Step |
Hyp |
Ref |
Expression |
1 |
|
ovex |
⊢ ( 𝑁 WWalksN 𝐺 ) ∈ V |
2 |
1
|
mptrabex |
⊢ ( 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ↦ ( 𝑤 prefix 𝑁 ) ) ∈ V |
3 |
2
|
resex |
⊢ ( ( 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ↦ ( 𝑤 prefix 𝑁 ) ) ↾ { 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ∣ ( 𝑤 ‘ 0 ) = 𝑋 } ) ∈ V |
4 |
|
eqid |
⊢ ( 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ↦ ( 𝑤 prefix 𝑁 ) ) = ( 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ↦ ( 𝑤 prefix 𝑁 ) ) |
5 |
|
eqid |
⊢ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } = { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } |
6 |
5 4
|
clwwlkf1o |
⊢ ( 𝑁 ∈ ℕ → ( 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ↦ ( 𝑤 prefix 𝑁 ) ) : { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } –1-1-onto→ ( 𝑁 ClWWalksN 𝐺 ) ) |
7 |
|
fveq1 |
⊢ ( 𝑦 = ( 𝑤 prefix 𝑁 ) → ( 𝑦 ‘ 0 ) = ( ( 𝑤 prefix 𝑁 ) ‘ 0 ) ) |
8 |
7
|
eqeq1d |
⊢ ( 𝑦 = ( 𝑤 prefix 𝑁 ) → ( ( 𝑦 ‘ 0 ) = 𝑋 ↔ ( ( 𝑤 prefix 𝑁 ) ‘ 0 ) = 𝑋 ) ) |
9 |
8
|
3ad2ant3 |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ∧ 𝑦 = ( 𝑤 prefix 𝑁 ) ) → ( ( 𝑦 ‘ 0 ) = 𝑋 ↔ ( ( 𝑤 prefix 𝑁 ) ‘ 0 ) = 𝑋 ) ) |
10 |
|
fveq2 |
⊢ ( 𝑥 = 𝑤 → ( lastS ‘ 𝑥 ) = ( lastS ‘ 𝑤 ) ) |
11 |
|
fveq1 |
⊢ ( 𝑥 = 𝑤 → ( 𝑥 ‘ 0 ) = ( 𝑤 ‘ 0 ) ) |
12 |
10 11
|
eqeq12d |
⊢ ( 𝑥 = 𝑤 → ( ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) ↔ ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) ) ) |
13 |
12
|
elrab |
⊢ ( 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ↔ ( 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∧ ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) ) ) |
14 |
|
eqid |
⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 ) |
15 |
|
eqid |
⊢ ( Edg ‘ 𝐺 ) = ( Edg ‘ 𝐺 ) |
16 |
14 15
|
wwlknp |
⊢ ( 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) → ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑁 ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) ) |
17 |
|
simpll |
⊢ ( ( ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ) ∧ 𝑁 ∈ ℕ ) → 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ) |
18 |
|
nnz |
⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℤ ) |
19 |
|
uzid |
⊢ ( 𝑁 ∈ ℤ → 𝑁 ∈ ( ℤ≥ ‘ 𝑁 ) ) |
20 |
|
peano2uz |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑁 ) → ( 𝑁 + 1 ) ∈ ( ℤ≥ ‘ 𝑁 ) ) |
21 |
18 19 20
|
3syl |
⊢ ( 𝑁 ∈ ℕ → ( 𝑁 + 1 ) ∈ ( ℤ≥ ‘ 𝑁 ) ) |
22 |
|
elfz1end |
⊢ ( 𝑁 ∈ ℕ ↔ 𝑁 ∈ ( 1 ... 𝑁 ) ) |
23 |
22
|
biimpi |
⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ( 1 ... 𝑁 ) ) |
24 |
|
fzss2 |
⊢ ( ( 𝑁 + 1 ) ∈ ( ℤ≥ ‘ 𝑁 ) → ( 1 ... 𝑁 ) ⊆ ( 1 ... ( 𝑁 + 1 ) ) ) |
25 |
24
|
sselda |
⊢ ( ( ( 𝑁 + 1 ) ∈ ( ℤ≥ ‘ 𝑁 ) ∧ 𝑁 ∈ ( 1 ... 𝑁 ) ) → 𝑁 ∈ ( 1 ... ( 𝑁 + 1 ) ) ) |
26 |
21 23 25
|
syl2anc |
⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ( 1 ... ( 𝑁 + 1 ) ) ) |
27 |
26
|
adantl |
⊢ ( ( ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ) ∧ 𝑁 ∈ ℕ ) → 𝑁 ∈ ( 1 ... ( 𝑁 + 1 ) ) ) |
28 |
|
oveq2 |
⊢ ( ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) → ( 1 ... ( ♯ ‘ 𝑤 ) ) = ( 1 ... ( 𝑁 + 1 ) ) ) |
29 |
28
|
eleq2d |
⊢ ( ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) → ( 𝑁 ∈ ( 1 ... ( ♯ ‘ 𝑤 ) ) ↔ 𝑁 ∈ ( 1 ... ( 𝑁 + 1 ) ) ) ) |
30 |
29
|
adantl |
⊢ ( ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ) → ( 𝑁 ∈ ( 1 ... ( ♯ ‘ 𝑤 ) ) ↔ 𝑁 ∈ ( 1 ... ( 𝑁 + 1 ) ) ) ) |
31 |
30
|
adantr |
⊢ ( ( ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ) ∧ 𝑁 ∈ ℕ ) → ( 𝑁 ∈ ( 1 ... ( ♯ ‘ 𝑤 ) ) ↔ 𝑁 ∈ ( 1 ... ( 𝑁 + 1 ) ) ) ) |
32 |
27 31
|
mpbird |
⊢ ( ( ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ) ∧ 𝑁 ∈ ℕ ) → 𝑁 ∈ ( 1 ... ( ♯ ‘ 𝑤 ) ) ) |
33 |
17 32
|
jca |
⊢ ( ( ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ) ∧ 𝑁 ∈ ℕ ) → ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑁 ∈ ( 1 ... ( ♯ ‘ 𝑤 ) ) ) ) |
34 |
33
|
ex |
⊢ ( ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ) → ( 𝑁 ∈ ℕ → ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑁 ∈ ( 1 ... ( ♯ ‘ 𝑤 ) ) ) ) ) |
35 |
34
|
3adant3 |
⊢ ( ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑁 ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) → ( 𝑁 ∈ ℕ → ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑁 ∈ ( 1 ... ( ♯ ‘ 𝑤 ) ) ) ) ) |
36 |
16 35
|
syl |
⊢ ( 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) → ( 𝑁 ∈ ℕ → ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑁 ∈ ( 1 ... ( ♯ ‘ 𝑤 ) ) ) ) ) |
37 |
36
|
adantr |
⊢ ( ( 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∧ ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) ) → ( 𝑁 ∈ ℕ → ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑁 ∈ ( 1 ... ( ♯ ‘ 𝑤 ) ) ) ) ) |
38 |
13 37
|
sylbi |
⊢ ( 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } → ( 𝑁 ∈ ℕ → ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑁 ∈ ( 1 ... ( ♯ ‘ 𝑤 ) ) ) ) ) |
39 |
38
|
impcom |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ) → ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑁 ∈ ( 1 ... ( ♯ ‘ 𝑤 ) ) ) ) |
40 |
|
pfxfv0 |
⊢ ( ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑁 ∈ ( 1 ... ( ♯ ‘ 𝑤 ) ) ) → ( ( 𝑤 prefix 𝑁 ) ‘ 0 ) = ( 𝑤 ‘ 0 ) ) |
41 |
39 40
|
syl |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ) → ( ( 𝑤 prefix 𝑁 ) ‘ 0 ) = ( 𝑤 ‘ 0 ) ) |
42 |
41
|
eqeq1d |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ) → ( ( ( 𝑤 prefix 𝑁 ) ‘ 0 ) = 𝑋 ↔ ( 𝑤 ‘ 0 ) = 𝑋 ) ) |
43 |
42
|
3adant3 |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ∧ 𝑦 = ( 𝑤 prefix 𝑁 ) ) → ( ( ( 𝑤 prefix 𝑁 ) ‘ 0 ) = 𝑋 ↔ ( 𝑤 ‘ 0 ) = 𝑋 ) ) |
44 |
9 43
|
bitrd |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ∧ 𝑦 = ( 𝑤 prefix 𝑁 ) ) → ( ( 𝑦 ‘ 0 ) = 𝑋 ↔ ( 𝑤 ‘ 0 ) = 𝑋 ) ) |
45 |
4 6 44
|
f1oresrab |
⊢ ( 𝑁 ∈ ℕ → ( ( 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ↦ ( 𝑤 prefix 𝑁 ) ) ↾ { 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ∣ ( 𝑤 ‘ 0 ) = 𝑋 } ) : { 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ∣ ( 𝑤 ‘ 0 ) = 𝑋 } –1-1-onto→ { 𝑦 ∈ ( 𝑁 ClWWalksN 𝐺 ) ∣ ( 𝑦 ‘ 0 ) = 𝑋 } ) |
46 |
45
|
adantl |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ( ( 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ↦ ( 𝑤 prefix 𝑁 ) ) ↾ { 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ∣ ( 𝑤 ‘ 0 ) = 𝑋 } ) : { 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ∣ ( 𝑤 ‘ 0 ) = 𝑋 } –1-1-onto→ { 𝑦 ∈ ( 𝑁 ClWWalksN 𝐺 ) ∣ ( 𝑦 ‘ 0 ) = 𝑋 } ) |
47 |
|
clwwlknon |
⊢ ( 𝑋 ( ClWWalksNOn ‘ 𝐺 ) 𝑁 ) = { 𝑦 ∈ ( 𝑁 ClWWalksN 𝐺 ) ∣ ( 𝑦 ‘ 0 ) = 𝑋 } |
48 |
47
|
a1i |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ( 𝑋 ( ClWWalksNOn ‘ 𝐺 ) 𝑁 ) = { 𝑦 ∈ ( 𝑁 ClWWalksN 𝐺 ) ∣ ( 𝑦 ‘ 0 ) = 𝑋 } ) |
49 |
48
|
f1oeq3d |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ( ( ( 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ↦ ( 𝑤 prefix 𝑁 ) ) ↾ { 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ∣ ( 𝑤 ‘ 0 ) = 𝑋 } ) : { 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ∣ ( 𝑤 ‘ 0 ) = 𝑋 } –1-1-onto→ ( 𝑋 ( ClWWalksNOn ‘ 𝐺 ) 𝑁 ) ↔ ( ( 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ↦ ( 𝑤 prefix 𝑁 ) ) ↾ { 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ∣ ( 𝑤 ‘ 0 ) = 𝑋 } ) : { 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ∣ ( 𝑤 ‘ 0 ) = 𝑋 } –1-1-onto→ { 𝑦 ∈ ( 𝑁 ClWWalksN 𝐺 ) ∣ ( 𝑦 ‘ 0 ) = 𝑋 } ) ) |
50 |
46 49
|
mpbird |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ( ( 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ↦ ( 𝑤 prefix 𝑁 ) ) ↾ { 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ∣ ( 𝑤 ‘ 0 ) = 𝑋 } ) : { 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ∣ ( 𝑤 ‘ 0 ) = 𝑋 } –1-1-onto→ ( 𝑋 ( ClWWalksNOn ‘ 𝐺 ) 𝑁 ) ) |
51 |
|
f1oeq1 |
⊢ ( 𝑓 = ( ( 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ↦ ( 𝑤 prefix 𝑁 ) ) ↾ { 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ∣ ( 𝑤 ‘ 0 ) = 𝑋 } ) → ( 𝑓 : { 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ∣ ( 𝑤 ‘ 0 ) = 𝑋 } –1-1-onto→ ( 𝑋 ( ClWWalksNOn ‘ 𝐺 ) 𝑁 ) ↔ ( ( 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ↦ ( 𝑤 prefix 𝑁 ) ) ↾ { 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ∣ ( 𝑤 ‘ 0 ) = 𝑋 } ) : { 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ∣ ( 𝑤 ‘ 0 ) = 𝑋 } –1-1-onto→ ( 𝑋 ( ClWWalksNOn ‘ 𝐺 ) 𝑁 ) ) ) |
52 |
51
|
spcegv |
⊢ ( ( ( 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ↦ ( 𝑤 prefix 𝑁 ) ) ↾ { 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ∣ ( 𝑤 ‘ 0 ) = 𝑋 } ) ∈ V → ( ( ( 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ↦ ( 𝑤 prefix 𝑁 ) ) ↾ { 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ∣ ( 𝑤 ‘ 0 ) = 𝑋 } ) : { 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ∣ ( 𝑤 ‘ 0 ) = 𝑋 } –1-1-onto→ ( 𝑋 ( ClWWalksNOn ‘ 𝐺 ) 𝑁 ) → ∃ 𝑓 𝑓 : { 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ∣ ( 𝑤 ‘ 0 ) = 𝑋 } –1-1-onto→ ( 𝑋 ( ClWWalksNOn ‘ 𝐺 ) 𝑁 ) ) ) |
53 |
3 50 52
|
mpsyl |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ∃ 𝑓 𝑓 : { 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ∣ ( 𝑤 ‘ 0 ) = 𝑋 } –1-1-onto→ ( 𝑋 ( ClWWalksNOn ‘ 𝐺 ) 𝑁 ) ) |
54 |
|
df-rab |
⊢ { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) ∧ ( 𝑤 ‘ 0 ) = 𝑋 ) } = { 𝑤 ∣ ( 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∧ ( ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) ∧ ( 𝑤 ‘ 0 ) = 𝑋 ) ) } |
55 |
|
anass |
⊢ ( ( ( 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∧ ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) ) ∧ ( 𝑤 ‘ 0 ) = 𝑋 ) ↔ ( 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∧ ( ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) ∧ ( 𝑤 ‘ 0 ) = 𝑋 ) ) ) |
56 |
55
|
bicomi |
⊢ ( ( 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∧ ( ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) ∧ ( 𝑤 ‘ 0 ) = 𝑋 ) ) ↔ ( ( 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∧ ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) ) ∧ ( 𝑤 ‘ 0 ) = 𝑋 ) ) |
57 |
56
|
abbii |
⊢ { 𝑤 ∣ ( 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∧ ( ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) ∧ ( 𝑤 ‘ 0 ) = 𝑋 ) ) } = { 𝑤 ∣ ( ( 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∧ ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) ) ∧ ( 𝑤 ‘ 0 ) = 𝑋 ) } |
58 |
13
|
bicomi |
⊢ ( ( 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∧ ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) ) ↔ 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ) |
59 |
58
|
anbi1i |
⊢ ( ( ( 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∧ ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) ) ∧ ( 𝑤 ‘ 0 ) = 𝑋 ) ↔ ( 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ∧ ( 𝑤 ‘ 0 ) = 𝑋 ) ) |
60 |
59
|
abbii |
⊢ { 𝑤 ∣ ( ( 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∧ ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) ) ∧ ( 𝑤 ‘ 0 ) = 𝑋 ) } = { 𝑤 ∣ ( 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ∧ ( 𝑤 ‘ 0 ) = 𝑋 ) } |
61 |
|
df-rab |
⊢ { 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ∣ ( 𝑤 ‘ 0 ) = 𝑋 } = { 𝑤 ∣ ( 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ∧ ( 𝑤 ‘ 0 ) = 𝑋 ) } |
62 |
60 61
|
eqtr4i |
⊢ { 𝑤 ∣ ( ( 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∧ ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) ) ∧ ( 𝑤 ‘ 0 ) = 𝑋 ) } = { 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ∣ ( 𝑤 ‘ 0 ) = 𝑋 } |
63 |
54 57 62
|
3eqtri |
⊢ { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) ∧ ( 𝑤 ‘ 0 ) = 𝑋 ) } = { 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ∣ ( 𝑤 ‘ 0 ) = 𝑋 } |
64 |
|
f1oeq2 |
⊢ ( { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) ∧ ( 𝑤 ‘ 0 ) = 𝑋 ) } = { 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ∣ ( 𝑤 ‘ 0 ) = 𝑋 } → ( 𝑓 : { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) ∧ ( 𝑤 ‘ 0 ) = 𝑋 ) } –1-1-onto→ ( 𝑋 ( ClWWalksNOn ‘ 𝐺 ) 𝑁 ) ↔ 𝑓 : { 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ∣ ( 𝑤 ‘ 0 ) = 𝑋 } –1-1-onto→ ( 𝑋 ( ClWWalksNOn ‘ 𝐺 ) 𝑁 ) ) ) |
65 |
63 64
|
mp1i |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ( 𝑓 : { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) ∧ ( 𝑤 ‘ 0 ) = 𝑋 ) } –1-1-onto→ ( 𝑋 ( ClWWalksNOn ‘ 𝐺 ) 𝑁 ) ↔ 𝑓 : { 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ∣ ( 𝑤 ‘ 0 ) = 𝑋 } –1-1-onto→ ( 𝑋 ( ClWWalksNOn ‘ 𝐺 ) 𝑁 ) ) ) |
66 |
65
|
exbidv |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ( ∃ 𝑓 𝑓 : { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) ∧ ( 𝑤 ‘ 0 ) = 𝑋 ) } –1-1-onto→ ( 𝑋 ( ClWWalksNOn ‘ 𝐺 ) 𝑁 ) ↔ ∃ 𝑓 𝑓 : { 𝑤 ∈ { 𝑥 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( lastS ‘ 𝑥 ) = ( 𝑥 ‘ 0 ) } ∣ ( 𝑤 ‘ 0 ) = 𝑋 } –1-1-onto→ ( 𝑋 ( ClWWalksNOn ‘ 𝐺 ) 𝑁 ) ) ) |
67 |
53 66
|
mpbird |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ) → ∃ 𝑓 𝑓 : { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( ( lastS ‘ 𝑤 ) = ( 𝑤 ‘ 0 ) ∧ ( 𝑤 ‘ 0 ) = 𝑋 ) } –1-1-onto→ ( 𝑋 ( ClWWalksNOn ‘ 𝐺 ) 𝑁 ) ) |