . . . . . . . . . . . 13
⊢ (∩ ran 𝑈 ⊆ (𝑡‘𝑎) ↔ (∩ ran
𝑈 ∪ ((𝑡‘𝑎) ∖ ∩ ran
𝑈)) = (𝑡‘𝑎)) |
| 54 | 52, 53 | sylib 218 |
. . . . . . . . . . . 12
⊢ ((𝑡:ω–1-1→V ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (∩ ran
𝑈 ∪ ((𝑡‘𝑎) ∖ ∩ ran
𝑈)) = (𝑡‘𝑎)) |
| 55 | | fveq2 6906 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑣 = 𝑏 → (𝑡‘𝑣) = (𝑡‘𝑏)) |
| 56 | 55 | sseq2d 4016 |
. . . . . . . . . . . . . . . 16
⊢ (𝑣 = 𝑏 → (∩ ran
𝑈 ⊆ (𝑡‘𝑣) ↔ ∩ ran
𝑈 ⊆ (𝑡‘𝑏))) |
| 57 | 56, 6 | elrab2 3695 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 ∈ 𝑃 ↔ (𝑏 ∈ ω ∧ ∩ ran 𝑈 ⊆ (𝑡‘𝑏))) |
| 58 | 57 | simprbi 496 |
. . . . . . . . . . . . . 14
⊢ (𝑏 ∈ 𝑃 → ∩ ran
𝑈 ⊆ (𝑡‘𝑏)) |
| 59 | 58 | ad2antll 729 |
. . . . . . . . . . . . 13
⊢ ((𝑡:ω–1-1→V ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ∩ ran
𝑈 ⊆ (𝑡‘𝑏)) |
| 60 | | undif 4482 |
. . . . . . . . . . . . 13
⊢ (∩ ran 𝑈 ⊆ (𝑡‘𝑏) ↔ (∩ ran
𝑈 ∪ ((𝑡‘𝑏) ∖ ∩ ran
𝑈)) = (𝑡‘𝑏)) |
| 61 | 59, 60 | sylib 218 |
. . . . . . . . . . . 12
⊢ ((𝑡:ω–1-1→V ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (∩ ran
𝑈 ∪ ((𝑡‘𝑏) ∖ ∩ ran
𝑈)) = (𝑡‘𝑏)) |
| 62 | 54, 61 | eqeq12d 2753 |
. . . . . . . . . . 11
⊢ ((𝑡:ω–1-1→V ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ((∩ ran
𝑈 ∪ ((𝑡‘𝑎) ∖ ∩ ran
𝑈)) = (∩ ran 𝑈 ∪ ((𝑡‘𝑏) ∖ ∩ ran
𝑈)) ↔ (𝑡‘𝑎) = (𝑡‘𝑏))) |
| 63 | 47, 62 | imbitrid 244 |
. . . . . . . . . 10
⊢ ((𝑡:ω–1-1→V ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (((𝑡‘𝑎) ∖ ∩ ran
𝑈) = ((𝑡‘𝑏) ∖ ∩ ran
𝑈) → (𝑡‘𝑎) = (𝑡‘𝑏))) |
| 64 | 7 | sseli 3979 |
. . . . . . . . . . . 12
⊢ (𝑎 ∈ 𝑃 → 𝑎 ∈ ω) |
| 65 | 7 | sseli 3979 |
. . . . . . . . . . . 12
⊢ (𝑏 ∈ 𝑃 → 𝑏 ∈ ω) |
| 66 | 64, 65 | anim12i 613 |
. . . . . . . . . . 11
⊢ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) → (𝑎 ∈ ω ∧ 𝑏 ∈ ω)) |
| 67 | | f1fveq 7282 |
. . . . . . . . . . 11
⊢ ((𝑡:ω–1-1→V ∧ (𝑎 ∈ ω ∧ 𝑏 ∈ ω)) → ((𝑡‘𝑎) = (𝑡‘𝑏) ↔ 𝑎 = 𝑏)) |
| 68 | 66, 67 | sylan2 593 |
. . . . . . . . . 10
⊢ ((𝑡:ω–1-1→V ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → ((𝑡‘𝑎) = (𝑡‘𝑏) ↔ 𝑎 = 𝑏)) |
| 69 | 63, 68 | sylibd 239 |
. . . . . . . . 9
⊢ ((𝑡:ω–1-1→V ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (((𝑡‘𝑎) ∖ ∩ ran
𝑈) = ((𝑡‘𝑏) ∖ ∩ ran
𝑈) → 𝑎 = 𝑏)) |
| 70 | 46, 69 | sylbid 240 |
. . . . . . . 8
⊢ ((𝑡:ω–1-1→V ∧ (𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃)) → (((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran
𝑈))‘𝑎) = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran
𝑈))‘𝑏) → 𝑎 = 𝑏)) |
| 71 | 70 | ralrimivva 3202 |
. . . . . . 7
⊢ (𝑡:ω–1-1→V → ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 (((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran
𝑈))‘𝑎) = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran
𝑈))‘𝑏) → 𝑎 = 𝑏)) |
| 72 | | dff13 7275 |
. . . . . . 7
⊢ ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran
𝑈)):𝑃–1-1→V ↔ ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran
𝑈)):𝑃⟶V ∧ ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 (((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran
𝑈))‘𝑎) = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran
𝑈))‘𝑏) → 𝑎 = 𝑏))) |
| 73 | 33, 71, 72 | sylanbrc 583 |
. . . . . 6
⊢ (𝑡:ω–1-1→V → (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran
𝑈)):𝑃–1-1→V) |
| 74 | | fin23lem.c |
. . . . . . . . 9
⊢ 𝑄 = (𝑤 ∈ ω ↦ (℩𝑥 ∈ 𝑃 (𝑥 ∩ 𝑃) ≈ 𝑤)) |
| 75 | 74 | fin23lem22 10367 |
. . . . . . . 8
⊢ ((𝑃 ⊆ ω ∧ ¬
𝑃 ∈ Fin) → 𝑄:ω–1-1-onto→𝑃) |
| 76 | | f1of1 6847 |
. . . . . . . 8
⊢ (𝑄:ω–1-1-onto→𝑃 → 𝑄:ω–1-1→𝑃) |
| 77 | 75, 76 | syl 17 |
. . . . . . 7
⊢ ((𝑃 ⊆ ω ∧ ¬
𝑃 ∈ Fin) → 𝑄:ω–1-1→𝑃) |
| 78 | 7, 77 | mpan 690 |
. . . . . 6
⊢ (¬
𝑃 ∈ Fin → 𝑄:ω–1-1→𝑃) |
| 79 | | f1co 6815 |
. . . . . 6
⊢ (((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran
𝑈)):𝑃–1-1→V ∧ 𝑄:ω–1-1→𝑃) → ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran
𝑈)) ∘ 𝑄):ω–1-1→V) |
| 80 | 73, 78, 79 | syl2an 596 |
. . . . 5
⊢ ((𝑡:ω–1-1→V ∧ ¬ 𝑃 ∈ Fin) → ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran
𝑈)) ∘ 𝑄):ω–1-1→V) |
| 81 | | f1eq1 6799 |
. . . . 5
⊢ (𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran
𝑈)) ∘ 𝑄) → (𝑍:ω–1-1→V ↔ ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran
𝑈)) ∘ 𝑄):ω–1-1→V)) |
| 82 | 80, 81 | syl5ibrcom 247 |
. . . 4
⊢ ((𝑡:ω–1-1→V ∧ ¬ 𝑃 ∈ Fin) → (𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran
𝑈)) ∘ 𝑄) → 𝑍:ω–1-1→V)) |
| 83 | 82 | impr 454 |
. . 3
⊢ ((𝑡:ω–1-1→V ∧ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran
𝑈)) ∘ 𝑄))) → 𝑍:ω–1-1→V) |
| 84 | 26, 83 | jaodan 960 |
. 2
⊢ ((𝑡:ω–1-1→V ∧ ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡 ∘ 𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran
𝑈)) ∘ 𝑄)))) → 𝑍:ω–1-1→V) |
| 85 | 3, 84 | mpan2 691 |
1
⊢ (𝑡:ω–1-1→V → 𝑍:ω–1-1→V) |