Step | Hyp | Ref
| Expression |
1 | | fnfvelrn 6901 |
. . . . . 6
⊢ ((𝐹 Fn 𝐴 ∧ 𝑜 ∈ 𝐴) → (𝐹‘𝑜) ∈ ran 𝐹) |
2 | 1 | ex 416 |
. . . . 5
⊢ (𝐹 Fn 𝐴 → (𝑜 ∈ 𝐴 → (𝐹‘𝑜) ∈ ran 𝐹)) |
3 | 2 | adantr 484 |
. . . 4
⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝑜 ∈ 𝐴 → (𝐹‘𝑜) ∈ ran 𝐹)) |
4 | | fnrnfv 6772 |
. . . . . . . . . 10
⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑝 ∈ 𝐴 𝑦 = (𝐹‘𝑝)}) |
5 | 4 | abeq2d 2871 |
. . . . . . . . 9
⊢ (𝐹 Fn 𝐴 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑝 ∈ 𝐴 𝑦 = (𝐹‘𝑝))) |
6 | 5 | adantr 484 |
. . . . . . . 8
⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝑦 ∈ ran 𝐹 ↔ ∃𝑝 ∈ 𝐴 𝑦 = (𝐹‘𝑝))) |
7 | | nfv 1922 |
. . . . . . . . . 10
⊢
Ⅎ𝑝 𝐹 Fn 𝐴 |
8 | | nfra1 3140 |
. . . . . . . . . 10
⊢
Ⅎ𝑝∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} |
9 | 7, 8 | nfan 1907 |
. . . . . . . . 9
⊢
Ⅎ𝑝(𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) |
10 | | nfv 1922 |
. . . . . . . . 9
⊢
Ⅎ𝑝∀𝑜 ∈ 𝐴 (𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜)) |
11 | | eleq2 2826 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 = (𝐹‘𝑝) → (𝑜 ∈ 𝑦 ↔ 𝑜 ∈ (𝐹‘𝑝))) |
12 | | elin 3882 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑜 ∈ ((𝐹‘𝑝) ∩ 𝐴) ↔ (𝑜 ∈ (𝐹‘𝑝) ∧ 𝑜 ∈ 𝐴)) |
13 | 12 | rbaib 542 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑜 ∈ 𝐴 → (𝑜 ∈ ((𝐹‘𝑝) ∩ 𝐴) ↔ 𝑜 ∈ (𝐹‘𝑝))) |
14 | 13 | ad2antll 729 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑜 ∈ 𝐴)) → (𝑜 ∈ ((𝐹‘𝑝) ∩ 𝐴) ↔ 𝑜 ∈ (𝐹‘𝑝))) |
15 | | rsp 3127 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(∀𝑝 ∈
𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → (𝑝 ∈ 𝐴 → ((𝐹‘𝑝) ∩ 𝐴) = {𝑝})) |
16 | | eleq2 2826 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → (𝑜 ∈ ((𝐹‘𝑝) ∩ 𝐴) ↔ 𝑜 ∈ {𝑝})) |
17 | | velsn 4557 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑜 ∈ {𝑝} ↔ 𝑜 = 𝑝) |
18 | | equcom 2026 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑜 = 𝑝 ↔ 𝑝 = 𝑜) |
19 | 17, 18 | bitri 278 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑜 ∈ {𝑝} ↔ 𝑝 = 𝑜) |
20 | 16, 19 | bitrdi 290 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → (𝑜 ∈ ((𝐹‘𝑝) ∩ 𝐴) ↔ 𝑝 = 𝑜)) |
21 | 15, 20 | syl6 35 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(∀𝑝 ∈
𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → (𝑝 ∈ 𝐴 → (𝑜 ∈ ((𝐹‘𝑝) ∩ 𝐴) ↔ 𝑝 = 𝑜))) |
22 | 21 | adantl 485 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝑝 ∈ 𝐴 → (𝑜 ∈ ((𝐹‘𝑝) ∩ 𝐴) ↔ 𝑝 = 𝑜))) |
23 | 22 | adantrd 495 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → ((𝑝 ∈ 𝐴 ∧ 𝑜 ∈ 𝐴) → (𝑜 ∈ ((𝐹‘𝑝) ∩ 𝐴) ↔ 𝑝 = 𝑜))) |
24 | 23 | imp 410 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑜 ∈ 𝐴)) → (𝑜 ∈ ((𝐹‘𝑝) ∩ 𝐴) ↔ 𝑝 = 𝑜)) |
25 | 14, 24 | bitr3d 284 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑜 ∈ 𝐴)) → (𝑜 ∈ (𝐹‘𝑝) ↔ 𝑝 = 𝑜)) |
26 | 11, 25 | sylan9bbr 514 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑜 ∈ 𝐴)) ∧ 𝑦 = (𝐹‘𝑝)) → (𝑜 ∈ 𝑦 ↔ 𝑝 = 𝑜)) |
27 | 26 | ex 416 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑜 ∈ 𝐴)) → (𝑦 = (𝐹‘𝑝) → (𝑜 ∈ 𝑦 ↔ 𝑝 = 𝑜))) |
28 | 27 | anass1rs 655 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑜 ∈ 𝐴) ∧ 𝑝 ∈ 𝐴) → (𝑦 = (𝐹‘𝑝) → (𝑜 ∈ 𝑦 ↔ 𝑝 = 𝑜))) |
29 | 28 | impr 458 |
. . . . . . . . . . . . . 14
⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑜 ∈ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑝))) → (𝑜 ∈ 𝑦 ↔ 𝑝 = 𝑜)) |
30 | 29 | an32s 652 |
. . . . . . . . . . . . 13
⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑝))) ∧ 𝑜 ∈ 𝐴) → (𝑜 ∈ 𝑦 ↔ 𝑝 = 𝑜)) |
31 | | eqeq1 2741 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 = (𝐹‘𝑝) → (𝑦 = (𝐹‘𝑜) ↔ (𝐹‘𝑝) = (𝐹‘𝑜))) |
32 | | dffn3 6558 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹) |
33 | | fvineqsnf1 35318 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝐹:𝐴⟶ran 𝐹 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → 𝐹:𝐴–1-1→ran 𝐹) |
34 | 32, 33 | sylanb 584 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → 𝐹:𝐴–1-1→ran 𝐹) |
35 | | dff13 7067 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝐹:𝐴–1-1→ran 𝐹 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ∀𝑝 ∈ 𝐴 ∀𝑜 ∈ 𝐴 ((𝐹‘𝑝) = (𝐹‘𝑜) → 𝑝 = 𝑜))) |
36 | 34, 35 | sylib 221 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝐹:𝐴⟶ran 𝐹 ∧ ∀𝑝 ∈ 𝐴 ∀𝑜 ∈ 𝐴 ((𝐹‘𝑝) = (𝐹‘𝑜) → 𝑝 = 𝑜))) |
37 | 36 | simprd 499 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → ∀𝑝 ∈ 𝐴 ∀𝑜 ∈ 𝐴 ((𝐹‘𝑝) = (𝐹‘𝑜) → 𝑝 = 𝑜)) |
38 | | rsp 3127 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(∀𝑝 ∈
𝐴 ∀𝑜 ∈ 𝐴 ((𝐹‘𝑝) = (𝐹‘𝑜) → 𝑝 = 𝑜) → (𝑝 ∈ 𝐴 → ∀𝑜 ∈ 𝐴 ((𝐹‘𝑝) = (𝐹‘𝑜) → 𝑝 = 𝑜))) |
39 | 37, 38 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝑝 ∈ 𝐴 → ∀𝑜 ∈ 𝐴 ((𝐹‘𝑝) = (𝐹‘𝑜) → 𝑝 = 𝑜))) |
40 | | rsp 3127 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(∀𝑜 ∈
𝐴 ((𝐹‘𝑝) = (𝐹‘𝑜) → 𝑝 = 𝑜) → (𝑜 ∈ 𝐴 → ((𝐹‘𝑝) = (𝐹‘𝑜) → 𝑝 = 𝑜))) |
41 | 39, 40 | syl6 35 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝑝 ∈ 𝐴 → (𝑜 ∈ 𝐴 → ((𝐹‘𝑝) = (𝐹‘𝑜) → 𝑝 = 𝑜)))) |
42 | 41 | imp32 422 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑜 ∈ 𝐴)) → ((𝐹‘𝑝) = (𝐹‘𝑜) → 𝑝 = 𝑜)) |
43 | | fveq2 6717 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑝 = 𝑜 → (𝐹‘𝑝) = (𝐹‘𝑜)) |
44 | 42, 43 | impbid1 228 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑜 ∈ 𝐴)) → ((𝐹‘𝑝) = (𝐹‘𝑜) ↔ 𝑝 = 𝑜)) |
45 | 31, 44 | sylan9bbr 514 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑜 ∈ 𝐴)) ∧ 𝑦 = (𝐹‘𝑝)) → (𝑦 = (𝐹‘𝑜) ↔ 𝑝 = 𝑜)) |
46 | 45 | ex 416 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑜 ∈ 𝐴)) → (𝑦 = (𝐹‘𝑝) → (𝑦 = (𝐹‘𝑜) ↔ 𝑝 = 𝑜))) |
47 | 46 | anass1rs 655 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑜 ∈ 𝐴) ∧ 𝑝 ∈ 𝐴) → (𝑦 = (𝐹‘𝑝) → (𝑦 = (𝐹‘𝑜) ↔ 𝑝 = 𝑜))) |
48 | 47 | impr 458 |
. . . . . . . . . . . . . 14
⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑜 ∈ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑝))) → (𝑦 = (𝐹‘𝑜) ↔ 𝑝 = 𝑜)) |
49 | 48 | an32s 652 |
. . . . . . . . . . . . 13
⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑝))) ∧ 𝑜 ∈ 𝐴) → (𝑦 = (𝐹‘𝑜) ↔ 𝑝 = 𝑜)) |
50 | 30, 49 | bitr4d 285 |
. . . . . . . . . . . 12
⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑝))) ∧ 𝑜 ∈ 𝐴) → (𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜))) |
51 | 50 | ex 416 |
. . . . . . . . . . 11
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑝))) → (𝑜 ∈ 𝐴 → (𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜)))) |
52 | 51 | ralrimiv 3104 |
. . . . . . . . . 10
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑝 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑝))) → ∀𝑜 ∈ 𝐴 (𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜))) |
53 | 52 | exp32 424 |
. . . . . . . . 9
⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝑝 ∈ 𝐴 → (𝑦 = (𝐹‘𝑝) → ∀𝑜 ∈ 𝐴 (𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜))))) |
54 | 9, 10, 53 | rexlimd 3236 |
. . . . . . . 8
⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (∃𝑝 ∈ 𝐴 𝑦 = (𝐹‘𝑝) → ∀𝑜 ∈ 𝐴 (𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜)))) |
55 | 6, 54 | sylbid 243 |
. . . . . . 7
⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝑦 ∈ ran 𝐹 → ∀𝑜 ∈ 𝐴 (𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜)))) |
56 | | rsp 3127 |
. . . . . . 7
⊢
(∀𝑜 ∈
𝐴 (𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜)) → (𝑜 ∈ 𝐴 → (𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜)))) |
57 | 55, 56 | syl6 35 |
. . . . . 6
⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝑦 ∈ ran 𝐹 → (𝑜 ∈ 𝐴 → (𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜))))) |
58 | 57 | com23 86 |
. . . . 5
⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝑜 ∈ 𝐴 → (𝑦 ∈ ran 𝐹 → (𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜))))) |
59 | 58 | ralrimdv 3109 |
. . . 4
⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝑜 ∈ 𝐴 → ∀𝑦 ∈ ran 𝐹(𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜)))) |
60 | | reu6i 3641 |
. . . . 5
⊢ (((𝐹‘𝑜) ∈ ran 𝐹 ∧ ∀𝑦 ∈ ran 𝐹(𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜))) → ∃!𝑦 ∈ ran 𝐹 𝑜 ∈ 𝑦) |
61 | 60 | ex 416 |
. . . 4
⊢ ((𝐹‘𝑜) ∈ ran 𝐹 → (∀𝑦 ∈ ran 𝐹(𝑜 ∈ 𝑦 ↔ 𝑦 = (𝐹‘𝑜)) → ∃!𝑦 ∈ ran 𝐹 𝑜 ∈ 𝑦)) |
62 | 3, 59, 61 | syl6c 70 |
. . 3
⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝑜 ∈ 𝐴 → ∃!𝑦 ∈ ran 𝐹 𝑜 ∈ 𝑦)) |
63 | 62 | ralrimiv 3104 |
. 2
⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → ∀𝑜 ∈ 𝐴 ∃!𝑦 ∈ ran 𝐹 𝑜 ∈ 𝑦) |
64 | | nfv 1922 |
. . . 4
⊢
Ⅎ𝑥 𝑞 = 𝑜 |
65 | | nfv 1922 |
. . . 4
⊢
Ⅎ𝑦 𝑞 = 𝑜 |
66 | | nfvd 1923 |
. . . 4
⊢ (𝑞 = 𝑜 → Ⅎ𝑦 𝑞 ∈ 𝑥) |
67 | | nfvd 1923 |
. . . 4
⊢ (𝑞 = 𝑜 → Ⅎ𝑥 𝑜 ∈ 𝑦) |
68 | | eleq12 2827 |
. . . . 5
⊢ ((𝑞 = 𝑜 ∧ 𝑥 = 𝑦) → (𝑞 ∈ 𝑥 ↔ 𝑜 ∈ 𝑦)) |
69 | 68 | ex 416 |
. . . 4
⊢ (𝑞 = 𝑜 → (𝑥 = 𝑦 → (𝑞 ∈ 𝑥 ↔ 𝑜 ∈ 𝑦))) |
70 | 64, 65, 66, 67, 69 | cbvreud 35281 |
. . 3
⊢ (𝑞 = 𝑜 → (∃!𝑥 ∈ ran 𝐹 𝑞 ∈ 𝑥 ↔ ∃!𝑦 ∈ ran 𝐹 𝑜 ∈ 𝑦)) |
71 | 70 | cbvralvw 3358 |
. 2
⊢
(∀𝑞 ∈
𝐴 ∃!𝑥 ∈ ran 𝐹 𝑞 ∈ 𝑥 ↔ ∀𝑜 ∈ 𝐴 ∃!𝑦 ∈ ran 𝐹 𝑜 ∈ 𝑦) |
72 | 63, 71 | sylibr 237 |
1
⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → ∀𝑞 ∈ 𝐴 ∃!𝑥 ∈ ran 𝐹 𝑞 ∈ 𝑥) |