Step |
Hyp |
Ref |
Expression |
1 |
|
isubgr3stgr.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
2 |
|
isubgr3stgr.u |
⊢ 𝑈 = ( 𝐺 NeighbVtx 𝑋 ) |
3 |
|
isubgr3stgr.c |
⊢ 𝐶 = ( 𝐺 ClNeighbVtx 𝑋 ) |
4 |
|
isubgr3stgr.n |
⊢ 𝑁 ∈ ℕ0 |
5 |
|
isubgr3stgr.s |
⊢ 𝑆 = ( StarGr ‘ 𝑁 ) |
6 |
|
isubgr3stgr.w |
⊢ 𝑊 = ( Vtx ‘ 𝑆 ) |
7 |
5 6
|
stgrorder |
⊢ ( 𝑁 ∈ ℕ0 → ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ) |
8 |
4 7
|
ax-mp |
⊢ ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) |
9 |
|
oveq1 |
⊢ ( ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) → ( ( ♯ ‘ 𝑊 ) − 1 ) = ( ( 𝑁 + 1 ) − 1 ) ) |
10 |
|
nn0cn |
⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ ) |
11 |
|
pncan1 |
⊢ ( 𝑁 ∈ ℂ → ( ( 𝑁 + 1 ) − 1 ) = 𝑁 ) |
12 |
10 11
|
syl |
⊢ ( 𝑁 ∈ ℕ0 → ( ( 𝑁 + 1 ) − 1 ) = 𝑁 ) |
13 |
4 12
|
mp1i |
⊢ ( ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) → ( ( 𝑁 + 1 ) − 1 ) = 𝑁 ) |
14 |
9 13
|
eqtrd |
⊢ ( ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) → ( ( ♯ ‘ 𝑊 ) − 1 ) = 𝑁 ) |
15 |
14
|
adantr |
⊢ ( ( ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ∧ ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ ( ♯ ‘ 𝑈 ) = 𝑁 ) ) → ( ( ♯ ‘ 𝑊 ) − 1 ) = 𝑁 ) |
16 |
|
peano2nn0 |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑁 + 1 ) ∈ ℕ0 ) |
17 |
4 16
|
ax-mp |
⊢ ( 𝑁 + 1 ) ∈ ℕ0 |
18 |
|
eleq1 |
⊢ ( ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) → ( ( ♯ ‘ 𝑊 ) ∈ ℕ0 ↔ ( 𝑁 + 1 ) ∈ ℕ0 ) ) |
19 |
17 18
|
mpbiri |
⊢ ( ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) → ( ♯ ‘ 𝑊 ) ∈ ℕ0 ) |
20 |
19
|
adantr |
⊢ ( ( ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ∧ ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ ( ♯ ‘ 𝑈 ) = 𝑁 ) ) → ( ♯ ‘ 𝑊 ) ∈ ℕ0 ) |
21 |
6
|
fvexi |
⊢ 𝑊 ∈ V |
22 |
|
hashclb |
⊢ ( 𝑊 ∈ V → ( 𝑊 ∈ Fin ↔ ( ♯ ‘ 𝑊 ) ∈ ℕ0 ) ) |
23 |
21 22
|
mp1i |
⊢ ( ( ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ∧ ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ ( ♯ ‘ 𝑈 ) = 𝑁 ) ) → ( 𝑊 ∈ Fin ↔ ( ♯ ‘ 𝑊 ) ∈ ℕ0 ) ) |
24 |
20 23
|
mpbird |
⊢ ( ( ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ∧ ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ ( ♯ ‘ 𝑈 ) = 𝑁 ) ) → 𝑊 ∈ Fin ) |
25 |
5 6
|
stgrvtx0 |
⊢ ( 𝑁 ∈ ℕ0 → 0 ∈ 𝑊 ) |
26 |
4 25
|
ax-mp |
⊢ 0 ∈ 𝑊 |
27 |
|
hashdifsn |
⊢ ( ( 𝑊 ∈ Fin ∧ 0 ∈ 𝑊 ) → ( ♯ ‘ ( 𝑊 ∖ { 0 } ) ) = ( ( ♯ ‘ 𝑊 ) − 1 ) ) |
28 |
24 26 27
|
sylancl |
⊢ ( ( ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ∧ ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ ( ♯ ‘ 𝑈 ) = 𝑁 ) ) → ( ♯ ‘ ( 𝑊 ∖ { 0 } ) ) = ( ( ♯ ‘ 𝑊 ) − 1 ) ) |
29 |
|
simpr3 |
⊢ ( ( ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ∧ ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ ( ♯ ‘ 𝑈 ) = 𝑁 ) ) → ( ♯ ‘ 𝑈 ) = 𝑁 ) |
30 |
15 28 29
|
3eqtr4rd |
⊢ ( ( ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ∧ ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ ( ♯ ‘ 𝑈 ) = 𝑁 ) ) → ( ♯ ‘ 𝑈 ) = ( ♯ ‘ ( 𝑊 ∖ { 0 } ) ) ) |
31 |
|
eleq1 |
⊢ ( ( ♯ ‘ 𝑈 ) = 𝑁 → ( ( ♯ ‘ 𝑈 ) ∈ ℕ0 ↔ 𝑁 ∈ ℕ0 ) ) |
32 |
4 31
|
mpbiri |
⊢ ( ( ♯ ‘ 𝑈 ) = 𝑁 → ( ♯ ‘ 𝑈 ) ∈ ℕ0 ) |
33 |
32
|
3ad2ant3 |
⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ ( ♯ ‘ 𝑈 ) = 𝑁 ) → ( ♯ ‘ 𝑈 ) ∈ ℕ0 ) |
34 |
2
|
ovexi |
⊢ 𝑈 ∈ V |
35 |
|
hashclb |
⊢ ( 𝑈 ∈ V → ( 𝑈 ∈ Fin ↔ ( ♯ ‘ 𝑈 ) ∈ ℕ0 ) ) |
36 |
34 35
|
mp1i |
⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ ( ♯ ‘ 𝑈 ) = 𝑁 ) → ( 𝑈 ∈ Fin ↔ ( ♯ ‘ 𝑈 ) ∈ ℕ0 ) ) |
37 |
33 36
|
mpbird |
⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ ( ♯ ‘ 𝑈 ) = 𝑁 ) → 𝑈 ∈ Fin ) |
38 |
|
diffi |
⊢ ( 𝑊 ∈ Fin → ( 𝑊 ∖ { 0 } ) ∈ Fin ) |
39 |
24 38
|
syl |
⊢ ( ( ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ∧ ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ ( ♯ ‘ 𝑈 ) = 𝑁 ) ) → ( 𝑊 ∖ { 0 } ) ∈ Fin ) |
40 |
|
hasheqf1o |
⊢ ( ( 𝑈 ∈ Fin ∧ ( 𝑊 ∖ { 0 } ) ∈ Fin ) → ( ( ♯ ‘ 𝑈 ) = ( ♯ ‘ ( 𝑊 ∖ { 0 } ) ) ↔ ∃ 𝑓 𝑓 : 𝑈 –1-1-onto→ ( 𝑊 ∖ { 0 } ) ) ) |
41 |
37 39 40
|
syl2an2 |
⊢ ( ( ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ∧ ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ ( ♯ ‘ 𝑈 ) = 𝑁 ) ) → ( ( ♯ ‘ 𝑈 ) = ( ♯ ‘ ( 𝑊 ∖ { 0 } ) ) ↔ ∃ 𝑓 𝑓 : 𝑈 –1-1-onto→ ( 𝑊 ∖ { 0 } ) ) ) |
42 |
30 41
|
mpbid |
⊢ ( ( ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ∧ ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ ( ♯ ‘ 𝑈 ) = 𝑁 ) ) → ∃ 𝑓 𝑓 : 𝑈 –1-1-onto→ ( 𝑊 ∖ { 0 } ) ) |
43 |
8 42
|
mpan |
⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ ( ♯ ‘ 𝑈 ) = 𝑁 ) → ∃ 𝑓 𝑓 : 𝑈 –1-1-onto→ ( 𝑊 ∖ { 0 } ) ) |