| 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 |
|
isubgr3stgr.e |
⊢ 𝐸 = ( Edg ‘ 𝐺 ) |
| 8 |
|
preq2 |
⊢ ( 𝑧 = ( 𝐹 ‘ 𝐵 ) → { 0 , 𝑧 } = { 0 , ( 𝐹 ‘ 𝐵 ) } ) |
| 9 |
8
|
eqeq2d |
⊢ ( 𝑧 = ( 𝐹 ‘ 𝐵 ) → ( ( 𝐹 “ { 𝑋 , 𝐵 } ) = { 0 , 𝑧 } ↔ ( 𝐹 “ { 𝑋 , 𝐵 } ) = { 0 , ( 𝐹 ‘ 𝐵 ) } ) ) |
| 10 |
|
f1of |
⊢ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 → 𝐹 : 𝐶 ⟶ 𝑊 ) |
| 11 |
10
|
adantr |
⊢ ( ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) → 𝐹 : 𝐶 ⟶ 𝑊 ) |
| 12 |
11
|
adantr |
⊢ ( ( ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ∧ ( 𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) ) → 𝐹 : 𝐶 ⟶ 𝑊 ) |
| 13 |
|
simpr3 |
⊢ ( ( ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ∧ ( 𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) ) → 𝐵 ∈ 𝐶 ) |
| 14 |
12 13
|
ffvelcdmd |
⊢ ( ( ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ∧ ( 𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑊 ) |
| 15 |
5
|
fveq2i |
⊢ ( Vtx ‘ 𝑆 ) = ( Vtx ‘ ( StarGr ‘ 𝑁 ) ) |
| 16 |
|
stgrvtx |
⊢ ( 𝑁 ∈ ℕ0 → ( Vtx ‘ ( StarGr ‘ 𝑁 ) ) = ( 0 ... 𝑁 ) ) |
| 17 |
4 16
|
ax-mp |
⊢ ( Vtx ‘ ( StarGr ‘ 𝑁 ) ) = ( 0 ... 𝑁 ) |
| 18 |
6 15 17
|
3eqtri |
⊢ 𝑊 = ( 0 ... 𝑁 ) |
| 19 |
18
|
eleq2i |
⊢ ( ( 𝐹 ‘ 𝐵 ) ∈ 𝑊 ↔ ( 𝐹 ‘ 𝐵 ) ∈ ( 0 ... 𝑁 ) ) |
| 20 |
|
fz0sn0fz1 |
⊢ ( 𝑁 ∈ ℕ0 → ( 0 ... 𝑁 ) = ( { 0 } ∪ ( 1 ... 𝑁 ) ) ) |
| 21 |
4 20
|
ax-mp |
⊢ ( 0 ... 𝑁 ) = ( { 0 } ∪ ( 1 ... 𝑁 ) ) |
| 22 |
21
|
eleq2i |
⊢ ( ( 𝐹 ‘ 𝐵 ) ∈ ( 0 ... 𝑁 ) ↔ ( 𝐹 ‘ 𝐵 ) ∈ ( { 0 } ∪ ( 1 ... 𝑁 ) ) ) |
| 23 |
|
elun |
⊢ ( ( 𝐹 ‘ 𝐵 ) ∈ ( { 0 } ∪ ( 1 ... 𝑁 ) ) ↔ ( ( 𝐹 ‘ 𝐵 ) ∈ { 0 } ∨ ( 𝐹 ‘ 𝐵 ) ∈ ( 1 ... 𝑁 ) ) ) |
| 24 |
|
fvex |
⊢ ( 𝐹 ‘ 𝐵 ) ∈ V |
| 25 |
24
|
elsn |
⊢ ( ( 𝐹 ‘ 𝐵 ) ∈ { 0 } ↔ ( 𝐹 ‘ 𝐵 ) = 0 ) |
| 26 |
25
|
orbi1i |
⊢ ( ( ( 𝐹 ‘ 𝐵 ) ∈ { 0 } ∨ ( 𝐹 ‘ 𝐵 ) ∈ ( 1 ... 𝑁 ) ) ↔ ( ( 𝐹 ‘ 𝐵 ) = 0 ∨ ( 𝐹 ‘ 𝐵 ) ∈ ( 1 ... 𝑁 ) ) ) |
| 27 |
23 26
|
bitri |
⊢ ( ( 𝐹 ‘ 𝐵 ) ∈ ( { 0 } ∪ ( 1 ... 𝑁 ) ) ↔ ( ( 𝐹 ‘ 𝐵 ) = 0 ∨ ( 𝐹 ‘ 𝐵 ) ∈ ( 1 ... 𝑁 ) ) ) |
| 28 |
19 22 27
|
3bitri |
⊢ ( ( 𝐹 ‘ 𝐵 ) ∈ 𝑊 ↔ ( ( 𝐹 ‘ 𝐵 ) = 0 ∨ ( 𝐹 ‘ 𝐵 ) ∈ ( 1 ... 𝑁 ) ) ) |
| 29 |
|
eqeq2 |
⊢ ( ( 𝐹 ‘ 𝑋 ) = 0 → ( ( 𝐹 ‘ 𝐵 ) = ( 𝐹 ‘ 𝑋 ) ↔ ( 𝐹 ‘ 𝐵 ) = 0 ) ) |
| 30 |
29
|
adantl |
⊢ ( ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) → ( ( 𝐹 ‘ 𝐵 ) = ( 𝐹 ‘ 𝑋 ) ↔ ( 𝐹 ‘ 𝐵 ) = 0 ) ) |
| 31 |
30
|
adantr |
⊢ ( ( ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ∧ ( 𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) ) → ( ( 𝐹 ‘ 𝐵 ) = ( 𝐹 ‘ 𝑋 ) ↔ ( 𝐹 ‘ 𝐵 ) = 0 ) ) |
| 32 |
|
f1of1 |
⊢ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 → 𝐹 : 𝐶 –1-1→ 𝑊 ) |
| 33 |
|
dff14a |
⊢ ( 𝐹 : 𝐶 –1-1→ 𝑊 ↔ ( 𝐹 : 𝐶 ⟶ 𝑊 ∧ ∀ 𝑎 ∈ 𝐶 ∀ 𝑏 ∈ 𝐶 ( 𝑎 ≠ 𝑏 → ( 𝐹 ‘ 𝑎 ) ≠ ( 𝐹 ‘ 𝑏 ) ) ) ) |
| 34 |
|
simpl |
⊢ ( ( 𝑎 = 𝑋 ∧ 𝑏 = 𝐵 ) → 𝑎 = 𝑋 ) |
| 35 |
|
simpr |
⊢ ( ( 𝑎 = 𝑋 ∧ 𝑏 = 𝐵 ) → 𝑏 = 𝐵 ) |
| 36 |
34 35
|
neeq12d |
⊢ ( ( 𝑎 = 𝑋 ∧ 𝑏 = 𝐵 ) → ( 𝑎 ≠ 𝑏 ↔ 𝑋 ≠ 𝐵 ) ) |
| 37 |
|
fveq2 |
⊢ ( 𝑎 = 𝑋 → ( 𝐹 ‘ 𝑎 ) = ( 𝐹 ‘ 𝑋 ) ) |
| 38 |
37
|
adantr |
⊢ ( ( 𝑎 = 𝑋 ∧ 𝑏 = 𝐵 ) → ( 𝐹 ‘ 𝑎 ) = ( 𝐹 ‘ 𝑋 ) ) |
| 39 |
|
fveq2 |
⊢ ( 𝑏 = 𝐵 → ( 𝐹 ‘ 𝑏 ) = ( 𝐹 ‘ 𝐵 ) ) |
| 40 |
39
|
adantl |
⊢ ( ( 𝑎 = 𝑋 ∧ 𝑏 = 𝐵 ) → ( 𝐹 ‘ 𝑏 ) = ( 𝐹 ‘ 𝐵 ) ) |
| 41 |
38 40
|
neeq12d |
⊢ ( ( 𝑎 = 𝑋 ∧ 𝑏 = 𝐵 ) → ( ( 𝐹 ‘ 𝑎 ) ≠ ( 𝐹 ‘ 𝑏 ) ↔ ( 𝐹 ‘ 𝑋 ) ≠ ( 𝐹 ‘ 𝐵 ) ) ) |
| 42 |
36 41
|
imbi12d |
⊢ ( ( 𝑎 = 𝑋 ∧ 𝑏 = 𝐵 ) → ( ( 𝑎 ≠ 𝑏 → ( 𝐹 ‘ 𝑎 ) ≠ ( 𝐹 ‘ 𝑏 ) ) ↔ ( 𝑋 ≠ 𝐵 → ( 𝐹 ‘ 𝑋 ) ≠ ( 𝐹 ‘ 𝐵 ) ) ) ) |
| 43 |
42
|
rspc2gv |
⊢ ( ( 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) → ( ∀ 𝑎 ∈ 𝐶 ∀ 𝑏 ∈ 𝐶 ( 𝑎 ≠ 𝑏 → ( 𝐹 ‘ 𝑎 ) ≠ ( 𝐹 ‘ 𝑏 ) ) → ( 𝑋 ≠ 𝐵 → ( 𝐹 ‘ 𝑋 ) ≠ ( 𝐹 ‘ 𝐵 ) ) ) ) |
| 44 |
43
|
3adant1 |
⊢ ( ( 𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) → ( ∀ 𝑎 ∈ 𝐶 ∀ 𝑏 ∈ 𝐶 ( 𝑎 ≠ 𝑏 → ( 𝐹 ‘ 𝑎 ) ≠ ( 𝐹 ‘ 𝑏 ) ) → ( 𝑋 ≠ 𝐵 → ( 𝐹 ‘ 𝑋 ) ≠ ( 𝐹 ‘ 𝐵 ) ) ) ) |
| 45 |
|
id |
⊢ ( ( 𝑋 ≠ 𝐵 → ( 𝐹 ‘ 𝑋 ) ≠ ( 𝐹 ‘ 𝐵 ) ) → ( 𝑋 ≠ 𝐵 → ( 𝐹 ‘ 𝑋 ) ≠ ( 𝐹 ‘ 𝐵 ) ) ) |
| 46 |
|
eqneqall |
⊢ ( ( 𝐹 ‘ 𝑋 ) = ( 𝐹 ‘ 𝐵 ) → ( ( 𝐹 ‘ 𝑋 ) ≠ ( 𝐹 ‘ 𝐵 ) → ( 𝐹 ‘ 𝐵 ) ∈ ( 1 ... 𝑁 ) ) ) |
| 47 |
46
|
eqcoms |
⊢ ( ( 𝐹 ‘ 𝐵 ) = ( 𝐹 ‘ 𝑋 ) → ( ( 𝐹 ‘ 𝑋 ) ≠ ( 𝐹 ‘ 𝐵 ) → ( 𝐹 ‘ 𝐵 ) ∈ ( 1 ... 𝑁 ) ) ) |
| 48 |
47
|
com12 |
⊢ ( ( 𝐹 ‘ 𝑋 ) ≠ ( 𝐹 ‘ 𝐵 ) → ( ( 𝐹 ‘ 𝐵 ) = ( 𝐹 ‘ 𝑋 ) → ( 𝐹 ‘ 𝐵 ) ∈ ( 1 ... 𝑁 ) ) ) |
| 49 |
45 48
|
syl6com |
⊢ ( 𝑋 ≠ 𝐵 → ( ( 𝑋 ≠ 𝐵 → ( 𝐹 ‘ 𝑋 ) ≠ ( 𝐹 ‘ 𝐵 ) ) → ( ( 𝐹 ‘ 𝐵 ) = ( 𝐹 ‘ 𝑋 ) → ( 𝐹 ‘ 𝐵 ) ∈ ( 1 ... 𝑁 ) ) ) ) |
| 50 |
49
|
3ad2ant1 |
⊢ ( ( 𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) → ( ( 𝑋 ≠ 𝐵 → ( 𝐹 ‘ 𝑋 ) ≠ ( 𝐹 ‘ 𝐵 ) ) → ( ( 𝐹 ‘ 𝐵 ) = ( 𝐹 ‘ 𝑋 ) → ( 𝐹 ‘ 𝐵 ) ∈ ( 1 ... 𝑁 ) ) ) ) |
| 51 |
44 50
|
syld |
⊢ ( ( 𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) → ( ∀ 𝑎 ∈ 𝐶 ∀ 𝑏 ∈ 𝐶 ( 𝑎 ≠ 𝑏 → ( 𝐹 ‘ 𝑎 ) ≠ ( 𝐹 ‘ 𝑏 ) ) → ( ( 𝐹 ‘ 𝐵 ) = ( 𝐹 ‘ 𝑋 ) → ( 𝐹 ‘ 𝐵 ) ∈ ( 1 ... 𝑁 ) ) ) ) |
| 52 |
51
|
adantld |
⊢ ( ( 𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) → ( ( 𝐹 : 𝐶 ⟶ 𝑊 ∧ ∀ 𝑎 ∈ 𝐶 ∀ 𝑏 ∈ 𝐶 ( 𝑎 ≠ 𝑏 → ( 𝐹 ‘ 𝑎 ) ≠ ( 𝐹 ‘ 𝑏 ) ) ) → ( ( 𝐹 ‘ 𝐵 ) = ( 𝐹 ‘ 𝑋 ) → ( 𝐹 ‘ 𝐵 ) ∈ ( 1 ... 𝑁 ) ) ) ) |
| 53 |
33 52
|
biimtrid |
⊢ ( ( 𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) → ( 𝐹 : 𝐶 –1-1→ 𝑊 → ( ( 𝐹 ‘ 𝐵 ) = ( 𝐹 ‘ 𝑋 ) → ( 𝐹 ‘ 𝐵 ) ∈ ( 1 ... 𝑁 ) ) ) ) |
| 54 |
32 53
|
syl5com |
⊢ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 → ( ( 𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) → ( ( 𝐹 ‘ 𝐵 ) = ( 𝐹 ‘ 𝑋 ) → ( 𝐹 ‘ 𝐵 ) ∈ ( 1 ... 𝑁 ) ) ) ) |
| 55 |
54
|
adantr |
⊢ ( ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) → ( ( 𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) → ( ( 𝐹 ‘ 𝐵 ) = ( 𝐹 ‘ 𝑋 ) → ( 𝐹 ‘ 𝐵 ) ∈ ( 1 ... 𝑁 ) ) ) ) |
| 56 |
55
|
imp |
⊢ ( ( ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ∧ ( 𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) ) → ( ( 𝐹 ‘ 𝐵 ) = ( 𝐹 ‘ 𝑋 ) → ( 𝐹 ‘ 𝐵 ) ∈ ( 1 ... 𝑁 ) ) ) |
| 57 |
31 56
|
sylbird |
⊢ ( ( ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ∧ ( 𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) ) → ( ( 𝐹 ‘ 𝐵 ) = 0 → ( 𝐹 ‘ 𝐵 ) ∈ ( 1 ... 𝑁 ) ) ) |
| 58 |
|
idd |
⊢ ( ( ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ∧ ( 𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) ) → ( ( 𝐹 ‘ 𝐵 ) ∈ ( 1 ... 𝑁 ) → ( 𝐹 ‘ 𝐵 ) ∈ ( 1 ... 𝑁 ) ) ) |
| 59 |
57 58
|
jaod |
⊢ ( ( ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ∧ ( 𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) ) → ( ( ( 𝐹 ‘ 𝐵 ) = 0 ∨ ( 𝐹 ‘ 𝐵 ) ∈ ( 1 ... 𝑁 ) ) → ( 𝐹 ‘ 𝐵 ) ∈ ( 1 ... 𝑁 ) ) ) |
| 60 |
28 59
|
biimtrid |
⊢ ( ( ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ∧ ( 𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) ) → ( ( 𝐹 ‘ 𝐵 ) ∈ 𝑊 → ( 𝐹 ‘ 𝐵 ) ∈ ( 1 ... 𝑁 ) ) ) |
| 61 |
14 60
|
mpd |
⊢ ( ( ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ∧ ( 𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) ) → ( 𝐹 ‘ 𝐵 ) ∈ ( 1 ... 𝑁 ) ) |
| 62 |
|
f1ofn |
⊢ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 → 𝐹 Fn 𝐶 ) |
| 63 |
62
|
adantr |
⊢ ( ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) → 𝐹 Fn 𝐶 ) |
| 64 |
|
3simpc |
⊢ ( ( 𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) → ( 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) ) |
| 65 |
63 64
|
anim12i |
⊢ ( ( ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ∧ ( 𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) ) → ( 𝐹 Fn 𝐶 ∧ ( 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) ) ) |
| 66 |
|
3anass |
⊢ ( ( 𝐹 Fn 𝐶 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) ↔ ( 𝐹 Fn 𝐶 ∧ ( 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) ) ) |
| 67 |
65 66
|
sylibr |
⊢ ( ( ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ∧ ( 𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) ) → ( 𝐹 Fn 𝐶 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) ) |
| 68 |
|
fnimapr |
⊢ ( ( 𝐹 Fn 𝐶 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) → ( 𝐹 “ { 𝑋 , 𝐵 } ) = { ( 𝐹 ‘ 𝑋 ) , ( 𝐹 ‘ 𝐵 ) } ) |
| 69 |
67 68
|
syl |
⊢ ( ( ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ∧ ( 𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) ) → ( 𝐹 “ { 𝑋 , 𝐵 } ) = { ( 𝐹 ‘ 𝑋 ) , ( 𝐹 ‘ 𝐵 ) } ) |
| 70 |
|
simpr |
⊢ ( ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) → ( 𝐹 ‘ 𝑋 ) = 0 ) |
| 71 |
70
|
adantr |
⊢ ( ( ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ∧ ( 𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) ) → ( 𝐹 ‘ 𝑋 ) = 0 ) |
| 72 |
71
|
preq1d |
⊢ ( ( ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ∧ ( 𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) ) → { ( 𝐹 ‘ 𝑋 ) , ( 𝐹 ‘ 𝐵 ) } = { 0 , ( 𝐹 ‘ 𝐵 ) } ) |
| 73 |
69 72
|
eqtrd |
⊢ ( ( ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ∧ ( 𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) ) → ( 𝐹 “ { 𝑋 , 𝐵 } ) = { 0 , ( 𝐹 ‘ 𝐵 ) } ) |
| 74 |
9 61 73
|
rspcedvdw |
⊢ ( ( ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ∧ ( 𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑋 , 𝐵 } ) = { 0 , 𝑧 } ) |
| 75 |
74
|
ex |
⊢ ( ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) → ( ( 𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑋 , 𝐵 } ) = { 0 , 𝑧 } ) ) |
| 76 |
|
neeq1 |
⊢ ( 𝐴 = 𝑋 → ( 𝐴 ≠ 𝐵 ↔ 𝑋 ≠ 𝐵 ) ) |
| 77 |
|
eleq1 |
⊢ ( 𝐴 = 𝑋 → ( 𝐴 ∈ 𝐶 ↔ 𝑋 ∈ 𝐶 ) ) |
| 78 |
76 77
|
3anbi12d |
⊢ ( 𝐴 = 𝑋 → ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) ↔ ( 𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) ) ) |
| 79 |
|
preq1 |
⊢ ( 𝐴 = 𝑋 → { 𝐴 , 𝐵 } = { 𝑋 , 𝐵 } ) |
| 80 |
79
|
imaeq2d |
⊢ ( 𝐴 = 𝑋 → ( 𝐹 “ { 𝐴 , 𝐵 } ) = ( 𝐹 “ { 𝑋 , 𝐵 } ) ) |
| 81 |
80
|
eqeq1d |
⊢ ( 𝐴 = 𝑋 → ( ( 𝐹 “ { 𝐴 , 𝐵 } ) = { 0 , 𝑧 } ↔ ( 𝐹 “ { 𝑋 , 𝐵 } ) = { 0 , 𝑧 } ) ) |
| 82 |
81
|
rexbidv |
⊢ ( 𝐴 = 𝑋 → ( ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝐴 , 𝐵 } ) = { 0 , 𝑧 } ↔ ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑋 , 𝐵 } ) = { 0 , 𝑧 } ) ) |
| 83 |
78 82
|
imbi12d |
⊢ ( 𝐴 = 𝑋 → ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝐴 , 𝐵 } ) = { 0 , 𝑧 } ) ↔ ( ( 𝑋 ≠ 𝐵 ∧ 𝑋 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝑋 , 𝐵 } ) = { 0 , 𝑧 } ) ) ) |
| 84 |
75 83
|
imbitrrid |
⊢ ( 𝐴 = 𝑋 → ( ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) → ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝐴 , 𝐵 } ) = { 0 , 𝑧 } ) ) ) |
| 85 |
84
|
3imp |
⊢ ( ( 𝐴 = 𝑋 ∧ ( 𝐹 : 𝐶 –1-1-onto→ 𝑊 ∧ ( 𝐹 ‘ 𝑋 ) = 0 ) ∧ ( 𝐴 ≠ 𝐵 ∧ 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) ) → ∃ 𝑧 ∈ ( 1 ... 𝑁 ) ( 𝐹 “ { 𝐴 , 𝐵 } ) = { 0 , 𝑧 } ) |