Step | Hyp | Ref
| Expression |
1 | | simpl 473 |
. . . . . . . 8
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝑉) |
2 | 1 | anim2i 593 |
. . . . . . 7
⊢ ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐺 ∈ UHGraph ∧ 𝐴 ∈ 𝑉)) |
3 | 2 | 3adant3 1081 |
. . . . . 6
⊢ ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒) → (𝐺 ∈ UHGraph ∧ 𝐴 ∈ 𝑉)) |
4 | 3 | adantl 482 |
. . . . 5
⊢ ((𝐴 = 𝐵 ∧ (𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒)) → (𝐺 ∈ UHGraph ∧ 𝐴 ∈ 𝑉)) |
5 | | 1pthon2v.v |
. . . . . . 7
⊢ 𝑉 = (Vtx‘𝐺) |
6 | 5 | 0pthonv 26990 |
. . . . . 6
⊢ (𝐴 ∈ 𝑉 → ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐴)𝑝) |
7 | 6 | adantl 482 |
. . . . 5
⊢ ((𝐺 ∈ UHGraph ∧ 𝐴 ∈ 𝑉) → ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐴)𝑝) |
8 | 4, 7 | syl 17 |
. . . 4
⊢ ((𝐴 = 𝐵 ∧ (𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒)) → ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐴)𝑝) |
9 | | oveq2 6658 |
. . . . . . . 8
⊢ (𝐵 = 𝐴 → (𝐴(PathsOn‘𝐺)𝐵) = (𝐴(PathsOn‘𝐺)𝐴)) |
10 | 9 | eqcoms 2630 |
. . . . . . 7
⊢ (𝐴 = 𝐵 → (𝐴(PathsOn‘𝐺)𝐵) = (𝐴(PathsOn‘𝐺)𝐴)) |
11 | 10 | breqd 4664 |
. . . . . 6
⊢ (𝐴 = 𝐵 → (𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝 ↔ 𝑓(𝐴(PathsOn‘𝐺)𝐴)𝑝)) |
12 | 11 | 2exbidv 1852 |
. . . . 5
⊢ (𝐴 = 𝐵 → (∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝 ↔ ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐴)𝑝)) |
13 | 12 | adantr 481 |
. . . 4
⊢ ((𝐴 = 𝐵 ∧ (𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒)) → (∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝 ↔ ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐴)𝑝)) |
14 | 8, 13 | mpbird 247 |
. . 3
⊢ ((𝐴 = 𝐵 ∧ (𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒)) → ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝) |
15 | 14 | ex 450 |
. 2
⊢ (𝐴 = 𝐵 → ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒) → ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝)) |
16 | | 1pthon2v.e |
. . . . . . . . . . 11
⊢ 𝐸 = (Edg‘𝐺) |
17 | 16 | eleq2i 2693 |
. . . . . . . . . 10
⊢ (𝑒 ∈ 𝐸 ↔ 𝑒 ∈ (Edg‘𝐺)) |
18 | | eqid 2622 |
. . . . . . . . . . 11
⊢
(iEdg‘𝐺) =
(iEdg‘𝐺) |
19 | 18 | uhgredgiedgb 26021 |
. . . . . . . . . 10
⊢ (𝐺 ∈ UHGraph → (𝑒 ∈ (Edg‘𝐺) ↔ ∃𝑖 ∈ dom (iEdg‘𝐺)𝑒 = ((iEdg‘𝐺)‘𝑖))) |
20 | 17, 19 | syl5bb 272 |
. . . . . . . . 9
⊢ (𝐺 ∈ UHGraph → (𝑒 ∈ 𝐸 ↔ ∃𝑖 ∈ dom (iEdg‘𝐺)𝑒 = ((iEdg‘𝐺)‘𝑖))) |
21 | 20 | 3ad2ant1 1082 |
. . . . . . . 8
⊢ ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → (𝑒 ∈ 𝐸 ↔ ∃𝑖 ∈ dom (iEdg‘𝐺)𝑒 = ((iEdg‘𝐺)‘𝑖))) |
22 | | s1cli 13384 |
. . . . . . . . . . . 12
⊢
〈“𝑖”〉 ∈ Word V |
23 | | s2cli 13625 |
. . . . . . . . . . . 12
⊢
〈“𝐴𝐵”〉 ∈ Word
V |
24 | 22, 23 | pm3.2i 471 |
. . . . . . . . . . 11
⊢
(〈“𝑖”〉 ∈ Word V ∧
〈“𝐴𝐵”〉 ∈ Word
V) |
25 | | eqid 2622 |
. . . . . . . . . . . 12
⊢
〈“𝐴𝐵”〉 =
〈“𝐴𝐵”〉 |
26 | | eqid 2622 |
. . . . . . . . . . . 12
⊢
〈“𝑖”〉 = 〈“𝑖”〉 |
27 | | simpl2l 1114 |
. . . . . . . . . . . 12
⊢ (((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) → 𝐴 ∈ 𝑉) |
28 | | simpl2r 1115 |
. . . . . . . . . . . 12
⊢ (((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) → 𝐵 ∈ 𝑉) |
29 | | eqneqall 2805 |
. . . . . . . . . . . . . . . 16
⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → ((iEdg‘𝐺)‘𝑖) = {𝐴})) |
30 | 29 | com12 32 |
. . . . . . . . . . . . . . 15
⊢ (𝐴 ≠ 𝐵 → (𝐴 = 𝐵 → ((iEdg‘𝐺)‘𝑖) = {𝐴})) |
31 | 30 | 3ad2ant3 1084 |
. . . . . . . . . . . . . 14
⊢ ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → (𝐴 = 𝐵 → ((iEdg‘𝐺)‘𝑖) = {𝐴})) |
32 | 31 | adantr 481 |
. . . . . . . . . . . . 13
⊢ (((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) → (𝐴 = 𝐵 → ((iEdg‘𝐺)‘𝑖) = {𝐴})) |
33 | 32 | imp 445 |
. . . . . . . . . . . 12
⊢ ((((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) ∧ 𝐴 = 𝐵) → ((iEdg‘𝐺)‘𝑖) = {𝐴}) |
34 | | sseq2 3627 |
. . . . . . . . . . . . . . . 16
⊢ (𝑒 = ((iEdg‘𝐺)‘𝑖) → ({𝐴, 𝐵} ⊆ 𝑒 ↔ {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖))) |
35 | 34 | adantl 482 |
. . . . . . . . . . . . . . 15
⊢ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) → ({𝐴, 𝐵} ⊆ 𝑒 ↔ {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖))) |
36 | 35 | biimpa 501 |
. . . . . . . . . . . . . 14
⊢ (((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒) → {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖)) |
37 | 36 | adantl 482 |
. . . . . . . . . . . . 13
⊢ (((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) → {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖)) |
38 | 37 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖)) |
39 | 25, 26, 27, 28, 33, 38, 5, 18 | 1pthond 27004 |
. . . . . . . . . . 11
⊢ (((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) → 〈“𝑖”〉(𝐴(PathsOn‘𝐺)𝐵)〈“𝐴𝐵”〉) |
40 | | breq12 4658 |
. . . . . . . . . . . 12
⊢ ((𝑓 = 〈“𝑖”〉 ∧ 𝑝 = 〈“𝐴𝐵”〉) → (𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝 ↔ 〈“𝑖”〉(𝐴(PathsOn‘𝐺)𝐵)〈“𝐴𝐵”〉)) |
41 | 40 | spc2egv 3295 |
. . . . . . . . . . 11
⊢
((〈“𝑖”〉 ∈ Word V ∧
〈“𝐴𝐵”〉 ∈ Word V)
→ (〈“𝑖”〉(𝐴(PathsOn‘𝐺)𝐵)〈“𝐴𝐵”〉 → ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝)) |
42 | 24, 39, 41 | mpsyl 68 |
. . . . . . . . . 10
⊢ (((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) → ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝) |
43 | 42 | exp44 641 |
. . . . . . . . 9
⊢ ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → (𝑖 ∈ dom (iEdg‘𝐺) → (𝑒 = ((iEdg‘𝐺)‘𝑖) → ({𝐴, 𝐵} ⊆ 𝑒 → ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝)))) |
44 | 43 | rexlimdv 3030 |
. . . . . . . 8
⊢ ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → (∃𝑖 ∈ dom (iEdg‘𝐺)𝑒 = ((iEdg‘𝐺)‘𝑖) → ({𝐴, 𝐵} ⊆ 𝑒 → ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝))) |
45 | 21, 44 | sylbid 230 |
. . . . . . 7
⊢ ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → (𝑒 ∈ 𝐸 → ({𝐴, 𝐵} ⊆ 𝑒 → ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝))) |
46 | 45 | rexlimdv 3030 |
. . . . . 6
⊢ ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → (∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒 → ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝)) |
47 | 46 | 3exp 1264 |
. . . . 5
⊢ (𝐺 ∈ UHGraph → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 ≠ 𝐵 → (∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒 → ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝)))) |
48 | 47 | com34 91 |
. . . 4
⊢ (𝐺 ∈ UHGraph → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒 → (𝐴 ≠ 𝐵 → ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝)))) |
49 | 48 | 3imp 1256 |
. . 3
⊢ ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒) → (𝐴 ≠ 𝐵 → ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝)) |
50 | 49 | com12 32 |
. 2
⊢ (𝐴 ≠ 𝐵 → ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒) → ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝)) |
51 | 15, 50 | pm2.61ine 2877 |
1
⊢ ((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒) → ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝) |