| Step | Hyp | Ref
| Expression |
| 1 | | lmff.5 |
. . . . . 6
⊢ (𝜑 → 𝐹 ∈ dom
(⇝𝑡‘𝐽)) |
| 2 | | eldm2g 5910 |
. . . . . . 7
⊢ (𝐹 ∈ dom
(⇝𝑡‘𝐽) → (𝐹 ∈ dom
(⇝𝑡‘𝐽) ↔ ∃𝑦〈𝐹, 𝑦〉 ∈
(⇝𝑡‘𝐽))) |
| 3 | 2 | ibi 267 |
. . . . . 6
⊢ (𝐹 ∈ dom
(⇝𝑡‘𝐽) → ∃𝑦〈𝐹, 𝑦〉 ∈
(⇝𝑡‘𝐽)) |
| 4 | 1, 3 | syl 17 |
. . . . 5
⊢ (𝜑 → ∃𝑦〈𝐹, 𝑦〉 ∈
(⇝𝑡‘𝐽)) |
| 5 | | df-br 5144 |
. . . . . 6
⊢ (𝐹(⇝𝑡‘𝐽)𝑦 ↔ 〈𝐹, 𝑦〉 ∈
(⇝𝑡‘𝐽)) |
| 6 | 5 | exbii 1848 |
. . . . 5
⊢
(∃𝑦 𝐹(⇝𝑡‘𝐽)𝑦 ↔ ∃𝑦〈𝐹, 𝑦〉 ∈
(⇝𝑡‘𝐽)) |
| 7 | 4, 6 | sylibr 234 |
. . . 4
⊢ (𝜑 → ∃𝑦 𝐹(⇝𝑡‘𝐽)𝑦) |
| 8 | | lmff.3 |
. . . . . 6
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
| 9 | | lmcl 23305 |
. . . . . 6
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡‘𝐽)𝑦) → 𝑦 ∈ 𝑋) |
| 10 | 8, 9 | sylan 580 |
. . . . 5
⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐽)𝑦) → 𝑦 ∈ 𝑋) |
| 11 | | eleq2 2830 |
. . . . . . 7
⊢ (𝑗 = 𝑋 → (𝑦 ∈ 𝑗 ↔ 𝑦 ∈ 𝑋)) |
| 12 | | feq3 6718 |
. . . . . . . 8
⊢ (𝑗 = 𝑋 → ((𝐹 ↾ 𝑥):𝑥⟶𝑗 ↔ (𝐹 ↾ 𝑥):𝑥⟶𝑋)) |
| 13 | 12 | rexbidv 3179 |
. . . . . . 7
⊢ (𝑗 = 𝑋 → (∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑗 ↔ ∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑋)) |
| 14 | 11, 13 | imbi12d 344 |
. . . . . 6
⊢ (𝑗 = 𝑋 → ((𝑦 ∈ 𝑗 → ∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑗) ↔ (𝑦 ∈ 𝑋 → ∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑋))) |
| 15 | 8 | lmbr 23266 |
. . . . . . . 8
⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑦 ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ 𝑦 ∈ 𝑋 ∧ ∀𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 → ∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑗)))) |
| 16 | 15 | biimpa 476 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐽)𝑦) → (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ 𝑦 ∈ 𝑋 ∧ ∀𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 → ∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑗))) |
| 17 | 16 | simp3d 1145 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐽)𝑦) → ∀𝑗 ∈ 𝐽 (𝑦 ∈ 𝑗 → ∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑗)) |
| 18 | | toponmax 22932 |
. . . . . . . 8
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽) |
| 19 | 8, 18 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑋 ∈ 𝐽) |
| 20 | 19 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐽)𝑦) → 𝑋 ∈ 𝐽) |
| 21 | 14, 17, 20 | rspcdva 3623 |
. . . . 5
⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐽)𝑦) → (𝑦 ∈ 𝑋 → ∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑋)) |
| 22 | 10, 21 | mpd 15 |
. . . 4
⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐽)𝑦) → ∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑋) |
| 23 | 7, 22 | exlimddv 1935 |
. . 3
⊢ (𝜑 → ∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑋) |
| 24 | | uzf 12881 |
. . . 4
⊢
ℤ≥:ℤ⟶𝒫 ℤ |
| 25 | | ffn 6736 |
. . . 4
⊢
(ℤ≥:ℤ⟶𝒫 ℤ →
ℤ≥ Fn ℤ) |
| 26 | | reseq2 5992 |
. . . . . 6
⊢ (𝑥 =
(ℤ≥‘𝑗) → (𝐹 ↾ 𝑥) = (𝐹 ↾ (ℤ≥‘𝑗))) |
| 27 | | id 22 |
. . . . . 6
⊢ (𝑥 =
(ℤ≥‘𝑗) → 𝑥 = (ℤ≥‘𝑗)) |
| 28 | 26, 27 | feq12d 6724 |
. . . . 5
⊢ (𝑥 =
(ℤ≥‘𝑗) → ((𝐹 ↾ 𝑥):𝑥⟶𝑋 ↔ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋)) |
| 29 | 28 | rexrn 7107 |
. . . 4
⊢
(ℤ≥ Fn ℤ → (∃𝑥 ∈ ran ℤ≥(𝐹 ↾ 𝑥):𝑥⟶𝑋 ↔ ∃𝑗 ∈ ℤ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋)) |
| 30 | 24, 25, 29 | mp2b 10 |
. . 3
⊢
(∃𝑥 ∈ ran
ℤ≥(𝐹
↾ 𝑥):𝑥⟶𝑋 ↔ ∃𝑗 ∈ ℤ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋) |
| 31 | 23, 30 | sylib 218 |
. 2
⊢ (𝜑 → ∃𝑗 ∈ ℤ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋) |
| 32 | | lmff.4 |
. . . 4
⊢ (𝜑 → 𝑀 ∈ ℤ) |
| 33 | | lmff.1 |
. . . . 5
⊢ 𝑍 =
(ℤ≥‘𝑀) |
| 34 | 33 | rexuz3 15387 |
. . . 4
⊢ (𝑀 ∈ ℤ →
(∃𝑗 ∈ 𝑍 ∀𝑥 ∈ (ℤ≥‘𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝑋) ↔ ∃𝑗 ∈ ℤ ∀𝑥 ∈ (ℤ≥‘𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝑋))) |
| 35 | 32, 34 | syl 17 |
. . 3
⊢ (𝜑 → (∃𝑗 ∈ 𝑍 ∀𝑥 ∈ (ℤ≥‘𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝑋) ↔ ∃𝑗 ∈ ℤ ∀𝑥 ∈ (ℤ≥‘𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝑋))) |
| 36 | 16 | simp1d 1143 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐹(⇝𝑡‘𝐽)𝑦) → 𝐹 ∈ (𝑋 ↑pm
ℂ)) |
| 37 | 7, 36 | exlimddv 1935 |
. . . . . 6
⊢ (𝜑 → 𝐹 ∈ (𝑋 ↑pm
ℂ)) |
| 38 | | pmfun 8887 |
. . . . . 6
⊢ (𝐹 ∈ (𝑋 ↑pm ℂ) → Fun
𝐹) |
| 39 | 37, 38 | syl 17 |
. . . . 5
⊢ (𝜑 → Fun 𝐹) |
| 40 | | ffvresb 7145 |
. . . . 5
⊢ (Fun
𝐹 → ((𝐹 ↾
(ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋 ↔ ∀𝑥 ∈ (ℤ≥‘𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝑋))) |
| 41 | 39, 40 | syl 17 |
. . . 4
⊢ (𝜑 → ((𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋 ↔ ∀𝑥 ∈ (ℤ≥‘𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝑋))) |
| 42 | 41 | rexbidv 3179 |
. . 3
⊢ (𝜑 → (∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋 ↔ ∃𝑗 ∈ 𝑍 ∀𝑥 ∈ (ℤ≥‘𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝑋))) |
| 43 | 41 | rexbidv 3179 |
. . 3
⊢ (𝜑 → (∃𝑗 ∈ ℤ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋 ↔ ∃𝑗 ∈ ℤ ∀𝑥 ∈ (ℤ≥‘𝑗)(𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝑋))) |
| 44 | 35, 42, 43 | 3bitr4d 311 |
. 2
⊢ (𝜑 → (∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋 ↔ ∃𝑗 ∈ ℤ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋)) |
| 45 | 31, 44 | mpbird 257 |
1
⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶𝑋) |