Step | Hyp | Ref
| Expression |
1 | | pssnel 4385 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑍 ⊊ ran 𝐹 → ∃𝑥(𝑥 ∈ ran 𝐹 ∧ ¬ 𝑥 ∈ 𝑍)) |
2 | 1 | adantl 485 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑥(𝑥 ∈ ran 𝐹 ∧ ¬ 𝑥 ∈ 𝑍)) |
3 | | df-rex 3067 |
. . . . . . . . . . . . . . . . . . 19
⊢
(∃𝑥 ∈ ran
𝐹 ¬ 𝑥 ∈ 𝑍 ↔ ∃𝑥(𝑥 ∈ ran 𝐹 ∧ ¬ 𝑥 ∈ 𝑍)) |
4 | 2, 3 | sylibr 237 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑥 ∈ ran 𝐹 ¬ 𝑥 ∈ 𝑍) |
5 | | fnrnfv 6772 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑥 ∣ ∃𝑝 ∈ 𝐴 𝑥 = (𝐹‘𝑝)}) |
6 | 5 | abeq2d 2871 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑝 ∈ 𝐴 𝑥 = (𝐹‘𝑝))) |
7 | 6 | biimpd 232 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ ran 𝐹 → ∃𝑝 ∈ 𝐴 𝑥 = (𝐹‘𝑝))) |
8 | 7 | ralrimiv 3104 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐹 Fn 𝐴 → ∀𝑥 ∈ ran 𝐹∃𝑝 ∈ 𝐴 𝑥 = (𝐹‘𝑝)) |
9 | 8 | adantr 484 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → ∀𝑥 ∈ ran 𝐹∃𝑝 ∈ 𝐴 𝑥 = (𝐹‘𝑝)) |
10 | 9 | adantr 484 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∀𝑥 ∈ ran 𝐹∃𝑝 ∈ 𝐴 𝑥 = (𝐹‘𝑝)) |
11 | | r19.29r 3177 |
. . . . . . . . . . . . . . . . . 18
⊢
((∃𝑥 ∈
ran 𝐹 ¬ 𝑥 ∈ 𝑍 ∧ ∀𝑥 ∈ ran 𝐹∃𝑝 ∈ 𝐴 𝑥 = (𝐹‘𝑝)) → ∃𝑥 ∈ ran 𝐹(¬ 𝑥 ∈ 𝑍 ∧ ∃𝑝 ∈ 𝐴 𝑥 = (𝐹‘𝑝))) |
12 | 4, 10, 11 | syl2anc 587 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑥 ∈ ran 𝐹(¬ 𝑥 ∈ 𝑍 ∧ ∃𝑝 ∈ 𝐴 𝑥 = (𝐹‘𝑝))) |
13 | | nfra1 3140 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑝∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} |
14 | | rsp 3127 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
(∀𝑝 ∈
𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → (𝑝 ∈ 𝐴 → ((𝐹‘𝑝) ∩ 𝐴) = {𝑝})) |
15 | | vsnid 4578 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ 𝑝 ∈ {𝑝} |
16 | | eleq2 2826 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → (𝑝 ∈ ((𝐹‘𝑝) ∩ 𝐴) ↔ 𝑝 ∈ {𝑝})) |
17 | 15, 16 | mpbiri 261 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → 𝑝 ∈ ((𝐹‘𝑝) ∩ 𝐴)) |
18 | 17 | elin1d 4112 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → 𝑝 ∈ (𝐹‘𝑝)) |
19 | 14, 18 | syl6 35 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
(∀𝑝 ∈
𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → (𝑝 ∈ 𝐴 → 𝑝 ∈ (𝐹‘𝑝))) |
20 | 19 | adantr 484 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
((∀𝑝 ∈
𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} ∧ 𝑥 = (𝐹‘𝑝)) → (𝑝 ∈ 𝐴 → 𝑝 ∈ (𝐹‘𝑝))) |
21 | | eleq2 2826 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑥 = (𝐹‘𝑝) → (𝑝 ∈ 𝑥 ↔ 𝑝 ∈ (𝐹‘𝑝))) |
22 | 21 | adantl 485 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
((∀𝑝 ∈
𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} ∧ 𝑥 = (𝐹‘𝑝)) → (𝑝 ∈ 𝑥 ↔ 𝑝 ∈ (𝐹‘𝑝))) |
23 | 20, 22 | sylibrd 262 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((∀𝑝 ∈
𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} ∧ 𝑥 = (𝐹‘𝑝)) → (𝑝 ∈ 𝐴 → 𝑝 ∈ 𝑥)) |
24 | 23 | ex 416 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(∀𝑝 ∈
𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → (𝑥 = (𝐹‘𝑝) → (𝑝 ∈ 𝐴 → 𝑝 ∈ 𝑥))) |
25 | 24 | com23 86 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(∀𝑝 ∈
𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → (𝑝 ∈ 𝐴 → (𝑥 = (𝐹‘𝑝) → 𝑝 ∈ 𝑥))) |
26 | 13, 25 | reximdai 3230 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(∀𝑝 ∈
𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝} → (∃𝑝 ∈ 𝐴 𝑥 = (𝐹‘𝑝) → ∃𝑝 ∈ 𝐴 𝑝 ∈ 𝑥)) |
27 | 26 | adantl 485 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (∃𝑝 ∈ 𝐴 𝑥 = (𝐹‘𝑝) → ∃𝑝 ∈ 𝐴 𝑝 ∈ 𝑥)) |
28 | 27 | adantr 484 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → (∃𝑝 ∈ 𝐴 𝑥 = (𝐹‘𝑝) → ∃𝑝 ∈ 𝐴 𝑝 ∈ 𝑥)) |
29 | 28 | anim2d 615 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ((¬ 𝑥 ∈ 𝑍 ∧ ∃𝑝 ∈ 𝐴 𝑥 = (𝐹‘𝑝)) → (¬ 𝑥 ∈ 𝑍 ∧ ∃𝑝 ∈ 𝐴 𝑝 ∈ 𝑥))) |
30 | 29 | reximdv 3192 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → (∃𝑥 ∈ ran 𝐹(¬ 𝑥 ∈ 𝑍 ∧ ∃𝑝 ∈ 𝐴 𝑥 = (𝐹‘𝑝)) → ∃𝑥 ∈ ran 𝐹(¬ 𝑥 ∈ 𝑍 ∧ ∃𝑝 ∈ 𝐴 𝑝 ∈ 𝑥))) |
31 | 12, 30 | mpd 15 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑥 ∈ ran 𝐹(¬ 𝑥 ∈ 𝑍 ∧ ∃𝑝 ∈ 𝐴 𝑝 ∈ 𝑥)) |
32 | | ancom 464 |
. . . . . . . . . . . . . . . . . 18
⊢ ((¬
𝑥 ∈ 𝑍 ∧ ∃𝑝 ∈ 𝐴 𝑝 ∈ 𝑥) ↔ (∃𝑝 ∈ 𝐴 𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍)) |
33 | | r19.41v 3260 |
. . . . . . . . . . . . . . . . . 18
⊢
(∃𝑝 ∈
𝐴 (𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) ↔ (∃𝑝 ∈ 𝐴 𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍)) |
34 | 32, 33 | bitr4i 281 |
. . . . . . . . . . . . . . . . 17
⊢ ((¬
𝑥 ∈ 𝑍 ∧ ∃𝑝 ∈ 𝐴 𝑝 ∈ 𝑥) ↔ ∃𝑝 ∈ 𝐴 (𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍)) |
35 | 34 | rexbii 3170 |
. . . . . . . . . . . . . . . 16
⊢
(∃𝑥 ∈ ran
𝐹(¬ 𝑥 ∈ 𝑍 ∧ ∃𝑝 ∈ 𝐴 𝑝 ∈ 𝑥) ↔ ∃𝑥 ∈ ran 𝐹∃𝑝 ∈ 𝐴 (𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍)) |
36 | 31, 35 | sylib 221 |
. . . . . . . . . . . . . . 15
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑥 ∈ ran 𝐹∃𝑝 ∈ 𝐴 (𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍)) |
37 | | rexcom 3268 |
. . . . . . . . . . . . . . 15
⊢
(∃𝑝 ∈
𝐴 ∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) ↔ ∃𝑥 ∈ ran 𝐹∃𝑝 ∈ 𝐴 (𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍)) |
38 | 36, 37 | sylibr 237 |
. . . . . . . . . . . . . 14
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑝 ∈ 𝐴 ∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍)) |
39 | | nfre1 3225 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑥∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) |
40 | 39 | 19.3 2200 |
. . . . . . . . . . . . . . . 16
⊢
(∀𝑥∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) ↔ ∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍)) |
41 | | alral 3077 |
. . . . . . . . . . . . . . . 16
⊢
(∀𝑥∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) → ∀𝑥 ∈ ran 𝐹∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍)) |
42 | 40, 41 | sylbir 238 |
. . . . . . . . . . . . . . 15
⊢
(∃𝑥 ∈ ran
𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) → ∀𝑥 ∈ ran 𝐹∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍)) |
43 | 42 | reximi 3166 |
. . . . . . . . . . . . . 14
⊢
(∃𝑝 ∈
𝐴 ∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) → ∃𝑝 ∈ 𝐴 ∀𝑥 ∈ ran 𝐹∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍)) |
44 | 38, 43 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑝 ∈ 𝐴 ∀𝑥 ∈ ran 𝐹∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍)) |
45 | | nfv 1922 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑝 𝐹 Fn 𝐴 |
46 | 45, 13 | nfan 1907 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑝(𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) |
47 | | nfv 1922 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑝 𝑍 ⊊ ran 𝐹 |
48 | 46, 47 | nfan 1907 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑝((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) |
49 | | nfv 1922 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑥(((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ 𝑝 ∈ 𝐴) |
50 | | fvineqsneu 35319 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → ∀𝑝 ∈ 𝐴 ∃!𝑥 ∈ ran 𝐹 𝑝 ∈ 𝑥) |
51 | 50 | adantr 484 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∀𝑝 ∈ 𝐴 ∃!𝑥 ∈ ran 𝐹 𝑝 ∈ 𝑥) |
52 | | rsp 3127 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(∀𝑝 ∈
𝐴 ∃!𝑥 ∈ ran 𝐹 𝑝 ∈ 𝑥 → (𝑝 ∈ 𝐴 → ∃!𝑥 ∈ ran 𝐹 𝑝 ∈ 𝑥)) |
53 | 51, 52 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → (𝑝 ∈ 𝐴 → ∃!𝑥 ∈ ran 𝐹 𝑝 ∈ 𝑥)) |
54 | 53 | adantrd 495 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ((𝑝 ∈ 𝐴 ∧ 𝑥 ∈ ran 𝐹) → ∃!𝑥 ∈ ran 𝐹 𝑝 ∈ 𝑥)) |
55 | 54 | imp 410 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ (𝑝 ∈ 𝐴 ∧ 𝑥 ∈ ran 𝐹)) → ∃!𝑥 ∈ ran 𝐹 𝑝 ∈ 𝑥) |
56 | | reupick3 4234 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((∃!𝑥 ∈
ran 𝐹 𝑝 ∈ 𝑥 ∧ ∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) ∧ 𝑥 ∈ ran 𝐹) → (𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍)) |
57 | 56 | 3expa 1120 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((∃!𝑥 ∈
ran 𝐹 𝑝 ∈ 𝑥 ∧ ∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍)) ∧ 𝑥 ∈ ran 𝐹) → (𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍)) |
58 | 57 | expcom 417 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ ran 𝐹 → ((∃!𝑥 ∈ ran 𝐹 𝑝 ∈ 𝑥 ∧ ∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍)) → (𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍))) |
59 | 58 | adantl 485 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑝 ∈ 𝐴 ∧ 𝑥 ∈ ran 𝐹) → ((∃!𝑥 ∈ ran 𝐹 𝑝 ∈ 𝑥 ∧ ∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍)) → (𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍))) |
60 | 59 | adantl 485 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ (𝑝 ∈ 𝐴 ∧ 𝑥 ∈ ran 𝐹)) → ((∃!𝑥 ∈ ran 𝐹 𝑝 ∈ 𝑥 ∧ ∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍)) → (𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍))) |
61 | 55, 60 | mpand 695 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ (𝑝 ∈ 𝐴 ∧ 𝑥 ∈ ran 𝐹)) → (∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) → (𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍))) |
62 | 61 | expr 460 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ 𝑝 ∈ 𝐴) → (𝑥 ∈ ran 𝐹 → (∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) → (𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍)))) |
63 | 49, 62 | ralrimi 3137 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ 𝑝 ∈ 𝐴) → ∀𝑥 ∈ ran 𝐹(∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) → (𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍))) |
64 | 63 | ex 416 |
. . . . . . . . . . . . . 14
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → (𝑝 ∈ 𝐴 → ∀𝑥 ∈ ran 𝐹(∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) → (𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍)))) |
65 | 48, 64 | ralrimi 3137 |
. . . . . . . . . . . . 13
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∀𝑝 ∈ 𝐴 ∀𝑥 ∈ ran 𝐹(∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) → (𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍))) |
66 | | r19.29r 3177 |
. . . . . . . . . . . . 13
⊢
((∃𝑝 ∈
𝐴 ∀𝑥 ∈ ran 𝐹∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) ∧ ∀𝑝 ∈ 𝐴 ∀𝑥 ∈ ran 𝐹(∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) → (𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍))) → ∃𝑝 ∈ 𝐴 (∀𝑥 ∈ ran 𝐹∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) ∧ ∀𝑥 ∈ ran 𝐹(∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) → (𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍)))) |
67 | 44, 65, 66 | syl2anc 587 |
. . . . . . . . . . . 12
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑝 ∈ 𝐴 (∀𝑥 ∈ ran 𝐹∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) ∧ ∀𝑥 ∈ ran 𝐹(∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) → (𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍)))) |
68 | | ralim 3085 |
. . . . . . . . . . . . . 14
⊢
(∀𝑥 ∈
ran 𝐹(∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) → (𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍)) → (∀𝑥 ∈ ran 𝐹∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) → ∀𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍))) |
69 | 68 | impcom 411 |
. . . . . . . . . . . . 13
⊢
((∀𝑥 ∈
ran 𝐹∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) ∧ ∀𝑥 ∈ ran 𝐹(∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) → (𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍))) → ∀𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍)) |
70 | 69 | reximi 3166 |
. . . . . . . . . . . 12
⊢
(∃𝑝 ∈
𝐴 (∀𝑥 ∈ ran 𝐹∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) ∧ ∀𝑥 ∈ ran 𝐹(∃𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 ∧ ¬ 𝑥 ∈ 𝑍) → (𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍))) → ∃𝑝 ∈ 𝐴 ∀𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍)) |
71 | 67, 70 | syl 17 |
. . . . . . . . . . 11
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑝 ∈ 𝐴 ∀𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍)) |
72 | | con2b 363 |
. . . . . . . . . . . . . . 15
⊢ ((𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍) ↔ (𝑥 ∈ 𝑍 → ¬ 𝑝 ∈ 𝑥)) |
73 | 72 | ralbii 3088 |
. . . . . . . . . . . . . 14
⊢
(∀𝑥 ∈
ran 𝐹(𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍) ↔ ∀𝑥 ∈ ran 𝐹(𝑥 ∈ 𝑍 → ¬ 𝑝 ∈ 𝑥)) |
74 | | df-ral 3066 |
. . . . . . . . . . . . . 14
⊢
(∀𝑥 ∈
ran 𝐹(𝑥 ∈ 𝑍 → ¬ 𝑝 ∈ 𝑥) ↔ ∀𝑥(𝑥 ∈ ran 𝐹 → (𝑥 ∈ 𝑍 → ¬ 𝑝 ∈ 𝑥))) |
75 | | bi2.04 392 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ran 𝐹 → (𝑥 ∈ 𝑍 → ¬ 𝑝 ∈ 𝑥)) ↔ (𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥))) |
76 | 75 | albii 1827 |
. . . . . . . . . . . . . 14
⊢
(∀𝑥(𝑥 ∈ ran 𝐹 → (𝑥 ∈ 𝑍 → ¬ 𝑝 ∈ 𝑥)) ↔ ∀𝑥(𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥))) |
77 | 73, 74, 76 | 3bitri 300 |
. . . . . . . . . . . . 13
⊢
(∀𝑥 ∈
ran 𝐹(𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍) ↔ ∀𝑥(𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥))) |
78 | 77 | a1i 11 |
. . . . . . . . . . . 12
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → (∀𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍) ↔ ∀𝑥(𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥)))) |
79 | 48, 78 | rexbid 3239 |
. . . . . . . . . . 11
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → (∃𝑝 ∈ 𝐴 ∀𝑥 ∈ ran 𝐹(𝑝 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑍) ↔ ∃𝑝 ∈ 𝐴 ∀𝑥(𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥)))) |
80 | 71, 79 | mpbid 235 |
. . . . . . . . . 10
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑝 ∈ 𝐴 ∀𝑥(𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥))) |
81 | | nfv 1922 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑥((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) |
82 | | nfa1 2152 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑥∀𝑥(𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥)) |
83 | 81, 82 | nfan 1907 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑥(((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ ∀𝑥(𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥))) |
84 | | pssss 4010 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑍 ⊊ ran 𝐹 → 𝑍 ⊆ ran 𝐹) |
85 | | dfss2 3886 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑍 ⊆ ran 𝐹 ↔ ∀𝑥(𝑥 ∈ 𝑍 → 𝑥 ∈ ran 𝐹)) |
86 | 84, 85 | sylib 221 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑍 ⊊ ran 𝐹 → ∀𝑥(𝑥 ∈ 𝑍 → 𝑥 ∈ ran 𝐹)) |
87 | 86 | adantl 485 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∀𝑥(𝑥 ∈ 𝑍 → 𝑥 ∈ ran 𝐹)) |
88 | | df-ral 3066 |
. . . . . . . . . . . . . . . . . 18
⊢
(∀𝑥 ∈
𝑍 𝑥 ∈ ran 𝐹 ↔ ∀𝑥(𝑥 ∈ 𝑍 → 𝑥 ∈ ran 𝐹)) |
89 | 87, 88 | sylibr 237 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∀𝑥 ∈ 𝑍 𝑥 ∈ ran 𝐹) |
90 | 89 | adantr 484 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ ∀𝑥(𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥))) → ∀𝑥 ∈ 𝑍 𝑥 ∈ ran 𝐹) |
91 | | rsp 3127 |
. . . . . . . . . . . . . . . 16
⊢
(∀𝑥 ∈
𝑍 𝑥 ∈ ran 𝐹 → (𝑥 ∈ 𝑍 → 𝑥 ∈ ran 𝐹)) |
92 | 90, 91 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ ∀𝑥(𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥))) → (𝑥 ∈ 𝑍 → 𝑥 ∈ ran 𝐹)) |
93 | | df-ral 3066 |
. . . . . . . . . . . . . . . . . 18
⊢
(∀𝑥 ∈
𝑍 (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥) ↔ ∀𝑥(𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥))) |
94 | 93 | biimpri 231 |
. . . . . . . . . . . . . . . . 17
⊢
(∀𝑥(𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥)) → ∀𝑥 ∈ 𝑍 (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥)) |
95 | 94 | adantl 485 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ ∀𝑥(𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥))) → ∀𝑥 ∈ 𝑍 (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥)) |
96 | | rsp 3127 |
. . . . . . . . . . . . . . . 16
⊢
(∀𝑥 ∈
𝑍 (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥) → (𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥))) |
97 | 95, 96 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ ∀𝑥(𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥))) → (𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥))) |
98 | 92, 97 | mpdd 43 |
. . . . . . . . . . . . . 14
⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ ∀𝑥(𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥))) → (𝑥 ∈ 𝑍 → ¬ 𝑝 ∈ 𝑥)) |
99 | 83, 98 | ralrimi 3137 |
. . . . . . . . . . . . 13
⊢ ((((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) ∧ ∀𝑥(𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥))) → ∀𝑥 ∈ 𝑍 ¬ 𝑝 ∈ 𝑥) |
100 | 99 | ex 416 |
. . . . . . . . . . . 12
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → (∀𝑥(𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥)) → ∀𝑥 ∈ 𝑍 ¬ 𝑝 ∈ 𝑥)) |
101 | 100 | a1d 25 |
. . . . . . . . . . 11
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → (𝑝 ∈ 𝐴 → (∀𝑥(𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥)) → ∀𝑥 ∈ 𝑍 ¬ 𝑝 ∈ 𝑥))) |
102 | 48, 101 | reximdai 3230 |
. . . . . . . . . 10
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → (∃𝑝 ∈ 𝐴 ∀𝑥(𝑥 ∈ 𝑍 → (𝑥 ∈ ran 𝐹 → ¬ 𝑝 ∈ 𝑥)) → ∃𝑝 ∈ 𝐴 ∀𝑥 ∈ 𝑍 ¬ 𝑝 ∈ 𝑥)) |
103 | 80, 102 | mpd 15 |
. . . . . . . . 9
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑝 ∈ 𝐴 ∀𝑥 ∈ 𝑍 ¬ 𝑝 ∈ 𝑥) |
104 | | ralnex 3158 |
. . . . . . . . . 10
⊢
(∀𝑥 ∈
𝑍 ¬ 𝑝 ∈ 𝑥 ↔ ¬ ∃𝑥 ∈ 𝑍 𝑝 ∈ 𝑥) |
105 | 104 | rexbii 3170 |
. . . . . . . . 9
⊢
(∃𝑝 ∈
𝐴 ∀𝑥 ∈ 𝑍 ¬ 𝑝 ∈ 𝑥 ↔ ∃𝑝 ∈ 𝐴 ¬ ∃𝑥 ∈ 𝑍 𝑝 ∈ 𝑥) |
106 | 103, 105 | sylib 221 |
. . . . . . . 8
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑝 ∈ 𝐴 ¬ ∃𝑥 ∈ 𝑍 𝑝 ∈ 𝑥) |
107 | | eluni2 4823 |
. . . . . . . . . 10
⊢ (𝑝 ∈ ∪ 𝑍
↔ ∃𝑥 ∈
𝑍 𝑝 ∈ 𝑥) |
108 | 107 | notbii 323 |
. . . . . . . . 9
⊢ (¬
𝑝 ∈ ∪ 𝑍
↔ ¬ ∃𝑥
∈ 𝑍 𝑝 ∈ 𝑥) |
109 | 108 | rexbii 3170 |
. . . . . . . 8
⊢
(∃𝑝 ∈
𝐴 ¬ 𝑝 ∈ ∪ 𝑍 ↔ ∃𝑝 ∈ 𝐴 ¬ ∃𝑥 ∈ 𝑍 𝑝 ∈ 𝑥) |
110 | 106, 109 | sylibr 237 |
. . . . . . 7
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ∃𝑝 ∈ 𝐴 ¬ 𝑝 ∈ ∪ 𝑍) |
111 | | dfss3 3888 |
. . . . . . . . 9
⊢ (𝐴 ⊆ ∪ 𝑍
↔ ∀𝑝 ∈
𝐴 𝑝 ∈ ∪ 𝑍) |
112 | | dfral2 3159 |
. . . . . . . . 9
⊢
(∀𝑝 ∈
𝐴 𝑝 ∈ ∪ 𝑍 ↔ ¬ ∃𝑝 ∈ 𝐴 ¬ 𝑝 ∈ ∪ 𝑍) |
113 | 111, 112 | bitri 278 |
. . . . . . . 8
⊢ (𝐴 ⊆ ∪ 𝑍
↔ ¬ ∃𝑝
∈ 𝐴 ¬ 𝑝 ∈ ∪ 𝑍) |
114 | 113 | con2bii2 35241 |
. . . . . . 7
⊢ (¬
𝐴 ⊆ ∪ 𝑍
↔ ∃𝑝 ∈
𝐴 ¬ 𝑝 ∈ ∪ 𝑍) |
115 | 110, 114 | sylibr 237 |
. . . . . 6
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ 𝑍 ⊊ ran 𝐹) → ¬ 𝐴 ⊆ ∪ 𝑍) |
116 | 115 | ex 416 |
. . . . 5
⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝑍 ⊊ ran 𝐹 → ¬ 𝐴 ⊆ ∪ 𝑍)) |
117 | 116 | con2d 136 |
. . . 4
⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝐴 ⊆ ∪ 𝑍 → ¬ 𝑍 ⊊ ran 𝐹)) |
118 | | npss 4025 |
. . . 4
⊢ (¬
𝑍 ⊊ ran 𝐹 ↔ (𝑍 ⊆ ran 𝐹 → 𝑍 = ran 𝐹)) |
119 | 117, 118 | syl6ib 254 |
. . 3
⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝐴 ⊆ ∪ 𝑍 → (𝑍 ⊆ ran 𝐹 → 𝑍 = ran 𝐹))) |
120 | 119 | com23 86 |
. 2
⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) → (𝑍 ⊆ ran 𝐹 → (𝐴 ⊆ ∪ 𝑍 → 𝑍 = ran 𝐹))) |
121 | 120 | imp32 422 |
1
⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑝 ∈ 𝐴 ((𝐹‘𝑝) ∩ 𝐴) = {𝑝}) ∧ (𝑍 ⊆ ran 𝐹 ∧ 𝐴 ⊆ ∪ 𝑍)) → 𝑍 = ran 𝐹) |