Step | Hyp | Ref
| Expression |
1 | | ssid 3937 |
. . 3
⊢ On
⊆ On |
2 | | onelon 6255 |
. . . . . . . . 9
⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On) |
3 | 2 | ancoms 462 |
. . . . . . . 8
⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ On) → 𝑦 ∈ On) |
4 | | tz7.48.1 |
. . . . . . . . . . 11
⊢ 𝐹 Fn On |
5 | 4 | fndmi 6500 |
. . . . . . . . . 10
⊢ dom 𝐹 = On |
6 | 5 | eleq2i 2830 |
. . . . . . . . 9
⊢ (𝑦 ∈ dom 𝐹 ↔ 𝑦 ∈ On) |
7 | | fnfun 6496 |
. . . . . . . . . . . . 13
⊢ (𝐹 Fn On → Fun 𝐹) |
8 | 4, 7 | ax-mp 5 |
. . . . . . . . . . . 12
⊢ Fun 𝐹 |
9 | | funfvima 7064 |
. . . . . . . . . . . 12
⊢ ((Fun
𝐹 ∧ 𝑦 ∈ dom 𝐹) → (𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹 “ 𝑥))) |
10 | 8, 9 | mpan 690 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ dom 𝐹 → (𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹 “ 𝑥))) |
11 | 10 | impcom 411 |
. . . . . . . . . 10
⊢ ((𝑦 ∈ 𝑥 ∧ 𝑦 ∈ dom 𝐹) → (𝐹‘𝑦) ∈ (𝐹 “ 𝑥)) |
12 | | eleq1a 2834 |
. . . . . . . . . . 11
⊢ ((𝐹‘𝑦) ∈ (𝐹 “ 𝑥) → ((𝐹‘𝑥) = (𝐹‘𝑦) → (𝐹‘𝑥) ∈ (𝐹 “ 𝑥))) |
13 | | eldifn 4056 |
. . . . . . . . . . 11
⊢ ((𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ (𝐹‘𝑥) ∈ (𝐹 “ 𝑥)) |
14 | 12, 13 | nsyli 160 |
. . . . . . . . . 10
⊢ ((𝐹‘𝑦) ∈ (𝐹 “ 𝑥) → ((𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))) |
15 | 11, 14 | syl 17 |
. . . . . . . . 9
⊢ ((𝑦 ∈ 𝑥 ∧ 𝑦 ∈ dom 𝐹) → ((𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))) |
16 | 6, 15 | sylan2br 598 |
. . . . . . . 8
⊢ ((𝑦 ∈ 𝑥 ∧ 𝑦 ∈ On) → ((𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))) |
17 | 3, 16 | syldan 594 |
. . . . . . 7
⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ On) → ((𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))) |
18 | 17 | expimpd 457 |
. . . . . 6
⊢ (𝑦 ∈ 𝑥 → ((𝑥 ∈ On ∧ (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))) |
19 | 18 | com12 32 |
. . . . 5
⊢ ((𝑥 ∈ On ∧ (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))) → (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))) |
20 | 19 | ralrimiv 3105 |
. . . 4
⊢ ((𝑥 ∈ On ∧ (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))) → ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) |
21 | 20 | ralimiaa 3083 |
. . 3
⊢
(∀𝑥 ∈ On
(𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ∀𝑥 ∈ On ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) |
22 | 4 | tz7.48lem 8197 |
. . 3
⊢ ((On
⊆ On ∧ ∀𝑥
∈ On ∀𝑦 ∈
𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) → Fun ◡(𝐹 ↾ On)) |
23 | 1, 21, 22 | sylancr 590 |
. 2
⊢
(∀𝑥 ∈ On
(𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → Fun ◡(𝐹 ↾ On)) |
24 | | fnrel 6498 |
. . . . . 6
⊢ (𝐹 Fn On → Rel 𝐹) |
25 | 4, 24 | ax-mp 5 |
. . . . 5
⊢ Rel 𝐹 |
26 | 5 | eqimssi 3973 |
. . . . 5
⊢ dom 𝐹 ⊆ On |
27 | | relssres 5906 |
. . . . 5
⊢ ((Rel
𝐹 ∧ dom 𝐹 ⊆ On) → (𝐹 ↾ On) = 𝐹) |
28 | 25, 26, 27 | mp2an 692 |
. . . 4
⊢ (𝐹 ↾ On) = 𝐹 |
29 | 28 | cnveqi 5757 |
. . 3
⊢ ◡(𝐹 ↾ On) = ◡𝐹 |
30 | 29 | funeqi 6418 |
. 2
⊢ (Fun
◡(𝐹 ↾ On) ↔ Fun ◡𝐹) |
31 | 23, 30 | sylib 221 |
1
⊢
(∀𝑥 ∈ On
(𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → Fun ◡𝐹) |