Step | Hyp | Ref
| Expression |
1 | | foelrn 6286 |
. . . . 5
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐴 𝑥 = (𝐹‘𝑦)) |
2 | 1 | ralrimiva 2949 |
. . . 4
⊢ (𝐹:𝐴–onto→𝐵 → ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑥 = (𝐹‘𝑦)) |
3 | | fveq2 6103 |
. . . . . 6
⊢ (𝑦 = (𝑓‘𝑥) → (𝐹‘𝑦) = (𝐹‘(𝑓‘𝑥))) |
4 | 3 | eqeq2d 2620 |
. . . . 5
⊢ (𝑦 = (𝑓‘𝑥) → (𝑥 = (𝐹‘𝑦) ↔ 𝑥 = (𝐹‘(𝑓‘𝑥)))) |
5 | 4 | acni3 8753 |
. . . 4
⊢ ((𝐴 ∈ AC 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑥 = (𝐹‘𝑦)) → ∃𝑓(𝑓:𝐵⟶𝐴 ∧ ∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)))) |
6 | 2, 5 | sylan2 490 |
. . 3
⊢ ((𝐴 ∈ AC 𝐵 ∧ 𝐹:𝐴–onto→𝐵) → ∃𝑓(𝑓:𝐵⟶𝐴 ∧ ∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)))) |
7 | | simpll 786 |
. . . . 5
⊢ (((𝐴 ∈ AC 𝐵 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑓:𝐵⟶𝐴 ∧ ∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)))) → 𝐴 ∈ AC 𝐵) |
8 | | acnrcl 8748 |
. . . . 5
⊢ (𝐴 ∈ AC 𝐵 → 𝐵 ∈ V) |
9 | 7, 8 | syl 17 |
. . . 4
⊢ (((𝐴 ∈ AC 𝐵 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑓:𝐵⟶𝐴 ∧ ∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)))) → 𝐵 ∈ V) |
10 | | simprl 790 |
. . . . 5
⊢ (((𝐴 ∈ AC 𝐵 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑓:𝐵⟶𝐴 ∧ ∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)))) → 𝑓:𝐵⟶𝐴) |
11 | | fveq2 6103 |
. . . . . . 7
⊢ ((𝑓‘𝑦) = (𝑓‘𝑧) → (𝐹‘(𝑓‘𝑦)) = (𝐹‘(𝑓‘𝑧))) |
12 | | simprr 792 |
. . . . . . . 8
⊢ (((𝐴 ∈ AC 𝐵 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑓:𝐵⟶𝐴 ∧ ∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)))) → ∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥))) |
13 | | id 22 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) |
14 | | fveq2 6103 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑦 → (𝑓‘𝑥) = (𝑓‘𝑦)) |
15 | 14 | fveq2d 6107 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑦 → (𝐹‘(𝑓‘𝑥)) = (𝐹‘(𝑓‘𝑦))) |
16 | 13, 15 | eqeq12d 2625 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑦 → (𝑥 = (𝐹‘(𝑓‘𝑥)) ↔ 𝑦 = (𝐹‘(𝑓‘𝑦)))) |
17 | 16 | rspccva 3281 |
. . . . . . . . . 10
⊢
((∀𝑥 ∈
𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)) ∧ 𝑦 ∈ 𝐵) → 𝑦 = (𝐹‘(𝑓‘𝑦))) |
18 | | id 22 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧) |
19 | | fveq2 6103 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑧 → (𝑓‘𝑥) = (𝑓‘𝑧)) |
20 | 19 | fveq2d 6107 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑧 → (𝐹‘(𝑓‘𝑥)) = (𝐹‘(𝑓‘𝑧))) |
21 | 18, 20 | eqeq12d 2625 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑧 → (𝑥 = (𝐹‘(𝑓‘𝑥)) ↔ 𝑧 = (𝐹‘(𝑓‘𝑧)))) |
22 | 21 | rspccva 3281 |
. . . . . . . . . 10
⊢
((∀𝑥 ∈
𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)) ∧ 𝑧 ∈ 𝐵) → 𝑧 = (𝐹‘(𝑓‘𝑧))) |
23 | 17, 22 | eqeqan12d 2626 |
. . . . . . . . 9
⊢
(((∀𝑥 ∈
𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)) ∧ 𝑦 ∈ 𝐵) ∧ (∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)) ∧ 𝑧 ∈ 𝐵)) → (𝑦 = 𝑧 ↔ (𝐹‘(𝑓‘𝑦)) = (𝐹‘(𝑓‘𝑧)))) |
24 | 23 | anandis 869 |
. . . . . . . 8
⊢
((∀𝑥 ∈
𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦 = 𝑧 ↔ (𝐹‘(𝑓‘𝑦)) = (𝐹‘(𝑓‘𝑧)))) |
25 | 12, 24 | sylan 487 |
. . . . . . 7
⊢ ((((𝐴 ∈ AC 𝐵 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑓:𝐵⟶𝐴 ∧ ∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑦 = 𝑧 ↔ (𝐹‘(𝑓‘𝑦)) = (𝐹‘(𝑓‘𝑧)))) |
26 | 11, 25 | syl5ibr 235 |
. . . . . 6
⊢ ((((𝐴 ∈ AC 𝐵 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑓:𝐵⟶𝐴 ∧ ∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑓‘𝑦) = (𝑓‘𝑧) → 𝑦 = 𝑧)) |
27 | 26 | ralrimivva 2954 |
. . . . 5
⊢ (((𝐴 ∈ AC 𝐵 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑓:𝐵⟶𝐴 ∧ ∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)))) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑓‘𝑦) = (𝑓‘𝑧) → 𝑦 = 𝑧)) |
28 | | dff13 6416 |
. . . . 5
⊢ (𝑓:𝐵–1-1→𝐴 ↔ (𝑓:𝐵⟶𝐴 ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑓‘𝑦) = (𝑓‘𝑧) → 𝑦 = 𝑧))) |
29 | 10, 27, 28 | sylanbrc 695 |
. . . 4
⊢ (((𝐴 ∈ AC 𝐵 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑓:𝐵⟶𝐴 ∧ ∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)))) → 𝑓:𝐵–1-1→𝐴) |
30 | | f1dom2g 7859 |
. . . 4
⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ AC 𝐵 ∧ 𝑓:𝐵–1-1→𝐴) → 𝐵 ≼ 𝐴) |
31 | 9, 7, 29, 30 | syl3anc 1318 |
. . 3
⊢ (((𝐴 ∈ AC 𝐵 ∧ 𝐹:𝐴–onto→𝐵) ∧ (𝑓:𝐵⟶𝐴 ∧ ∀𝑥 ∈ 𝐵 𝑥 = (𝐹‘(𝑓‘𝑥)))) → 𝐵 ≼ 𝐴) |
32 | 6, 31 | exlimddv 1850 |
. 2
⊢ ((𝐴 ∈ AC 𝐵 ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ≼ 𝐴) |
33 | 32 | ex 449 |
1
⊢ (𝐴 ∈ AC 𝐵 → (𝐹:𝐴–onto→𝐵 → 𝐵 ≼ 𝐴)) |