Proof of Theorem isotilem
Step | Hyp | Ref
| Expression |
1 | | isof1o 5754 |
. . . . . 6
⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴–1-1-onto→𝐵) |
2 | | f1of 5413 |
. . . . . 6
⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴⟶𝐵) |
3 | | ffvelrn 5599 |
. . . . . . . 8
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑢 ∈ 𝐴) → (𝐹‘𝑢) ∈ 𝐵) |
4 | 3 | ex 114 |
. . . . . . 7
⊢ (𝐹:𝐴⟶𝐵 → (𝑢 ∈ 𝐴 → (𝐹‘𝑢) ∈ 𝐵)) |
5 | | ffvelrn 5599 |
. . . . . . . 8
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑣 ∈ 𝐴) → (𝐹‘𝑣) ∈ 𝐵) |
6 | 5 | ex 114 |
. . . . . . 7
⊢ (𝐹:𝐴⟶𝐵 → (𝑣 ∈ 𝐴 → (𝐹‘𝑣) ∈ 𝐵)) |
7 | 4, 6 | anim12d 333 |
. . . . . 6
⊢ (𝐹:𝐴⟶𝐵 → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → ((𝐹‘𝑢) ∈ 𝐵 ∧ (𝐹‘𝑣) ∈ 𝐵))) |
8 | 1, 2, 7 | 3syl 17 |
. . . . 5
⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → ((𝐹‘𝑢) ∈ 𝐵 ∧ (𝐹‘𝑣) ∈ 𝐵))) |
9 | 8 | imp 123 |
. . . 4
⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → ((𝐹‘𝑢) ∈ 𝐵 ∧ (𝐹‘𝑣) ∈ 𝐵)) |
10 | | eqeq1 2164 |
. . . . . 6
⊢ (𝑥 = (𝐹‘𝑢) → (𝑥 = 𝑦 ↔ (𝐹‘𝑢) = 𝑦)) |
11 | | breq1 3968 |
. . . . . . . 8
⊢ (𝑥 = (𝐹‘𝑢) → (𝑥𝑆𝑦 ↔ (𝐹‘𝑢)𝑆𝑦)) |
12 | 11 | notbid 657 |
. . . . . . 7
⊢ (𝑥 = (𝐹‘𝑢) → (¬ 𝑥𝑆𝑦 ↔ ¬ (𝐹‘𝑢)𝑆𝑦)) |
13 | | breq2 3969 |
. . . . . . . 8
⊢ (𝑥 = (𝐹‘𝑢) → (𝑦𝑆𝑥 ↔ 𝑦𝑆(𝐹‘𝑢))) |
14 | 13 | notbid 657 |
. . . . . . 7
⊢ (𝑥 = (𝐹‘𝑢) → (¬ 𝑦𝑆𝑥 ↔ ¬ 𝑦𝑆(𝐹‘𝑢))) |
15 | 12, 14 | anbi12d 465 |
. . . . . 6
⊢ (𝑥 = (𝐹‘𝑢) → ((¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥) ↔ (¬ (𝐹‘𝑢)𝑆𝑦 ∧ ¬ 𝑦𝑆(𝐹‘𝑢)))) |
16 | 10, 15 | bibi12d 234 |
. . . . 5
⊢ (𝑥 = (𝐹‘𝑢) → ((𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥)) ↔ ((𝐹‘𝑢) = 𝑦 ↔ (¬ (𝐹‘𝑢)𝑆𝑦 ∧ ¬ 𝑦𝑆(𝐹‘𝑢))))) |
17 | | eqeq2 2167 |
. . . . . 6
⊢ (𝑦 = (𝐹‘𝑣) → ((𝐹‘𝑢) = 𝑦 ↔ (𝐹‘𝑢) = (𝐹‘𝑣))) |
18 | | breq2 3969 |
. . . . . . . 8
⊢ (𝑦 = (𝐹‘𝑣) → ((𝐹‘𝑢)𝑆𝑦 ↔ (𝐹‘𝑢)𝑆(𝐹‘𝑣))) |
19 | 18 | notbid 657 |
. . . . . . 7
⊢ (𝑦 = (𝐹‘𝑣) → (¬ (𝐹‘𝑢)𝑆𝑦 ↔ ¬ (𝐹‘𝑢)𝑆(𝐹‘𝑣))) |
20 | | breq1 3968 |
. . . . . . . 8
⊢ (𝑦 = (𝐹‘𝑣) → (𝑦𝑆(𝐹‘𝑢) ↔ (𝐹‘𝑣)𝑆(𝐹‘𝑢))) |
21 | 20 | notbid 657 |
. . . . . . 7
⊢ (𝑦 = (𝐹‘𝑣) → (¬ 𝑦𝑆(𝐹‘𝑢) ↔ ¬ (𝐹‘𝑣)𝑆(𝐹‘𝑢))) |
22 | 19, 21 | anbi12d 465 |
. . . . . 6
⊢ (𝑦 = (𝐹‘𝑣) → ((¬ (𝐹‘𝑢)𝑆𝑦 ∧ ¬ 𝑦𝑆(𝐹‘𝑢)) ↔ (¬ (𝐹‘𝑢)𝑆(𝐹‘𝑣) ∧ ¬ (𝐹‘𝑣)𝑆(𝐹‘𝑢)))) |
23 | 17, 22 | bibi12d 234 |
. . . . 5
⊢ (𝑦 = (𝐹‘𝑣) → (((𝐹‘𝑢) = 𝑦 ↔ (¬ (𝐹‘𝑢)𝑆𝑦 ∧ ¬ 𝑦𝑆(𝐹‘𝑢))) ↔ ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ (¬ (𝐹‘𝑢)𝑆(𝐹‘𝑣) ∧ ¬ (𝐹‘𝑣)𝑆(𝐹‘𝑢))))) |
24 | 16, 23 | rspc2v 2829 |
. . . 4
⊢ (((𝐹‘𝑢) ∈ 𝐵 ∧ (𝐹‘𝑣) ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥)) → ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ (¬ (𝐹‘𝑢)𝑆(𝐹‘𝑣) ∧ ¬ (𝐹‘𝑣)𝑆(𝐹‘𝑢))))) |
25 | 9, 24 | syl 14 |
. . 3
⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥)) → ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ (¬ (𝐹‘𝑢)𝑆(𝐹‘𝑣) ∧ ¬ (𝐹‘𝑣)𝑆(𝐹‘𝑢))))) |
26 | | f1of1 5412 |
. . . . . . 7
⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–1-1→𝐵) |
27 | 1, 26 | syl 14 |
. . . . . 6
⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴–1-1→𝐵) |
28 | | f1fveq 5719 |
. . . . . 6
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ 𝑢 = 𝑣)) |
29 | 27, 28 | sylan 281 |
. . . . 5
⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ 𝑢 = 𝑣)) |
30 | 29 | bicomd 140 |
. . . 4
⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (𝐹‘𝑢) = (𝐹‘𝑣))) |
31 | | isorel 5755 |
. . . . . 6
⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢𝑅𝑣 ↔ (𝐹‘𝑢)𝑆(𝐹‘𝑣))) |
32 | 31 | notbid 657 |
. . . . 5
⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (¬ 𝑢𝑅𝑣 ↔ ¬ (𝐹‘𝑢)𝑆(𝐹‘𝑣))) |
33 | | isorel 5755 |
. . . . . . 7
⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑣 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴)) → (𝑣𝑅𝑢 ↔ (𝐹‘𝑣)𝑆(𝐹‘𝑢))) |
34 | 33 | notbid 657 |
. . . . . 6
⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑣 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴)) → (¬ 𝑣𝑅𝑢 ↔ ¬ (𝐹‘𝑣)𝑆(𝐹‘𝑢))) |
35 | 34 | ancom2s 556 |
. . . . 5
⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (¬ 𝑣𝑅𝑢 ↔ ¬ (𝐹‘𝑣)𝑆(𝐹‘𝑢))) |
36 | 32, 35 | anbi12d 465 |
. . . 4
⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → ((¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢) ↔ (¬ (𝐹‘𝑢)𝑆(𝐹‘𝑣) ∧ ¬ (𝐹‘𝑣)𝑆(𝐹‘𝑢)))) |
37 | 30, 36 | bibi12d 234 |
. . 3
⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → ((𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)) ↔ ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ (¬ (𝐹‘𝑢)𝑆(𝐹‘𝑣) ∧ ¬ (𝐹‘𝑣)𝑆(𝐹‘𝑢))))) |
38 | 25, 37 | sylibrd 168 |
. 2
⊢ ((𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))) |
39 | 38 | ralrimdvva 2542 |
1
⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥)) → ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))) |