Step | Hyp | Ref
| Expression |
1 | | frgrncvvdeq.v1 |
. . 3
⊢ 𝑉 = (Vtx‘𝐺) |
2 | | frgrncvvdeq.e |
. . 3
⊢ 𝐸 = (Edg‘𝐺) |
3 | | frgrncvvdeq.nx |
. . 3
⊢ 𝐷 = (𝐺 NeighbVtx 𝑋) |
4 | | frgrncvvdeq.ny |
. . 3
⊢ 𝑁 = (𝐺 NeighbVtx 𝑌) |
5 | | frgrncvvdeq.x |
. . 3
⊢ (𝜑 → 𝑋 ∈ 𝑉) |
6 | | frgrncvvdeq.y |
. . 3
⊢ (𝜑 → 𝑌 ∈ 𝑉) |
7 | | frgrncvvdeq.ne |
. . 3
⊢ (𝜑 → 𝑋 ≠ 𝑌) |
8 | | frgrncvvdeq.xy |
. . 3
⊢ (𝜑 → 𝑌 ∉ 𝐷) |
9 | | frgrncvvdeq.f |
. . 3
⊢ (𝜑 → 𝐺 ∈ FriendGraph ) |
10 | | frgrncvvdeq.a |
. . 3
⊢ 𝐴 = (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)) |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | frgrncvvdeqlem5 41473 |
. 2
⊢ (𝜑 → 𝐴:𝐷⟶𝑁) |
12 | | ffrn 5968 |
. . . . 5
⊢ (𝐴:𝐷⟶𝑁 → 𝐴:𝐷⟶ran 𝐴) |
13 | 12 | adantl 481 |
. . . 4
⊢ ((𝜑 ∧ 𝐴:𝐷⟶𝑁) → 𝐴:𝐷⟶ran 𝐴) |
14 | | ffvelrn 6265 |
. . . . . . . . . 10
⊢ ((𝐴:𝐷⟶𝑁 ∧ 𝑢 ∈ 𝐷) → (𝐴‘𝑢) ∈ 𝑁) |
15 | 14 | ad5ant23 1296 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝐴:𝐷⟶𝑁) ∧ 𝑢 ∈ 𝐷) ∧ 𝑤 ∈ 𝐷) ∧ (𝐴‘𝑢) = (𝐴‘𝑤)) → (𝐴‘𝑢) ∈ 𝑁) |
16 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | frgrncvvdeqlem2 41470 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑋 ∉ 𝑁) |
17 | | preq1 4212 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = 𝑢 → {𝑥, 𝑦} = {𝑢, 𝑦}) |
18 | 17 | eleq1d 2672 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = 𝑢 → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑢, 𝑦} ∈ 𝐸)) |
19 | 18 | riotabidv 6513 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = 𝑢 → (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸) = (℩𝑦 ∈ 𝑁 {𝑢, 𝑦} ∈ 𝐸)) |
20 | 19 | cbvmptv 4678 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)) = (𝑢 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑢, 𝑦} ∈ 𝐸)) |
21 | 10, 20 | eqtri 2632 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝐴 = (𝑢 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑢, 𝑦} ∈ 𝐸)) |
22 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 21 | frgrncvvdeqlem7 41475 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑢 ∈ 𝐷) → {𝑢, (𝐴‘𝑢)} ∈ 𝐸) |
23 | | preq1 4212 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = 𝑤 → {𝑥, 𝑦} = {𝑤, 𝑦}) |
24 | 23 | eleq1d 2672 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = 𝑤 → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑤, 𝑦} ∈ 𝐸)) |
25 | 24 | riotabidv 6513 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = 𝑤 → (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸) = (℩𝑦 ∈ 𝑁 {𝑤, 𝑦} ∈ 𝐸)) |
26 | 25 | cbvmptv 4678 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)) = (𝑤 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑤, 𝑦} ∈ 𝐸)) |
27 | 10, 26 | eqtri 2632 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝐴 = (𝑤 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑤, 𝑦} ∈ 𝐸)) |
28 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 27 | frgrncvvdeqlem7 41475 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑤 ∈ 𝐷) → {𝑤, (𝐴‘𝑤)} ∈ 𝐸) |
29 | 22, 28 | anim12dan 878 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → ({𝑢, (𝐴‘𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴‘𝑤)} ∈ 𝐸)) |
30 | | preq2 4213 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝐴‘𝑤) = (𝐴‘𝑢) → {𝑤, (𝐴‘𝑤)} = {𝑤, (𝐴‘𝑢)}) |
31 | 30 | eleq1d 2672 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝐴‘𝑤) = (𝐴‘𝑢) → ({𝑤, (𝐴‘𝑤)} ∈ 𝐸 ↔ {𝑤, (𝐴‘𝑢)} ∈ 𝐸)) |
32 | 31 | anbi2d 736 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐴‘𝑤) = (𝐴‘𝑢) → (({𝑢, (𝐴‘𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴‘𝑤)} ∈ 𝐸) ↔ ({𝑢, (𝐴‘𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴‘𝑢)} ∈ 𝐸))) |
33 | 32 | eqcoms 2618 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐴‘𝑢) = (𝐴‘𝑤) → (({𝑢, (𝐴‘𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴‘𝑤)} ∈ 𝐸) ↔ ({𝑢, (𝐴‘𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴‘𝑢)} ∈ 𝐸))) |
34 | 33 | biimpa 500 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐴‘𝑢) = (𝐴‘𝑤) ∧ ({𝑢, (𝐴‘𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴‘𝑤)} ∈ 𝐸)) → ({𝑢, (𝐴‘𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴‘𝑢)} ∈ 𝐸)) |
35 | | df-ne 2782 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑢 ≠ 𝑤 ↔ ¬ 𝑢 = 𝑤) |
36 | 2, 3 | frgrnbnb 41463 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝐺 ∈ FriendGraph ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷) ∧ 𝑢 ≠ 𝑤) → (({𝑢, (𝐴‘𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴‘𝑢)} ∈ 𝐸) → (𝐴‘𝑢) = 𝑋)) |
37 | 9, 36 | syl3an1 1351 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷) ∧ 𝑢 ≠ 𝑤) → (({𝑢, (𝐴‘𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴‘𝑢)} ∈ 𝐸) → (𝐴‘𝑢) = 𝑋)) |
38 | 37 | 3expa 1257 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) ∧ 𝑢 ≠ 𝑤) → (({𝑢, (𝐴‘𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴‘𝑢)} ∈ 𝐸) → (𝐴‘𝑢) = 𝑋)) |
39 | | df-nel 2783 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑋 ∉ 𝑁 ↔ ¬ 𝑋 ∈ 𝑁) |
40 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝐴‘𝑢) = 𝑋 → ((𝐴‘𝑢) ∈ 𝑁 ↔ 𝑋 ∈ 𝑁)) |
41 | 40 | biimpa 500 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (((𝐴‘𝑢) = 𝑋 ∧ (𝐴‘𝑢) ∈ 𝑁) → 𝑋 ∈ 𝑁) |
42 | 41 | pm2.24d 146 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (((𝐴‘𝑢) = 𝑋 ∧ (𝐴‘𝑢) ∈ 𝑁) → (¬ 𝑋 ∈ 𝑁 → 𝑢 = 𝑤)) |
43 | 42 | expcom 450 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝐴‘𝑢) ∈ 𝑁 → ((𝐴‘𝑢) = 𝑋 → (¬ 𝑋 ∈ 𝑁 → 𝑢 = 𝑤))) |
44 | 43 | com13 86 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (¬
𝑋 ∈ 𝑁 → ((𝐴‘𝑢) = 𝑋 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤))) |
45 | 39, 44 | sylbi 206 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑋 ∉ 𝑁 → ((𝐴‘𝑢) = 𝑋 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤))) |
46 | 45 | com12 32 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝐴‘𝑢) = 𝑋 → (𝑋 ∉ 𝑁 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤))) |
47 | 38, 46 | syl6 34 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) ∧ 𝑢 ≠ 𝑤) → (({𝑢, (𝐴‘𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴‘𝑢)} ∈ 𝐸) → (𝑋 ∉ 𝑁 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤)))) |
48 | 47 | expcom 450 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑢 ≠ 𝑤 → ((𝜑 ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → (({𝑢, (𝐴‘𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴‘𝑢)} ∈ 𝐸) → (𝑋 ∉ 𝑁 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤))))) |
49 | 48 | com23 84 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑢 ≠ 𝑤 → (({𝑢, (𝐴‘𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴‘𝑢)} ∈ 𝐸) → ((𝜑 ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → (𝑋 ∉ 𝑁 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤))))) |
50 | 35, 49 | sylbir 224 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (¬
𝑢 = 𝑤 → (({𝑢, (𝐴‘𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴‘𝑢)} ∈ 𝐸) → ((𝜑 ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → (𝑋 ∉ 𝑁 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤))))) |
51 | 34, 50 | syl5com 31 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐴‘𝑢) = (𝐴‘𝑤) ∧ ({𝑢, (𝐴‘𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴‘𝑤)} ∈ 𝐸)) → (¬ 𝑢 = 𝑤 → ((𝜑 ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → (𝑋 ∉ 𝑁 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤))))) |
52 | 51 | expcom 450 |
. . . . . . . . . . . . . . . . . 18
⊢ (({𝑢, (𝐴‘𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴‘𝑤)} ∈ 𝐸) → ((𝐴‘𝑢) = (𝐴‘𝑤) → (¬ 𝑢 = 𝑤 → ((𝜑 ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → (𝑋 ∉ 𝑁 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤)))))) |
53 | 52 | com24 93 |
. . . . . . . . . . . . . . . . 17
⊢ (({𝑢, (𝐴‘𝑢)} ∈ 𝐸 ∧ {𝑤, (𝐴‘𝑤)} ∈ 𝐸) → ((𝜑 ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → (¬ 𝑢 = 𝑤 → ((𝐴‘𝑢) = (𝐴‘𝑤) → (𝑋 ∉ 𝑁 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤)))))) |
54 | 29, 53 | mpcom 37 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷)) → (¬ 𝑢 = 𝑤 → ((𝐴‘𝑢) = (𝐴‘𝑤) → (𝑋 ∉ 𝑁 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤))))) |
55 | 54 | ex 449 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷) → (¬ 𝑢 = 𝑤 → ((𝐴‘𝑢) = (𝐴‘𝑤) → (𝑋 ∉ 𝑁 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤)))))) |
56 | 55 | com3r 85 |
. . . . . . . . . . . . . 14
⊢ (¬
𝑢 = 𝑤 → (𝜑 → ((𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷) → ((𝐴‘𝑢) = (𝐴‘𝑤) → (𝑋 ∉ 𝑁 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤)))))) |
57 | 56 | com15 99 |
. . . . . . . . . . . . 13
⊢ (𝑋 ∉ 𝑁 → (𝜑 → ((𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷) → ((𝐴‘𝑢) = (𝐴‘𝑤) → (¬ 𝑢 = 𝑤 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤)))))) |
58 | 16, 57 | mpcom 37 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑢 ∈ 𝐷 ∧ 𝑤 ∈ 𝐷) → ((𝐴‘𝑢) = (𝐴‘𝑤) → (¬ 𝑢 = 𝑤 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤))))) |
59 | 58 | expd 451 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑢 ∈ 𝐷 → (𝑤 ∈ 𝐷 → ((𝐴‘𝑢) = (𝐴‘𝑤) → (¬ 𝑢 = 𝑤 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤)))))) |
60 | 59 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐴:𝐷⟶𝑁) → (𝑢 ∈ 𝐷 → (𝑤 ∈ 𝐷 → ((𝐴‘𝑢) = (𝐴‘𝑤) → (¬ 𝑢 = 𝑤 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤)))))) |
61 | 60 | imp41 617 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝐴:𝐷⟶𝑁) ∧ 𝑢 ∈ 𝐷) ∧ 𝑤 ∈ 𝐷) ∧ (𝐴‘𝑢) = (𝐴‘𝑤)) → (¬ 𝑢 = 𝑤 → ((𝐴‘𝑢) ∈ 𝑁 → 𝑢 = 𝑤))) |
62 | 15, 61 | mpid 43 |
. . . . . . . 8
⊢
(((((𝜑 ∧ 𝐴:𝐷⟶𝑁) ∧ 𝑢 ∈ 𝐷) ∧ 𝑤 ∈ 𝐷) ∧ (𝐴‘𝑢) = (𝐴‘𝑤)) → (¬ 𝑢 = 𝑤 → 𝑢 = 𝑤)) |
63 | 62 | pm2.18d 123 |
. . . . . . 7
⊢
(((((𝜑 ∧ 𝐴:𝐷⟶𝑁) ∧ 𝑢 ∈ 𝐷) ∧ 𝑤 ∈ 𝐷) ∧ (𝐴‘𝑢) = (𝐴‘𝑤)) → 𝑢 = 𝑤) |
64 | 63 | ex 449 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝐴:𝐷⟶𝑁) ∧ 𝑢 ∈ 𝐷) ∧ 𝑤 ∈ 𝐷) → ((𝐴‘𝑢) = (𝐴‘𝑤) → 𝑢 = 𝑤)) |
65 | 64 | ralrimiva 2949 |
. . . . 5
⊢ (((𝜑 ∧ 𝐴:𝐷⟶𝑁) ∧ 𝑢 ∈ 𝐷) → ∀𝑤 ∈ 𝐷 ((𝐴‘𝑢) = (𝐴‘𝑤) → 𝑢 = 𝑤)) |
66 | 65 | ralrimiva 2949 |
. . . 4
⊢ ((𝜑 ∧ 𝐴:𝐷⟶𝑁) → ∀𝑢 ∈ 𝐷 ∀𝑤 ∈ 𝐷 ((𝐴‘𝑢) = (𝐴‘𝑤) → 𝑢 = 𝑤)) |
67 | | dff13 6416 |
. . . 4
⊢ (𝐴:𝐷–1-1→ran 𝐴 ↔ (𝐴:𝐷⟶ran 𝐴 ∧ ∀𝑢 ∈ 𝐷 ∀𝑤 ∈ 𝐷 ((𝐴‘𝑢) = (𝐴‘𝑤) → 𝑢 = 𝑤))) |
68 | 13, 66, 67 | sylanbrc 695 |
. . 3
⊢ ((𝜑 ∧ 𝐴:𝐷⟶𝑁) → 𝐴:𝐷–1-1→ran 𝐴) |
69 | 68 | expcom 450 |
. 2
⊢ (𝐴:𝐷⟶𝑁 → (𝜑 → 𝐴:𝐷–1-1→ran 𝐴)) |
70 | 11, 69 | mpcom 37 |
1
⊢ (𝜑 → 𝐴:𝐷–1-1→ran 𝐴) |