Step | Hyp | Ref
| Expression |
1 | | eldifi 4061 |
. . . . 5
⊢ (𝑥 ∈ (𝑉 ∖ {𝑁}) → 𝑥 ∈ 𝑉) |
2 | | id 22 |
. . . . 5
⊢ (𝑁 ∈ 𝑉 → 𝑁 ∈ 𝑉) |
3 | | prelpwi 5363 |
. . . . 5
⊢ ((𝑥 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) → {𝑥, 𝑁} ∈ 𝒫 𝑉) |
4 | 1, 2, 3 | syl2anr 597 |
. . . 4
⊢ ((𝑁 ∈ 𝑉 ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → {𝑥, 𝑁} ∈ 𝒫 𝑉) |
5 | 1 | adantl 482 |
. . . . 5
⊢ ((𝑁 ∈ 𝑉 ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → 𝑥 ∈ 𝑉) |
6 | | eldifsni 4723 |
. . . . . . 7
⊢ (𝑥 ∈ (𝑉 ∖ {𝑁}) → 𝑥 ≠ 𝑁) |
7 | 6 | adantl 482 |
. . . . . 6
⊢ ((𝑁 ∈ 𝑉 ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → 𝑥 ≠ 𝑁) |
8 | | eqidd 2739 |
. . . . . 6
⊢ ((𝑁 ∈ 𝑉 ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → {𝑥, 𝑁} = {𝑥, 𝑁}) |
9 | 7, 8 | jca 512 |
. . . . 5
⊢ ((𝑁 ∈ 𝑉 ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑥 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑥, 𝑁})) |
10 | | id 22 |
. . . . . 6
⊢ (𝑥 ∈ 𝑉 → 𝑥 ∈ 𝑉) |
11 | | neeq1 3006 |
. . . . . . . 8
⊢ (𝑎 = 𝑥 → (𝑎 ≠ 𝑁 ↔ 𝑥 ≠ 𝑁)) |
12 | | preq1 4669 |
. . . . . . . . 9
⊢ (𝑎 = 𝑥 → {𝑎, 𝑁} = {𝑥, 𝑁}) |
13 | 12 | eqeq2d 2749 |
. . . . . . . 8
⊢ (𝑎 = 𝑥 → ({𝑥, 𝑁} = {𝑎, 𝑁} ↔ {𝑥, 𝑁} = {𝑥, 𝑁})) |
14 | 11, 13 | anbi12d 631 |
. . . . . . 7
⊢ (𝑎 = 𝑥 → ((𝑎 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁}) ↔ (𝑥 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑥, 𝑁}))) |
15 | 14 | adantl 482 |
. . . . . 6
⊢ ((𝑥 ∈ 𝑉 ∧ 𝑎 = 𝑥) → ((𝑎 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁}) ↔ (𝑥 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑥, 𝑁}))) |
16 | 10, 15 | rspcedv 3554 |
. . . . 5
⊢ (𝑥 ∈ 𝑉 → ((𝑥 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑥, 𝑁}) → ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁}))) |
17 | 5, 9, 16 | sylc 65 |
. . . 4
⊢ ((𝑁 ∈ 𝑉 ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁})) |
18 | | cusgrfi.p |
. . . . . 6
⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁})} |
19 | 18 | eleq2i 2830 |
. . . . 5
⊢ ({𝑥, 𝑁} ∈ 𝑃 ↔ {𝑥, 𝑁} ∈ {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁})}) |
20 | | eqeq1 2742 |
. . . . . . . 8
⊢ (𝑣 = {𝑥, 𝑁} → (𝑣 = {𝑎, 𝑁} ↔ {𝑥, 𝑁} = {𝑎, 𝑁})) |
21 | 20 | anbi2d 629 |
. . . . . . 7
⊢ (𝑣 = {𝑥, 𝑁} → ((𝑎 ≠ 𝑁 ∧ 𝑣 = {𝑎, 𝑁}) ↔ (𝑎 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁}))) |
22 | 21 | rexbidv 3226 |
. . . . . 6
⊢ (𝑣 = {𝑥, 𝑁} → (∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑣 = {𝑎, 𝑁}) ↔ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁}))) |
23 | | eqeq1 2742 |
. . . . . . . . 9
⊢ (𝑥 = 𝑣 → (𝑥 = {𝑎, 𝑁} ↔ 𝑣 = {𝑎, 𝑁})) |
24 | 23 | anbi2d 629 |
. . . . . . . 8
⊢ (𝑥 = 𝑣 → ((𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁}) ↔ (𝑎 ≠ 𝑁 ∧ 𝑣 = {𝑎, 𝑁}))) |
25 | 24 | rexbidv 3226 |
. . . . . . 7
⊢ (𝑥 = 𝑣 → (∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁}) ↔ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑣 = {𝑎, 𝑁}))) |
26 | 25 | cbvrabv 3426 |
. . . . . 6
⊢ {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁})} = {𝑣 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑣 = {𝑎, 𝑁})} |
27 | 22, 26 | elrab2 3627 |
. . . . 5
⊢ ({𝑥, 𝑁} ∈ {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁})} ↔ ({𝑥, 𝑁} ∈ 𝒫 𝑉 ∧ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁}))) |
28 | 19, 27 | bitri 274 |
. . . 4
⊢ ({𝑥, 𝑁} ∈ 𝑃 ↔ ({𝑥, 𝑁} ∈ 𝒫 𝑉 ∧ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ {𝑥, 𝑁} = {𝑎, 𝑁}))) |
29 | 4, 17, 28 | sylanbrc 583 |
. . 3
⊢ ((𝑁 ∈ 𝑉 ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → {𝑥, 𝑁} ∈ 𝑃) |
30 | 29 | ralrimiva 3103 |
. 2
⊢ (𝑁 ∈ 𝑉 → ∀𝑥 ∈ (𝑉 ∖ {𝑁}){𝑥, 𝑁} ∈ 𝑃) |
31 | | simpl 483 |
. . . . . . . . . . 11
⊢ ((𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}) → 𝑎 ≠ 𝑁) |
32 | 31 | anim2i 617 |
. . . . . . . . . 10
⊢ ((𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁})) → (𝑎 ∈ 𝑉 ∧ 𝑎 ≠ 𝑁)) |
33 | 32 | adantl 482 |
. . . . . . . . 9
⊢ (((𝑁 ∈ 𝑉 ∧ 𝑒 ∈ 𝒫 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}))) → (𝑎 ∈ 𝑉 ∧ 𝑎 ≠ 𝑁)) |
34 | | eldifsn 4720 |
. . . . . . . . 9
⊢ (𝑎 ∈ (𝑉 ∖ {𝑁}) ↔ (𝑎 ∈ 𝑉 ∧ 𝑎 ≠ 𝑁)) |
35 | 33, 34 | sylibr 233 |
. . . . . . . 8
⊢ (((𝑁 ∈ 𝑉 ∧ 𝑒 ∈ 𝒫 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}))) → 𝑎 ∈ (𝑉 ∖ {𝑁})) |
36 | | eqeq1 2742 |
. . . . . . . . . . . . . 14
⊢ (𝑒 = {𝑎, 𝑁} → (𝑒 = {𝑥, 𝑁} ↔ {𝑎, 𝑁} = {𝑥, 𝑁})) |
37 | 36 | adantl 482 |
. . . . . . . . . . . . 13
⊢ ((𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}) → (𝑒 = {𝑥, 𝑁} ↔ {𝑎, 𝑁} = {𝑥, 𝑁})) |
38 | 37 | ad2antlr 724 |
. . . . . . . . . . . 12
⊢ (((𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁})) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑒 = {𝑥, 𝑁} ↔ {𝑎, 𝑁} = {𝑥, 𝑁})) |
39 | | vex 3436 |
. . . . . . . . . . . . . 14
⊢ 𝑎 ∈ V |
40 | | vex 3436 |
. . . . . . . . . . . . . 14
⊢ 𝑥 ∈ V |
41 | 39, 40 | preqr1 4779 |
. . . . . . . . . . . . 13
⊢ ({𝑎, 𝑁} = {𝑥, 𝑁} → 𝑎 = 𝑥) |
42 | 41 | equcomd 2022 |
. . . . . . . . . . . 12
⊢ ({𝑎, 𝑁} = {𝑥, 𝑁} → 𝑥 = 𝑎) |
43 | 38, 42 | syl6bi 252 |
. . . . . . . . . . 11
⊢ (((𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁})) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑒 = {𝑥, 𝑁} → 𝑥 = 𝑎)) |
44 | 43 | adantll 711 |
. . . . . . . . . 10
⊢ ((((𝑁 ∈ 𝑉 ∧ 𝑒 ∈ 𝒫 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}))) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑒 = {𝑥, 𝑁} → 𝑥 = 𝑎)) |
45 | 12 | equcoms 2023 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑎 → {𝑎, 𝑁} = {𝑥, 𝑁}) |
46 | 45 | eqeq2d 2749 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑎 → (𝑒 = {𝑎, 𝑁} ↔ 𝑒 = {𝑥, 𝑁})) |
47 | 46 | biimpcd 248 |
. . . . . . . . . . . . 13
⊢ (𝑒 = {𝑎, 𝑁} → (𝑥 = 𝑎 → 𝑒 = {𝑥, 𝑁})) |
48 | 47 | adantl 482 |
. . . . . . . . . . . 12
⊢ ((𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}) → (𝑥 = 𝑎 → 𝑒 = {𝑥, 𝑁})) |
49 | 48 | adantl 482 |
. . . . . . . . . . 11
⊢ ((𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁})) → (𝑥 = 𝑎 → 𝑒 = {𝑥, 𝑁})) |
50 | 49 | ad2antlr 724 |
. . . . . . . . . 10
⊢ ((((𝑁 ∈ 𝑉 ∧ 𝑒 ∈ 𝒫 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}))) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑥 = 𝑎 → 𝑒 = {𝑥, 𝑁})) |
51 | 44, 50 | impbid 211 |
. . . . . . . . 9
⊢ ((((𝑁 ∈ 𝑉 ∧ 𝑒 ∈ 𝒫 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}))) ∧ 𝑥 ∈ (𝑉 ∖ {𝑁})) → (𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎)) |
52 | 51 | ralrimiva 3103 |
. . . . . . . 8
⊢ (((𝑁 ∈ 𝑉 ∧ 𝑒 ∈ 𝒫 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}))) → ∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎)) |
53 | 35, 52 | jca 512 |
. . . . . . 7
⊢ (((𝑁 ∈ 𝑉 ∧ 𝑒 ∈ 𝒫 𝑉) ∧ (𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}))) → (𝑎 ∈ (𝑉 ∖ {𝑁}) ∧ ∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎))) |
54 | 53 | ex 413 |
. . . . . 6
⊢ ((𝑁 ∈ 𝑉 ∧ 𝑒 ∈ 𝒫 𝑉) → ((𝑎 ∈ 𝑉 ∧ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁})) → (𝑎 ∈ (𝑉 ∖ {𝑁}) ∧ ∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎)))) |
55 | 54 | reximdv2 3199 |
. . . . 5
⊢ ((𝑁 ∈ 𝑉 ∧ 𝑒 ∈ 𝒫 𝑉) → (∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}) → ∃𝑎 ∈ (𝑉 ∖ {𝑁})∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎))) |
56 | 55 | expimpd 454 |
. . . 4
⊢ (𝑁 ∈ 𝑉 → ((𝑒 ∈ 𝒫 𝑉 ∧ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁})) → ∃𝑎 ∈ (𝑉 ∖ {𝑁})∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎))) |
57 | | eqeq1 2742 |
. . . . . . 7
⊢ (𝑥 = 𝑒 → (𝑥 = {𝑎, 𝑁} ↔ 𝑒 = {𝑎, 𝑁})) |
58 | 57 | anbi2d 629 |
. . . . . 6
⊢ (𝑥 = 𝑒 → ((𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁}) ↔ (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}))) |
59 | 58 | rexbidv 3226 |
. . . . 5
⊢ (𝑥 = 𝑒 → (∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑥 = {𝑎, 𝑁}) ↔ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}))) |
60 | 59, 18 | elrab2 3627 |
. . . 4
⊢ (𝑒 ∈ 𝑃 ↔ (𝑒 ∈ 𝒫 𝑉 ∧ ∃𝑎 ∈ 𝑉 (𝑎 ≠ 𝑁 ∧ 𝑒 = {𝑎, 𝑁}))) |
61 | | reu6 3661 |
. . . 4
⊢
(∃!𝑥 ∈
(𝑉 ∖ {𝑁})𝑒 = {𝑥, 𝑁} ↔ ∃𝑎 ∈ (𝑉 ∖ {𝑁})∀𝑥 ∈ (𝑉 ∖ {𝑁})(𝑒 = {𝑥, 𝑁} ↔ 𝑥 = 𝑎)) |
62 | 56, 60, 61 | 3imtr4g 296 |
. . 3
⊢ (𝑁 ∈ 𝑉 → (𝑒 ∈ 𝑃 → ∃!𝑥 ∈ (𝑉 ∖ {𝑁})𝑒 = {𝑥, 𝑁})) |
63 | 62 | ralrimiv 3102 |
. 2
⊢ (𝑁 ∈ 𝑉 → ∀𝑒 ∈ 𝑃 ∃!𝑥 ∈ (𝑉 ∖ {𝑁})𝑒 = {𝑥, 𝑁}) |
64 | | cusgrfi.f |
. . 3
⊢ 𝐹 = (𝑥 ∈ (𝑉 ∖ {𝑁}) ↦ {𝑥, 𝑁}) |
65 | 64 | f1ompt 6985 |
. 2
⊢ (𝐹:(𝑉 ∖ {𝑁})–1-1-onto→𝑃 ↔ (∀𝑥 ∈ (𝑉 ∖ {𝑁}){𝑥, 𝑁} ∈ 𝑃 ∧ ∀𝑒 ∈ 𝑃 ∃!𝑥 ∈ (𝑉 ∖ {𝑁})𝑒 = {𝑥, 𝑁})) |
66 | 30, 63, 65 | sylanbrc 583 |
1
⊢ (𝑁 ∈ 𝑉 → 𝐹:(𝑉 ∖ {𝑁})–1-1-onto→𝑃) |