| Step | Hyp | Ref
| Expression |
| 1 | | qtopf1.1 |
. . 3
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
| 2 | | qtopf1.2 |
. . . 4
⊢ (𝜑 → 𝐹:𝑋–1-1→𝑌) |
| 3 | | f1fn 6805 |
. . . 4
⊢ (𝐹:𝑋–1-1→𝑌 → 𝐹 Fn 𝑋) |
| 4 | 2, 3 | syl 17 |
. . 3
⊢ (𝜑 → 𝐹 Fn 𝑋) |
| 5 | | qtopid 23713 |
. . 3
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹))) |
| 6 | 1, 4, 5 | syl2anc 584 |
. 2
⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹))) |
| 7 | | f1f1orn 6859 |
. . . 4
⊢ (𝐹:𝑋–1-1→𝑌 → 𝐹:𝑋–1-1-onto→ran
𝐹) |
| 8 | | f1ocnv 6860 |
. . . 4
⊢ (𝐹:𝑋–1-1-onto→ran
𝐹 → ◡𝐹:ran 𝐹–1-1-onto→𝑋) |
| 9 | | f1of 6848 |
. . . 4
⊢ (◡𝐹:ran 𝐹–1-1-onto→𝑋 → ◡𝐹:ran 𝐹⟶𝑋) |
| 10 | 2, 7, 8, 9 | 4syl 19 |
. . 3
⊢ (𝜑 → ◡𝐹:ran 𝐹⟶𝑋) |
| 11 | | imacnvcnv 6226 |
. . . . 5
⊢ (◡◡𝐹 “ 𝑥) = (𝐹 “ 𝑥) |
| 12 | | imassrn 6089 |
. . . . . . 7
⊢ (𝐹 “ 𝑥) ⊆ ran 𝐹 |
| 13 | 12 | a1i 11 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ⊆ ran 𝐹) |
| 14 | 2 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝐹:𝑋–1-1→𝑌) |
| 15 | | toponss 22933 |
. . . . . . . . 9
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ 𝑋) |
| 16 | 1, 15 | sylan 580 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ 𝑋) |
| 17 | | f1imacnv 6864 |
. . . . . . . 8
⊢ ((𝐹:𝑋–1-1→𝑌 ∧ 𝑥 ⊆ 𝑋) → (◡𝐹 “ (𝐹 “ 𝑥)) = 𝑥) |
| 18 | 14, 16, 17 | syl2anc 584 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (◡𝐹 “ (𝐹 “ 𝑥)) = 𝑥) |
| 19 | | simpr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝑥 ∈ 𝐽) |
| 20 | 18, 19 | eqeltrd 2841 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (◡𝐹 “ (𝐹 “ 𝑥)) ∈ 𝐽) |
| 21 | | dffn4 6826 |
. . . . . . . . 9
⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–onto→ran 𝐹) |
| 22 | 4, 21 | sylib 218 |
. . . . . . . 8
⊢ (𝜑 → 𝐹:𝑋–onto→ran 𝐹) |
| 23 | | elqtop3 23711 |
. . . . . . . 8
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→ran 𝐹) → ((𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹 “ 𝑥) ⊆ ran 𝐹 ∧ (◡𝐹 “ (𝐹 “ 𝑥)) ∈ 𝐽))) |
| 24 | 1, 22, 23 | syl2anc 584 |
. . . . . . 7
⊢ (𝜑 → ((𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹 “ 𝑥) ⊆ ran 𝐹 ∧ (◡𝐹 “ (𝐹 “ 𝑥)) ∈ 𝐽))) |
| 25 | 24 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ((𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹 “ 𝑥) ⊆ ran 𝐹 ∧ (◡𝐹 “ (𝐹 “ 𝑥)) ∈ 𝐽))) |
| 26 | 13, 20, 25 | mpbir2and 713 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹)) |
| 27 | 11, 26 | eqeltrid 2845 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (◡◡𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹)) |
| 28 | 27 | ralrimiva 3146 |
. . 3
⊢ (𝜑 → ∀𝑥 ∈ 𝐽 (◡◡𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹)) |
| 29 | | qtoptopon 23712 |
. . . . 5
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→ran 𝐹) → (𝐽 qTop 𝐹) ∈ (TopOn‘ran 𝐹)) |
| 30 | 1, 22, 29 | syl2anc 584 |
. . . 4
⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ (TopOn‘ran 𝐹)) |
| 31 | | iscn 23243 |
. . . 4
⊢ (((𝐽 qTop 𝐹) ∈ (TopOn‘ran 𝐹) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (◡𝐹 ∈ ((𝐽 qTop 𝐹) Cn 𝐽) ↔ (◡𝐹:ran 𝐹⟶𝑋 ∧ ∀𝑥 ∈ 𝐽 (◡◡𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹)))) |
| 32 | 30, 1, 31 | syl2anc 584 |
. . 3
⊢ (𝜑 → (◡𝐹 ∈ ((𝐽 qTop 𝐹) Cn 𝐽) ↔ (◡𝐹:ran 𝐹⟶𝑋 ∧ ∀𝑥 ∈ 𝐽 (◡◡𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹)))) |
| 33 | 10, 28, 32 | mpbir2and 713 |
. 2
⊢ (𝜑 → ◡𝐹 ∈ ((𝐽 qTop 𝐹) Cn 𝐽)) |
| 34 | | ishmeo 23767 |
. 2
⊢ (𝐹 ∈ (𝐽Homeo(𝐽 qTop 𝐹)) ↔ (𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)) ∧ ◡𝐹 ∈ ((𝐽 qTop 𝐹) Cn 𝐽))) |
| 35 | 6, 33, 34 | sylanbrc 583 |
1
⊢ (𝜑 → 𝐹 ∈ (𝐽Homeo(𝐽 qTop 𝐹))) |