Step | Hyp | Ref
| Expression |
1 | | wlkiswwlkbij.t |
. . . 4
⊢ 𝑇 = {𝑝 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st
‘𝑝)) = 𝑁 ∧ ((2nd
‘𝑝)‘0) = 𝑃)} |
2 | | wlkiswwlkbij.w |
. . . 4
⊢ 𝑊 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃} |
3 | | wlkiswwlkbij.f |
. . . 4
⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (2nd ‘𝑡)) |
4 | 1, 2, 3 | wlkiswwlkfun 26245 |
. . 3
⊢ ((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇⟶𝑊) |
5 | 4 | 3adant1 1072 |
. 2
⊢ ((𝑉 USGrph 𝐸 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇⟶𝑊) |
6 | | fveq1 6102 |
. . . . . . 7
⊢ (𝑤 = 𝑝 → (𝑤‘0) = (𝑝‘0)) |
7 | 6 | eqeq1d 2612 |
. . . . . 6
⊢ (𝑤 = 𝑝 → ((𝑤‘0) = 𝑃 ↔ (𝑝‘0) = 𝑃)) |
8 | 7, 2 | elrab2 3333 |
. . . . 5
⊢ (𝑝 ∈ 𝑊 ↔ (𝑝 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑝‘0) = 𝑃)) |
9 | | wlklniswwlkn 26229 |
. . . . . . . . . . 11
⊢ (𝑉 USGrph 𝐸 → (∃𝑓(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 𝑁) ↔ 𝑝 ∈ ((𝑉 WWalksN 𝐸)‘𝑁))) |
10 | | df-br 4584 |
. . . . . . . . . . . . 13
⊢ (𝑓(𝑉 Walks 𝐸)𝑝 ↔ 〈𝑓, 𝑝〉 ∈ (𝑉 Walks 𝐸)) |
11 | | vex 3176 |
. . . . . . . . . . . . . . . . 17
⊢ 𝑓 ∈ V |
12 | | vex 3176 |
. . . . . . . . . . . . . . . . 17
⊢ 𝑝 ∈ V |
13 | 11, 12 | op1st 7067 |
. . . . . . . . . . . . . . . 16
⊢
(1st ‘〈𝑓, 𝑝〉) = 𝑓 |
14 | 13 | eqcomi 2619 |
. . . . . . . . . . . . . . 15
⊢ 𝑓 = (1st
‘〈𝑓, 𝑝〉) |
15 | 14 | fveq2i 6106 |
. . . . . . . . . . . . . 14
⊢
(#‘𝑓) =
(#‘(1st ‘〈𝑓, 𝑝〉)) |
16 | 15 | eqeq1i 2615 |
. . . . . . . . . . . . 13
⊢
((#‘𝑓) = 𝑁 ↔ (#‘(1st
‘〈𝑓, 𝑝〉)) = 𝑁) |
17 | 11, 12 | op2nd 7068 |
. . . . . . . . . . . . . . . . 17
⊢
(2nd ‘〈𝑓, 𝑝〉) = 𝑝 |
18 | 17 | eqcomi 2619 |
. . . . . . . . . . . . . . . 16
⊢ 𝑝 = (2nd
‘〈𝑓, 𝑝〉) |
19 | 18 | fveq1i 6104 |
. . . . . . . . . . . . . . 15
⊢ (𝑝‘0) = ((2nd
‘〈𝑓, 𝑝〉)‘0) |
20 | 19 | eqeq1i 2615 |
. . . . . . . . . . . . . 14
⊢ ((𝑝‘0) = 𝑃 ↔ ((2nd ‘〈𝑓, 𝑝〉)‘0) = 𝑃) |
21 | | opex 4859 |
. . . . . . . . . . . . . . . 16
⊢
〈𝑓, 𝑝〉 ∈ V |
22 | 21 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢
((〈𝑓, 𝑝〉 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st
‘〈𝑓, 𝑝〉)) = 𝑁) → 〈𝑓, 𝑝〉 ∈ V) |
23 | | simpll 786 |
. . . . . . . . . . . . . . . . . 18
⊢
(((〈𝑓, 𝑝〉 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st
‘〈𝑓, 𝑝〉)) = 𝑁) ∧ ((2nd ‘〈𝑓, 𝑝〉)‘0) = 𝑃) → 〈𝑓, 𝑝〉 ∈ (𝑉 Walks 𝐸)) |
24 | | simpr 476 |
. . . . . . . . . . . . . . . . . . 19
⊢
((〈𝑓, 𝑝〉 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st
‘〈𝑓, 𝑝〉)) = 𝑁) → (#‘(1st
‘〈𝑓, 𝑝〉)) = 𝑁) |
25 | 24 | anim1i 590 |
. . . . . . . . . . . . . . . . . 18
⊢
(((〈𝑓, 𝑝〉 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st
‘〈𝑓, 𝑝〉)) = 𝑁) ∧ ((2nd ‘〈𝑓, 𝑝〉)‘0) = 𝑃) → ((#‘(1st
‘〈𝑓, 𝑝〉)) = 𝑁 ∧ ((2nd ‘〈𝑓, 𝑝〉)‘0) = 𝑃)) |
26 | 18 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢
(((〈𝑓, 𝑝〉 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st
‘〈𝑓, 𝑝〉)) = 𝑁) ∧ ((2nd ‘〈𝑓, 𝑝〉)‘0) = 𝑃) → 𝑝 = (2nd ‘〈𝑓, 𝑝〉)) |
27 | 23, 25, 26 | jca31 555 |
. . . . . . . . . . . . . . . . 17
⊢
(((〈𝑓, 𝑝〉 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st
‘〈𝑓, 𝑝〉)) = 𝑁) ∧ ((2nd ‘〈𝑓, 𝑝〉)‘0) = 𝑃) → ((〈𝑓, 𝑝〉 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st
‘〈𝑓, 𝑝〉)) = 𝑁 ∧ ((2nd ‘〈𝑓, 𝑝〉)‘0) = 𝑃)) ∧ 𝑝 = (2nd ‘〈𝑓, 𝑝〉))) |
28 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑢 = 〈𝑓, 𝑝〉 → (𝑢 ∈ (𝑉 Walks 𝐸) ↔ 〈𝑓, 𝑝〉 ∈ (𝑉 Walks 𝐸))) |
29 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑢 = 〈𝑓, 𝑝〉 → (1st ‘𝑢) = (1st
‘〈𝑓, 𝑝〉)) |
30 | 29 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑢 = 〈𝑓, 𝑝〉 → (#‘(1st
‘𝑢)) =
(#‘(1st ‘〈𝑓, 𝑝〉))) |
31 | 30 | eqeq1d 2612 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑢 = 〈𝑓, 𝑝〉 → ((#‘(1st
‘𝑢)) = 𝑁 ↔ (#‘(1st
‘〈𝑓, 𝑝〉)) = 𝑁)) |
32 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑢 = 〈𝑓, 𝑝〉 → (2nd ‘𝑢) = (2nd
‘〈𝑓, 𝑝〉)) |
33 | 32 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑢 = 〈𝑓, 𝑝〉 → ((2nd ‘𝑢)‘0) = ((2nd
‘〈𝑓, 𝑝〉)‘0)) |
34 | 33 | eqeq1d 2612 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑢 = 〈𝑓, 𝑝〉 → (((2nd ‘𝑢)‘0) = 𝑃 ↔ ((2nd ‘〈𝑓, 𝑝〉)‘0) = 𝑃)) |
35 | 31, 34 | anbi12d 743 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑢 = 〈𝑓, 𝑝〉 → (((#‘(1st
‘𝑢)) = 𝑁 ∧ ((2nd
‘𝑢)‘0) = 𝑃) ↔
((#‘(1st ‘〈𝑓, 𝑝〉)) = 𝑁 ∧ ((2nd ‘〈𝑓, 𝑝〉)‘0) = 𝑃))) |
36 | 28, 35 | anbi12d 743 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑢 = 〈𝑓, 𝑝〉 → ((𝑢 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st
‘𝑢)) = 𝑁 ∧ ((2nd
‘𝑢)‘0) = 𝑃)) ↔ (〈𝑓, 𝑝〉 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st
‘〈𝑓, 𝑝〉)) = 𝑁 ∧ ((2nd ‘〈𝑓, 𝑝〉)‘0) = 𝑃)))) |
37 | 32 | eqeq2d 2620 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑢 = 〈𝑓, 𝑝〉 → (𝑝 = (2nd ‘𝑢) ↔ 𝑝 = (2nd ‘〈𝑓, 𝑝〉))) |
38 | 36, 37 | anbi12d 743 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑢 = 〈𝑓, 𝑝〉 → (((𝑢 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st
‘𝑢)) = 𝑁 ∧ ((2nd
‘𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd ‘𝑢)) ↔ ((〈𝑓, 𝑝〉 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st
‘〈𝑓, 𝑝〉)) = 𝑁 ∧ ((2nd ‘〈𝑓, 𝑝〉)‘0) = 𝑃)) ∧ 𝑝 = (2nd ‘〈𝑓, 𝑝〉)))) |
39 | 27, 38 | syl5ibrcom 236 |
. . . . . . . . . . . . . . . 16
⊢
(((〈𝑓, 𝑝〉 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st
‘〈𝑓, 𝑝〉)) = 𝑁) ∧ ((2nd ‘〈𝑓, 𝑝〉)‘0) = 𝑃) → (𝑢 = 〈𝑓, 𝑝〉 → ((𝑢 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st
‘𝑢)) = 𝑁 ∧ ((2nd
‘𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd ‘𝑢)))) |
40 | 39 | impancom 455 |
. . . . . . . . . . . . . . 15
⊢
(((〈𝑓, 𝑝〉 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st
‘〈𝑓, 𝑝〉)) = 𝑁) ∧ 𝑢 = 〈𝑓, 𝑝〉) → (((2nd
‘〈𝑓, 𝑝〉)‘0) = 𝑃 → ((𝑢 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st
‘𝑢)) = 𝑁 ∧ ((2nd
‘𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd ‘𝑢)))) |
41 | 22, 40 | spcimedv 3265 |
. . . . . . . . . . . . . 14
⊢
((〈𝑓, 𝑝〉 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st
‘〈𝑓, 𝑝〉)) = 𝑁) → (((2nd
‘〈𝑓, 𝑝〉)‘0) = 𝑃 → ∃𝑢((𝑢 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st
‘𝑢)) = 𝑁 ∧ ((2nd
‘𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd ‘𝑢)))) |
42 | 20, 41 | syl5bi 231 |
. . . . . . . . . . . . 13
⊢
((〈𝑓, 𝑝〉 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st
‘〈𝑓, 𝑝〉)) = 𝑁) → ((𝑝‘0) = 𝑃 → ∃𝑢((𝑢 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st
‘𝑢)) = 𝑁 ∧ ((2nd
‘𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd ‘𝑢)))) |
43 | 10, 16, 42 | syl2anb 495 |
. . . . . . . . . . . 12
⊢ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 𝑁) → ((𝑝‘0) = 𝑃 → ∃𝑢((𝑢 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st
‘𝑢)) = 𝑁 ∧ ((2nd
‘𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd ‘𝑢)))) |
44 | 43 | exlimiv 1845 |
. . . . . . . . . . 11
⊢
(∃𝑓(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 𝑁) → ((𝑝‘0) = 𝑃 → ∃𝑢((𝑢 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st
‘𝑢)) = 𝑁 ∧ ((2nd
‘𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd ‘𝑢)))) |
45 | 9, 44 | syl6bir 243 |
. . . . . . . . . 10
⊢ (𝑉 USGrph 𝐸 → (𝑝 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ((𝑝‘0) = 𝑃 → ∃𝑢((𝑢 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st
‘𝑢)) = 𝑁 ∧ ((2nd
‘𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd ‘𝑢))))) |
46 | 45 | imp32 448 |
. . . . . . . . 9
⊢ ((𝑉 USGrph 𝐸 ∧ (𝑝 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑝‘0) = 𝑃)) → ∃𝑢((𝑢 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st
‘𝑢)) = 𝑁 ∧ ((2nd
‘𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd ‘𝑢))) |
47 | | fveq2 6103 |
. . . . . . . . . . . . . . 15
⊢ (𝑝 = 𝑢 → (1st ‘𝑝) = (1st ‘𝑢)) |
48 | 47 | fveq2d 6107 |
. . . . . . . . . . . . . 14
⊢ (𝑝 = 𝑢 → (#‘(1st ‘𝑝)) = (#‘(1st
‘𝑢))) |
49 | 48 | eqeq1d 2612 |
. . . . . . . . . . . . 13
⊢ (𝑝 = 𝑢 → ((#‘(1st
‘𝑝)) = 𝑁 ↔ (#‘(1st
‘𝑢)) = 𝑁)) |
50 | | fveq2 6103 |
. . . . . . . . . . . . . . 15
⊢ (𝑝 = 𝑢 → (2nd ‘𝑝) = (2nd ‘𝑢)) |
51 | 50 | fveq1d 6105 |
. . . . . . . . . . . . . 14
⊢ (𝑝 = 𝑢 → ((2nd ‘𝑝)‘0) = ((2nd
‘𝑢)‘0)) |
52 | 51 | eqeq1d 2612 |
. . . . . . . . . . . . 13
⊢ (𝑝 = 𝑢 → (((2nd ‘𝑝)‘0) = 𝑃 ↔ ((2nd ‘𝑢)‘0) = 𝑃)) |
53 | 49, 52 | anbi12d 743 |
. . . . . . . . . . . 12
⊢ (𝑝 = 𝑢 → (((#‘(1st
‘𝑝)) = 𝑁 ∧ ((2nd
‘𝑝)‘0) = 𝑃) ↔
((#‘(1st ‘𝑢)) = 𝑁 ∧ ((2nd ‘𝑢)‘0) = 𝑃))) |
54 | 53 | elrab 3331 |
. . . . . . . . . . 11
⊢ (𝑢 ∈ {𝑝 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st
‘𝑝)) = 𝑁 ∧ ((2nd
‘𝑝)‘0) = 𝑃)} ↔ (𝑢 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st
‘𝑢)) = 𝑁 ∧ ((2nd
‘𝑢)‘0) = 𝑃))) |
55 | 54 | anbi1i 727 |
. . . . . . . . . 10
⊢ ((𝑢 ∈ {𝑝 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st
‘𝑝)) = 𝑁 ∧ ((2nd
‘𝑝)‘0) = 𝑃)} ∧ 𝑝 = (2nd ‘𝑢)) ↔ ((𝑢 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st
‘𝑢)) = 𝑁 ∧ ((2nd
‘𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd ‘𝑢))) |
56 | 55 | exbii 1764 |
. . . . . . . . 9
⊢
(∃𝑢(𝑢 ∈ {𝑝 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st
‘𝑝)) = 𝑁 ∧ ((2nd
‘𝑝)‘0) = 𝑃)} ∧ 𝑝 = (2nd ‘𝑢)) ↔ ∃𝑢((𝑢 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st
‘𝑢)) = 𝑁 ∧ ((2nd
‘𝑢)‘0) = 𝑃)) ∧ 𝑝 = (2nd ‘𝑢))) |
57 | 46, 56 | sylibr 223 |
. . . . . . . 8
⊢ ((𝑉 USGrph 𝐸 ∧ (𝑝 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑝‘0) = 𝑃)) → ∃𝑢(𝑢 ∈ {𝑝 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st
‘𝑝)) = 𝑁 ∧ ((2nd
‘𝑝)‘0) = 𝑃)} ∧ 𝑝 = (2nd ‘𝑢))) |
58 | | df-rex 2902 |
. . . . . . . 8
⊢
(∃𝑢 ∈
{𝑝 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st
‘𝑝)) = 𝑁 ∧ ((2nd
‘𝑝)‘0) = 𝑃)}𝑝 = (2nd ‘𝑢) ↔ ∃𝑢(𝑢 ∈ {𝑝 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st
‘𝑝)) = 𝑁 ∧ ((2nd
‘𝑝)‘0) = 𝑃)} ∧ 𝑝 = (2nd ‘𝑢))) |
59 | 57, 58 | sylibr 223 |
. . . . . . 7
⊢ ((𝑉 USGrph 𝐸 ∧ (𝑝 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑝‘0) = 𝑃)) → ∃𝑢 ∈ {𝑝 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st
‘𝑝)) = 𝑁 ∧ ((2nd
‘𝑝)‘0) = 𝑃)}𝑝 = (2nd ‘𝑢)) |
60 | 1 | rexeqi 3120 |
. . . . . . 7
⊢
(∃𝑢 ∈
𝑇 𝑝 = (2nd ‘𝑢) ↔ ∃𝑢 ∈ {𝑝 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st
‘𝑝)) = 𝑁 ∧ ((2nd
‘𝑝)‘0) = 𝑃)}𝑝 = (2nd ‘𝑢)) |
61 | 59, 60 | sylibr 223 |
. . . . . 6
⊢ ((𝑉 USGrph 𝐸 ∧ (𝑝 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑝‘0) = 𝑃)) → ∃𝑢 ∈ 𝑇 𝑝 = (2nd ‘𝑢)) |
62 | | fveq2 6103 |
. . . . . . . . 9
⊢ (𝑡 = 𝑢 → (2nd ‘𝑡) = (2nd ‘𝑢)) |
63 | | fvex 6113 |
. . . . . . . . 9
⊢
(2nd ‘𝑢) ∈ V |
64 | 62, 3, 63 | fvmpt 6191 |
. . . . . . . 8
⊢ (𝑢 ∈ 𝑇 → (𝐹‘𝑢) = (2nd ‘𝑢)) |
65 | 64 | eqeq2d 2620 |
. . . . . . 7
⊢ (𝑢 ∈ 𝑇 → (𝑝 = (𝐹‘𝑢) ↔ 𝑝 = (2nd ‘𝑢))) |
66 | 65 | rexbiia 3022 |
. . . . . 6
⊢
(∃𝑢 ∈
𝑇 𝑝 = (𝐹‘𝑢) ↔ ∃𝑢 ∈ 𝑇 𝑝 = (2nd ‘𝑢)) |
67 | 61, 66 | sylibr 223 |
. . . . 5
⊢ ((𝑉 USGrph 𝐸 ∧ (𝑝 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑝‘0) = 𝑃)) → ∃𝑢 ∈ 𝑇 𝑝 = (𝐹‘𝑢)) |
68 | 8, 67 | sylan2b 491 |
. . . 4
⊢ ((𝑉 USGrph 𝐸 ∧ 𝑝 ∈ 𝑊) → ∃𝑢 ∈ 𝑇 𝑝 = (𝐹‘𝑢)) |
69 | 68 | ralrimiva 2949 |
. . 3
⊢ (𝑉 USGrph 𝐸 → ∀𝑝 ∈ 𝑊 ∃𝑢 ∈ 𝑇 𝑝 = (𝐹‘𝑢)) |
70 | 69 | 3ad2ant1 1075 |
. 2
⊢ ((𝑉 USGrph 𝐸 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) →
∀𝑝 ∈ 𝑊 ∃𝑢 ∈ 𝑇 𝑝 = (𝐹‘𝑢)) |
71 | | dffo3 6282 |
. 2
⊢ (𝐹:𝑇–onto→𝑊 ↔ (𝐹:𝑇⟶𝑊 ∧ ∀𝑝 ∈ 𝑊 ∃𝑢 ∈ 𝑇 𝑝 = (𝐹‘𝑢))) |
72 | 5, 70, 71 | sylanbrc 695 |
1
⊢ ((𝑉 USGrph 𝐸 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇–onto→𝑊) |