| Step |
Hyp |
Ref |
Expression |
| 1 |
|
grimedg.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
| 2 |
|
grimedg.i |
⊢ 𝐼 = ( Edg ‘ 𝐺 ) |
| 3 |
|
grimedg.e |
⊢ 𝐸 = ( Edg ‘ 𝐻 ) |
| 4 |
|
eqid |
⊢ ( Vtx ‘ 𝐻 ) = ( Vtx ‘ 𝐻 ) |
| 5 |
|
eqid |
⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 ) |
| 6 |
|
eqid |
⊢ ( iEdg ‘ 𝐻 ) = ( iEdg ‘ 𝐻 ) |
| 7 |
1 4 5 6
|
grimprop |
⊢ ( 𝐹 ∈ ( 𝐺 GraphIso 𝐻 ) → ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ ∃ 𝑗 ( 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) ) ) ) |
| 8 |
2
|
eleq2i |
⊢ ( 𝐾 ∈ 𝐼 ↔ 𝐾 ∈ ( Edg ‘ 𝐺 ) ) |
| 9 |
5
|
uhgredgiedgb |
⊢ ( 𝐺 ∈ UHGraph → ( 𝐾 ∈ ( Edg ‘ 𝐺 ) ↔ ∃ 𝑘 ∈ dom ( iEdg ‘ 𝐺 ) 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ) ) |
| 10 |
9
|
ad2antll |
⊢ ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ ( 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) ) ) ∧ ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ) → ( 𝐾 ∈ ( Edg ‘ 𝐺 ) ↔ ∃ 𝑘 ∈ dom ( iEdg ‘ 𝐺 ) 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ) ) |
| 11 |
8 10
|
bitrid |
⊢ ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ ( 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) ) ) ∧ ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ) → ( 𝐾 ∈ 𝐼 ↔ ∃ 𝑘 ∈ dom ( iEdg ‘ 𝐺 ) 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ) ) |
| 12 |
|
2fveq3 |
⊢ ( 𝑖 = 𝑘 → ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑘 ) ) ) |
| 13 |
|
fveq2 |
⊢ ( 𝑖 = 𝑘 → ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) = ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ) |
| 14 |
13
|
imaeq2d |
⊢ ( 𝑖 = 𝑘 → ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ) ) |
| 15 |
12 14
|
eqeq12d |
⊢ ( 𝑖 = 𝑘 → ( ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) ↔ ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑘 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ) ) ) |
| 16 |
15
|
rspcv |
⊢ ( 𝑘 ∈ dom ( iEdg ‘ 𝐺 ) → ( ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) → ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑘 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ) ) ) |
| 17 |
16
|
adantl |
⊢ ( ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐺 ) ) → ( ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) → ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑘 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ) ) ) |
| 18 |
6
|
uhgrfun |
⊢ ( 𝐻 ∈ UHGraph → Fun ( iEdg ‘ 𝐻 ) ) |
| 19 |
18
|
ad2antrr |
⊢ ( ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐺 ) ) → Fun ( iEdg ‘ 𝐻 ) ) |
| 20 |
|
f1of |
⊢ ( 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) → 𝑗 : dom ( iEdg ‘ 𝐺 ) ⟶ dom ( iEdg ‘ 𝐻 ) ) |
| 21 |
20
|
ad2antll |
⊢ ( ( ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐺 ) ) ∧ ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ) → 𝑗 : dom ( iEdg ‘ 𝐺 ) ⟶ dom ( iEdg ‘ 𝐻 ) ) |
| 22 |
|
simplr |
⊢ ( ( ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐺 ) ) ∧ ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ) → 𝑘 ∈ dom ( iEdg ‘ 𝐺 ) ) |
| 23 |
21 22
|
ffvelcdmd |
⊢ ( ( ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐺 ) ) ∧ ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ) → ( 𝑗 ‘ 𝑘 ) ∈ dom ( iEdg ‘ 𝐻 ) ) |
| 24 |
6
|
iedgedg |
⊢ ( ( Fun ( iEdg ‘ 𝐻 ) ∧ ( 𝑗 ‘ 𝑘 ) ∈ dom ( iEdg ‘ 𝐻 ) ) → ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑘 ) ) ∈ ( Edg ‘ 𝐻 ) ) |
| 25 |
24 3
|
eleqtrrdi |
⊢ ( ( Fun ( iEdg ‘ 𝐻 ) ∧ ( 𝑗 ‘ 𝑘 ) ∈ dom ( iEdg ‘ 𝐻 ) ) → ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑘 ) ) ∈ 𝐸 ) |
| 26 |
19 23 25
|
syl2an2r |
⊢ ( ( ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐺 ) ) ∧ ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ) → ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑘 ) ) ∈ 𝐸 ) |
| 27 |
|
eleq1 |
⊢ ( ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ) = ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑘 ) ) → ( ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ) ∈ 𝐸 ↔ ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑘 ) ) ∈ 𝐸 ) ) |
| 28 |
27
|
eqcoms |
⊢ ( ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑘 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ) → ( ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ) ∈ 𝐸 ↔ ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑘 ) ) ∈ 𝐸 ) ) |
| 29 |
26 28
|
syl5ibrcom |
⊢ ( ( ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐺 ) ) ∧ ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ) → ( ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑘 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ) → ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ) ∈ 𝐸 ) ) |
| 30 |
29
|
ex |
⊢ ( ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐺 ) ) → ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) → ( ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑘 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ) → ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ) ∈ 𝐸 ) ) ) |
| 31 |
30
|
com23 |
⊢ ( ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐺 ) ) → ( ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑘 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ) → ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) → ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ) ∈ 𝐸 ) ) ) |
| 32 |
17 31
|
syld |
⊢ ( ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐺 ) ) → ( ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) → ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) → ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ) ∈ 𝐸 ) ) ) |
| 33 |
32
|
com13 |
⊢ ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) → ( ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) → ( ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐺 ) ) → ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ) ∈ 𝐸 ) ) ) |
| 34 |
33
|
impr |
⊢ ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ ( 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) ) ) → ( ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐺 ) ) → ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ) ∈ 𝐸 ) ) |
| 35 |
34
|
impl |
⊢ ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ ( 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) ) ) ∧ ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐺 ) ) → ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ) ∈ 𝐸 ) |
| 36 |
35
|
adantr |
⊢ ( ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ ( 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) ) ) ∧ ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐺 ) ) ∧ 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ) → ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ) ∈ 𝐸 ) |
| 37 |
|
imaeq2 |
⊢ ( 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) → ( 𝐹 “ 𝐾 ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ) ) |
| 38 |
37
|
eleq1d |
⊢ ( 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) → ( ( 𝐹 “ 𝐾 ) ∈ 𝐸 ↔ ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ) ∈ 𝐸 ) ) |
| 39 |
38
|
adantl |
⊢ ( ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ ( 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) ) ) ∧ ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐺 ) ) ∧ 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ) → ( ( 𝐹 “ 𝐾 ) ∈ 𝐸 ↔ ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ) ∈ 𝐸 ) ) |
| 40 |
36 39
|
mpbird |
⊢ ( ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ ( 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) ) ) ∧ ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐺 ) ) ∧ 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ) → ( 𝐹 “ 𝐾 ) ∈ 𝐸 ) |
| 41 |
1 5
|
uhgrss |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐺 ) ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ⊆ 𝑉 ) |
| 42 |
41
|
ex |
⊢ ( 𝐺 ∈ UHGraph → ( 𝑘 ∈ dom ( iEdg ‘ 𝐺 ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ⊆ 𝑉 ) ) |
| 43 |
42
|
ad2antll |
⊢ ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ ( 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) ) ) ∧ ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ) → ( 𝑘 ∈ dom ( iEdg ‘ 𝐺 ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ⊆ 𝑉 ) ) |
| 44 |
43
|
imp |
⊢ ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ ( 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) ) ) ∧ ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐺 ) ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ⊆ 𝑉 ) |
| 45 |
44
|
adantr |
⊢ ( ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ ( 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) ) ) ∧ ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐺 ) ) ∧ 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ⊆ 𝑉 ) |
| 46 |
|
sseq1 |
⊢ ( 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) → ( 𝐾 ⊆ 𝑉 ↔ ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ⊆ 𝑉 ) ) |
| 47 |
46
|
adantl |
⊢ ( ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ ( 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) ) ) ∧ ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐺 ) ) ∧ 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ) → ( 𝐾 ⊆ 𝑉 ↔ ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ⊆ 𝑉 ) ) |
| 48 |
45 47
|
mpbird |
⊢ ( ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ ( 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) ) ) ∧ ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐺 ) ) ∧ 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ) → 𝐾 ⊆ 𝑉 ) |
| 49 |
40 48
|
jca |
⊢ ( ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ ( 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) ) ) ∧ ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐺 ) ) ∧ 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) ) → ( ( 𝐹 “ 𝐾 ) ∈ 𝐸 ∧ 𝐾 ⊆ 𝑉 ) ) |
| 50 |
49
|
ex |
⊢ ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ ( 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) ) ) ∧ ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐺 ) ) → ( 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) → ( ( 𝐹 “ 𝐾 ) ∈ 𝐸 ∧ 𝐾 ⊆ 𝑉 ) ) ) |
| 51 |
50
|
rexlimdva |
⊢ ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ ( 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) ) ) ∧ ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ) → ( ∃ 𝑘 ∈ dom ( iEdg ‘ 𝐺 ) 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑘 ) → ( ( 𝐹 “ 𝐾 ) ∈ 𝐸 ∧ 𝐾 ⊆ 𝑉 ) ) ) |
| 52 |
11 51
|
sylbid |
⊢ ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ ( 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) ) ) ∧ ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ) → ( 𝐾 ∈ 𝐼 → ( ( 𝐹 “ 𝐾 ) ∈ 𝐸 ∧ 𝐾 ⊆ 𝑉 ) ) ) |
| 53 |
3
|
eleq2i |
⊢ ( ( 𝐹 “ 𝐾 ) ∈ 𝐸 ↔ ( 𝐹 “ 𝐾 ) ∈ ( Edg ‘ 𝐻 ) ) |
| 54 |
6
|
uhgredgiedgb |
⊢ ( 𝐻 ∈ UHGraph → ( ( 𝐹 “ 𝐾 ) ∈ ( Edg ‘ 𝐻 ) ↔ ∃ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ( 𝐹 “ 𝐾 ) = ( ( iEdg ‘ 𝐻 ) ‘ 𝑘 ) ) ) |
| 55 |
54
|
ad2antrl |
⊢ ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ ( 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) ) ) ∧ ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ) → ( ( 𝐹 “ 𝐾 ) ∈ ( Edg ‘ 𝐻 ) ↔ ∃ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ( 𝐹 “ 𝐾 ) = ( ( iEdg ‘ 𝐻 ) ‘ 𝑘 ) ) ) |
| 56 |
53 55
|
bitrid |
⊢ ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ ( 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) ) ) ∧ ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ) → ( ( 𝐹 “ 𝐾 ) ∈ 𝐸 ↔ ∃ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ( 𝐹 “ 𝐾 ) = ( ( iEdg ‘ 𝐻 ) ‘ 𝑘 ) ) ) |
| 57 |
|
f1ofo |
⊢ ( 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) → 𝑗 : dom ( iEdg ‘ 𝐺 ) –onto→ dom ( iEdg ‘ 𝐻 ) ) |
| 58 |
57
|
adantr |
⊢ ( ( 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) ) → 𝑗 : dom ( iEdg ‘ 𝐺 ) –onto→ dom ( iEdg ‘ 𝐻 ) ) |
| 59 |
58
|
ad2antlr |
⊢ ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ ( 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) ) ) ∧ ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ) → 𝑗 : dom ( iEdg ‘ 𝐺 ) –onto→ dom ( iEdg ‘ 𝐻 ) ) |
| 60 |
|
foelrn |
⊢ ( ( 𝑗 : dom ( iEdg ‘ 𝐺 ) –onto→ dom ( iEdg ‘ 𝐻 ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) → ∃ 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) 𝑘 = ( 𝑗 ‘ 𝑙 ) ) |
| 61 |
59 60
|
sylan |
⊢ ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ ( 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) ) ) ∧ ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) → ∃ 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) 𝑘 = ( 𝑗 ‘ 𝑙 ) ) |
| 62 |
|
2fveq3 |
⊢ ( 𝑖 = 𝑙 → ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) ) |
| 63 |
|
fveq2 |
⊢ ( 𝑖 = 𝑙 → ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) = ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) |
| 64 |
63
|
imaeq2d |
⊢ ( 𝑖 = 𝑙 → ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) ) |
| 65 |
62 64
|
eqeq12d |
⊢ ( 𝑖 = 𝑙 → ( ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) ↔ ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) ) ) |
| 66 |
65
|
rspcv |
⊢ ( 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) → ( ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) → ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) ) ) |
| 67 |
66
|
adantl |
⊢ ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( 𝐹 “ 𝐾 ) = ( ( iEdg ‘ 𝐻 ) ‘ 𝑘 ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) ∧ 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ) → ( ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) → ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) ) ) |
| 68 |
|
fveq2 |
⊢ ( 𝑘 = ( 𝑗 ‘ 𝑙 ) → ( ( iEdg ‘ 𝐻 ) ‘ 𝑘 ) = ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) ) |
| 69 |
68
|
eqeq2d |
⊢ ( 𝑘 = ( 𝑗 ‘ 𝑙 ) → ( ( 𝐹 “ 𝐾 ) = ( ( iEdg ‘ 𝐻 ) ‘ 𝑘 ) ↔ ( 𝐹 “ 𝐾 ) = ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) ) ) |
| 70 |
69
|
ad2antll |
⊢ ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) ∧ ( 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝑘 = ( 𝑗 ‘ 𝑙 ) ) ) → ( ( 𝐹 “ 𝐾 ) = ( ( iEdg ‘ 𝐻 ) ‘ 𝑘 ) ↔ ( 𝐹 “ 𝐾 ) = ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) ) ) |
| 71 |
|
simpl |
⊢ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) → 𝐻 ∈ UHGraph ) |
| 72 |
71
|
ad2antrl |
⊢ ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) → 𝐻 ∈ UHGraph ) |
| 73 |
|
simplrr |
⊢ ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) ∧ ( 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝑘 = ( 𝑗 ‘ 𝑙 ) ) ) → 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) |
| 74 |
|
eleq1 |
⊢ ( 𝑘 = ( 𝑗 ‘ 𝑙 ) → ( 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ↔ ( 𝑗 ‘ 𝑙 ) ∈ dom ( iEdg ‘ 𝐻 ) ) ) |
| 75 |
74
|
ad2antll |
⊢ ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) ∧ ( 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝑘 = ( 𝑗 ‘ 𝑙 ) ) ) → ( 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ↔ ( 𝑗 ‘ 𝑙 ) ∈ dom ( iEdg ‘ 𝐻 ) ) ) |
| 76 |
73 75
|
mpbid |
⊢ ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) ∧ ( 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝑘 = ( 𝑗 ‘ 𝑙 ) ) ) → ( 𝑗 ‘ 𝑙 ) ∈ dom ( iEdg ‘ 𝐻 ) ) |
| 77 |
4 6
|
uhgrss |
⊢ ( ( 𝐻 ∈ UHGraph ∧ ( 𝑗 ‘ 𝑙 ) ∈ dom ( iEdg ‘ 𝐻 ) ) → ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) ⊆ ( Vtx ‘ 𝐻 ) ) |
| 78 |
72 76 77
|
syl2an2r |
⊢ ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) ∧ ( 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝑘 = ( 𝑗 ‘ 𝑙 ) ) ) → ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) ⊆ ( Vtx ‘ 𝐻 ) ) |
| 79 |
78
|
ad2antrr |
⊢ ( ( ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) ∧ ( 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝑘 = ( 𝑗 ‘ 𝑙 ) ) ) ∧ ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) ) ∧ ( 𝐹 “ 𝐾 ) = ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) ) → ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) ⊆ ( Vtx ‘ 𝐻 ) ) |
| 80 |
|
sseq1 |
⊢ ( ( 𝐹 “ 𝐾 ) = ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) → ( ( 𝐹 “ 𝐾 ) ⊆ ( Vtx ‘ 𝐻 ) ↔ ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) ⊆ ( Vtx ‘ 𝐻 ) ) ) |
| 81 |
80
|
adantl |
⊢ ( ( ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) ∧ ( 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝑘 = ( 𝑗 ‘ 𝑙 ) ) ) ∧ ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) ) ∧ ( 𝐹 “ 𝐾 ) = ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) ) → ( ( 𝐹 “ 𝐾 ) ⊆ ( Vtx ‘ 𝐻 ) ↔ ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) ⊆ ( Vtx ‘ 𝐻 ) ) ) |
| 82 |
79 81
|
mpbird |
⊢ ( ( ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) ∧ ( 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝑘 = ( 𝑗 ‘ 𝑙 ) ) ) ∧ ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) ) ∧ ( 𝐹 “ 𝐾 ) = ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) ) → ( 𝐹 “ 𝐾 ) ⊆ ( Vtx ‘ 𝐻 ) ) |
| 83 |
|
eqeq2 |
⊢ ( ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) → ( ( 𝐹 “ 𝐾 ) = ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) ↔ ( 𝐹 “ 𝐾 ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) ) ) |
| 84 |
83
|
adantl |
⊢ ( ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) ∧ ( 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝑘 = ( 𝑗 ‘ 𝑙 ) ) ) ∧ ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) ) → ( ( 𝐹 “ 𝐾 ) = ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) ↔ ( 𝐹 “ 𝐾 ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) ) ) |
| 85 |
|
f1of1 |
⊢ ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) → 𝐹 : 𝑉 –1-1→ ( Vtx ‘ 𝐻 ) ) |
| 86 |
85
|
ad3antrrr |
⊢ ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) ∧ ( 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝑘 = ( 𝑗 ‘ 𝑙 ) ) ) → 𝐹 : 𝑉 –1-1→ ( Vtx ‘ 𝐻 ) ) |
| 87 |
86
|
ad3antrrr |
⊢ ( ( ( ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) ∧ ( 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝑘 = ( 𝑗 ‘ 𝑙 ) ) ) ∧ ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) ) ∧ ( 𝐹 “ 𝐾 ) ⊆ ( Vtx ‘ 𝐻 ) ) ∧ 𝐾 ⊆ 𝑉 ) → 𝐹 : 𝑉 –1-1→ ( Vtx ‘ 𝐻 ) ) |
| 88 |
|
simplr |
⊢ ( ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) → 𝐺 ∈ UHGraph ) |
| 89 |
88
|
adantl |
⊢ ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) → 𝐺 ∈ UHGraph ) |
| 90 |
|
simpl |
⊢ ( ( 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝑘 = ( 𝑗 ‘ 𝑙 ) ) → 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ) |
| 91 |
1 5
|
uhgrss |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ⊆ 𝑉 ) |
| 92 |
89 90 91
|
syl2an |
⊢ ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) ∧ ( 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝑘 = ( 𝑗 ‘ 𝑙 ) ) ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ⊆ 𝑉 ) |
| 93 |
92
|
ad2antrr |
⊢ ( ( ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) ∧ ( 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝑘 = ( 𝑗 ‘ 𝑙 ) ) ) ∧ ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) ) ∧ ( 𝐹 “ 𝐾 ) ⊆ ( Vtx ‘ 𝐻 ) ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ⊆ 𝑉 ) |
| 94 |
93
|
anim1ci |
⊢ ( ( ( ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) ∧ ( 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝑘 = ( 𝑗 ‘ 𝑙 ) ) ) ∧ ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) ) ∧ ( 𝐹 “ 𝐾 ) ⊆ ( Vtx ‘ 𝐻 ) ) ∧ 𝐾 ⊆ 𝑉 ) → ( 𝐾 ⊆ 𝑉 ∧ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ⊆ 𝑉 ) ) |
| 95 |
|
f1imaeq |
⊢ ( ( 𝐹 : 𝑉 –1-1→ ( Vtx ‘ 𝐻 ) ∧ ( 𝐾 ⊆ 𝑉 ∧ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ⊆ 𝑉 ) ) → ( ( 𝐹 “ 𝐾 ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) ↔ 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) ) |
| 96 |
87 94 95
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) ∧ ( 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝑘 = ( 𝑗 ‘ 𝑙 ) ) ) ∧ ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) ) ∧ ( 𝐹 “ 𝐾 ) ⊆ ( Vtx ‘ 𝐻 ) ) ∧ 𝐾 ⊆ 𝑉 ) → ( ( 𝐹 “ 𝐾 ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) ↔ 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) ) |
| 97 |
5
|
uhgrfun |
⊢ ( 𝐺 ∈ UHGraph → Fun ( iEdg ‘ 𝐺 ) ) |
| 98 |
97
|
ad2antlr |
⊢ ( ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) → Fun ( iEdg ‘ 𝐺 ) ) |
| 99 |
98
|
adantl |
⊢ ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) → Fun ( iEdg ‘ 𝐺 ) ) |
| 100 |
5
|
iedgedg |
⊢ ( ( Fun ( iEdg ‘ 𝐺 ) ∧ 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ∈ ( Edg ‘ 𝐺 ) ) |
| 101 |
99 90 100
|
syl2an |
⊢ ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) ∧ ( 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝑘 = ( 𝑗 ‘ 𝑙 ) ) ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ∈ ( Edg ‘ 𝐺 ) ) |
| 102 |
101 2
|
eleqtrrdi |
⊢ ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) ∧ ( 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝑘 = ( 𝑗 ‘ 𝑙 ) ) ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ∈ 𝐼 ) |
| 103 |
|
eleq1 |
⊢ ( 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) → ( 𝐾 ∈ 𝐼 ↔ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ∈ 𝐼 ) ) |
| 104 |
102 103
|
syl5ibrcom |
⊢ ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) ∧ ( 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝑘 = ( 𝑗 ‘ 𝑙 ) ) ) → ( 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) → 𝐾 ∈ 𝐼 ) ) |
| 105 |
104
|
ad3antrrr |
⊢ ( ( ( ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) ∧ ( 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝑘 = ( 𝑗 ‘ 𝑙 ) ) ) ∧ ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) ) ∧ ( 𝐹 “ 𝐾 ) ⊆ ( Vtx ‘ 𝐻 ) ) ∧ 𝐾 ⊆ 𝑉 ) → ( 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) → 𝐾 ∈ 𝐼 ) ) |
| 106 |
96 105
|
sylbid |
⊢ ( ( ( ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) ∧ ( 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝑘 = ( 𝑗 ‘ 𝑙 ) ) ) ∧ ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) ) ∧ ( 𝐹 “ 𝐾 ) ⊆ ( Vtx ‘ 𝐻 ) ) ∧ 𝐾 ⊆ 𝑉 ) → ( ( 𝐹 “ 𝐾 ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) → 𝐾 ∈ 𝐼 ) ) |
| 107 |
106
|
ex |
⊢ ( ( ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) ∧ ( 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝑘 = ( 𝑗 ‘ 𝑙 ) ) ) ∧ ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) ) ∧ ( 𝐹 “ 𝐾 ) ⊆ ( Vtx ‘ 𝐻 ) ) → ( 𝐾 ⊆ 𝑉 → ( ( 𝐹 “ 𝐾 ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) → 𝐾 ∈ 𝐼 ) ) ) |
| 108 |
107
|
com23 |
⊢ ( ( ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) ∧ ( 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝑘 = ( 𝑗 ‘ 𝑙 ) ) ) ∧ ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) ) ∧ ( 𝐹 “ 𝐾 ) ⊆ ( Vtx ‘ 𝐻 ) ) → ( ( 𝐹 “ 𝐾 ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) → ( 𝐾 ⊆ 𝑉 → 𝐾 ∈ 𝐼 ) ) ) |
| 109 |
108
|
ex |
⊢ ( ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) ∧ ( 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝑘 = ( 𝑗 ‘ 𝑙 ) ) ) ∧ ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) ) → ( ( 𝐹 “ 𝐾 ) ⊆ ( Vtx ‘ 𝐻 ) → ( ( 𝐹 “ 𝐾 ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) → ( 𝐾 ⊆ 𝑉 → 𝐾 ∈ 𝐼 ) ) ) ) |
| 110 |
109
|
com23 |
⊢ ( ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) ∧ ( 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝑘 = ( 𝑗 ‘ 𝑙 ) ) ) ∧ ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) ) → ( ( 𝐹 “ 𝐾 ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) → ( ( 𝐹 “ 𝐾 ) ⊆ ( Vtx ‘ 𝐻 ) → ( 𝐾 ⊆ 𝑉 → 𝐾 ∈ 𝐼 ) ) ) ) |
| 111 |
84 110
|
sylbid |
⊢ ( ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) ∧ ( 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝑘 = ( 𝑗 ‘ 𝑙 ) ) ) ∧ ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) ) → ( ( 𝐹 “ 𝐾 ) = ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) → ( ( 𝐹 “ 𝐾 ) ⊆ ( Vtx ‘ 𝐻 ) → ( 𝐾 ⊆ 𝑉 → 𝐾 ∈ 𝐼 ) ) ) ) |
| 112 |
111
|
imp |
⊢ ( ( ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) ∧ ( 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝑘 = ( 𝑗 ‘ 𝑙 ) ) ) ∧ ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) ) ∧ ( 𝐹 “ 𝐾 ) = ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) ) → ( ( 𝐹 “ 𝐾 ) ⊆ ( Vtx ‘ 𝐻 ) → ( 𝐾 ⊆ 𝑉 → 𝐾 ∈ 𝐼 ) ) ) |
| 113 |
82 112
|
mpd |
⊢ ( ( ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) ∧ ( 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝑘 = ( 𝑗 ‘ 𝑙 ) ) ) ∧ ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) ) ∧ ( 𝐹 “ 𝐾 ) = ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) ) → ( 𝐾 ⊆ 𝑉 → 𝐾 ∈ 𝐼 ) ) |
| 114 |
113
|
exp31 |
⊢ ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) ∧ ( 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝑘 = ( 𝑗 ‘ 𝑙 ) ) ) → ( ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) → ( ( 𝐹 “ 𝐾 ) = ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) → ( 𝐾 ⊆ 𝑉 → 𝐾 ∈ 𝐼 ) ) ) ) |
| 115 |
114
|
com23 |
⊢ ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) ∧ ( 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝑘 = ( 𝑗 ‘ 𝑙 ) ) ) → ( ( 𝐹 “ 𝐾 ) = ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) → ( ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) → ( 𝐾 ⊆ 𝑉 → 𝐾 ∈ 𝐼 ) ) ) ) |
| 116 |
70 115
|
sylbid |
⊢ ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) ∧ ( 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝑘 = ( 𝑗 ‘ 𝑙 ) ) ) → ( ( 𝐹 “ 𝐾 ) = ( ( iEdg ‘ 𝐻 ) ‘ 𝑘 ) → ( ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) → ( 𝐾 ⊆ 𝑉 → 𝐾 ∈ 𝐼 ) ) ) ) |
| 117 |
116
|
exp31 |
⊢ ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) → ( ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) → ( ( 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝑘 = ( 𝑗 ‘ 𝑙 ) ) → ( ( 𝐹 “ 𝐾 ) = ( ( iEdg ‘ 𝐻 ) ‘ 𝑘 ) → ( ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) → ( 𝐾 ⊆ 𝑉 → 𝐾 ∈ 𝐼 ) ) ) ) ) ) |
| 118 |
117
|
com23 |
⊢ ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) → ( ( 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝑘 = ( 𝑗 ‘ 𝑙 ) ) → ( ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) → ( ( 𝐹 “ 𝐾 ) = ( ( iEdg ‘ 𝐻 ) ‘ 𝑘 ) → ( ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) → ( 𝐾 ⊆ 𝑉 → 𝐾 ∈ 𝐼 ) ) ) ) ) ) |
| 119 |
118
|
com24 |
⊢ ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) → ( ( 𝐹 “ 𝐾 ) = ( ( iEdg ‘ 𝐻 ) ‘ 𝑘 ) → ( ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) → ( ( 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝑘 = ( 𝑗 ‘ 𝑙 ) ) → ( ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) → ( 𝐾 ⊆ 𝑉 → 𝐾 ∈ 𝐼 ) ) ) ) ) ) |
| 120 |
119
|
3imp |
⊢ ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( 𝐹 “ 𝐾 ) = ( ( iEdg ‘ 𝐻 ) ‘ 𝑘 ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) → ( ( 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝑘 = ( 𝑗 ‘ 𝑙 ) ) → ( ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) → ( 𝐾 ⊆ 𝑉 → 𝐾 ∈ 𝐼 ) ) ) ) |
| 121 |
120
|
expdimp |
⊢ ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( 𝐹 “ 𝐾 ) = ( ( iEdg ‘ 𝐻 ) ‘ 𝑘 ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) ∧ 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ) → ( 𝑘 = ( 𝑗 ‘ 𝑙 ) → ( ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑙 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑙 ) ) → ( 𝐾 ⊆ 𝑉 → 𝐾 ∈ 𝐼 ) ) ) ) |
| 122 |
67 121
|
syl5d |
⊢ ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( 𝐹 “ 𝐾 ) = ( ( iEdg ‘ 𝐻 ) ‘ 𝑘 ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) ∧ 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) ) → ( 𝑘 = ( 𝑗 ‘ 𝑙 ) → ( ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) → ( 𝐾 ⊆ 𝑉 → 𝐾 ∈ 𝐼 ) ) ) ) |
| 123 |
122
|
rexlimdva |
⊢ ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) ∧ ( 𝐹 “ 𝐾 ) = ( ( iEdg ‘ 𝐻 ) ‘ 𝑘 ) ∧ ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) ) → ( ∃ 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) 𝑘 = ( 𝑗 ‘ 𝑙 ) → ( ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) → ( 𝐾 ⊆ 𝑉 → 𝐾 ∈ 𝐼 ) ) ) ) |
| 124 |
123
|
3exp |
⊢ ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) → ( ( 𝐹 “ 𝐾 ) = ( ( iEdg ‘ 𝐻 ) ‘ 𝑘 ) → ( ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) → ( ∃ 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) 𝑘 = ( 𝑗 ‘ 𝑙 ) → ( ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) → ( 𝐾 ⊆ 𝑉 → 𝐾 ∈ 𝐼 ) ) ) ) ) ) |
| 125 |
124
|
com25 |
⊢ ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ) → ( ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) → ( ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) → ( ∃ 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) 𝑘 = ( 𝑗 ‘ 𝑙 ) → ( ( 𝐹 “ 𝐾 ) = ( ( iEdg ‘ 𝐻 ) ‘ 𝑘 ) → ( 𝐾 ⊆ 𝑉 → 𝐾 ∈ 𝐼 ) ) ) ) ) ) |
| 126 |
125
|
impr |
⊢ ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ ( 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) ) ) → ( ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) → ( ∃ 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) 𝑘 = ( 𝑗 ‘ 𝑙 ) → ( ( 𝐹 “ 𝐾 ) = ( ( iEdg ‘ 𝐻 ) ‘ 𝑘 ) → ( 𝐾 ⊆ 𝑉 → 𝐾 ∈ 𝐼 ) ) ) ) ) |
| 127 |
126
|
impl |
⊢ ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ ( 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) ) ) ∧ ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) → ( ∃ 𝑙 ∈ dom ( iEdg ‘ 𝐺 ) 𝑘 = ( 𝑗 ‘ 𝑙 ) → ( ( 𝐹 “ 𝐾 ) = ( ( iEdg ‘ 𝐻 ) ‘ 𝑘 ) → ( 𝐾 ⊆ 𝑉 → 𝐾 ∈ 𝐼 ) ) ) ) |
| 128 |
61 127
|
mpd |
⊢ ( ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ ( 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) ) ) ∧ ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ) ∧ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ) → ( ( 𝐹 “ 𝐾 ) = ( ( iEdg ‘ 𝐻 ) ‘ 𝑘 ) → ( 𝐾 ⊆ 𝑉 → 𝐾 ∈ 𝐼 ) ) ) |
| 129 |
128
|
rexlimdva |
⊢ ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ ( 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) ) ) ∧ ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ) → ( ∃ 𝑘 ∈ dom ( iEdg ‘ 𝐻 ) ( 𝐹 “ 𝐾 ) = ( ( iEdg ‘ 𝐻 ) ‘ 𝑘 ) → ( 𝐾 ⊆ 𝑉 → 𝐾 ∈ 𝐼 ) ) ) |
| 130 |
56 129
|
sylbid |
⊢ ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ ( 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) ) ) ∧ ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ) → ( ( 𝐹 “ 𝐾 ) ∈ 𝐸 → ( 𝐾 ⊆ 𝑉 → 𝐾 ∈ 𝐼 ) ) ) |
| 131 |
130
|
impd |
⊢ ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ ( 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) ) ) ∧ ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ) → ( ( ( 𝐹 “ 𝐾 ) ∈ 𝐸 ∧ 𝐾 ⊆ 𝑉 ) → 𝐾 ∈ 𝐼 ) ) |
| 132 |
52 131
|
impbid |
⊢ ( ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ ( 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) ) ) ∧ ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) ) → ( 𝐾 ∈ 𝐼 ↔ ( ( 𝐹 “ 𝐾 ) ∈ 𝐸 ∧ 𝐾 ⊆ 𝑉 ) ) ) |
| 133 |
132
|
exp31 |
⊢ ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) → ( ( 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) ) → ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) → ( 𝐾 ∈ 𝐼 ↔ ( ( 𝐹 “ 𝐾 ) ∈ 𝐸 ∧ 𝐾 ⊆ 𝑉 ) ) ) ) ) |
| 134 |
133
|
exlimdv |
⊢ ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) → ( ∃ 𝑗 ( 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) ) → ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) → ( 𝐾 ∈ 𝐼 ↔ ( ( 𝐹 “ 𝐾 ) ∈ 𝐸 ∧ 𝐾 ⊆ 𝑉 ) ) ) ) ) |
| 135 |
134
|
imp |
⊢ ( ( 𝐹 : 𝑉 –1-1-onto→ ( Vtx ‘ 𝐻 ) ∧ ∃ 𝑗 ( 𝑗 : dom ( iEdg ‘ 𝐺 ) –1-1-onto→ dom ( iEdg ‘ 𝐻 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ( ( iEdg ‘ 𝐻 ) ‘ ( 𝑗 ‘ 𝑖 ) ) = ( 𝐹 “ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ) ) ) → ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) → ( 𝐾 ∈ 𝐼 ↔ ( ( 𝐹 “ 𝐾 ) ∈ 𝐸 ∧ 𝐾 ⊆ 𝑉 ) ) ) ) |
| 136 |
7 135
|
syl |
⊢ ( 𝐹 ∈ ( 𝐺 GraphIso 𝐻 ) → ( ( 𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph ) → ( 𝐾 ∈ 𝐼 ↔ ( ( 𝐹 “ 𝐾 ) ∈ 𝐸 ∧ 𝐾 ⊆ 𝑉 ) ) ) ) |
| 137 |
136
|
expd |
⊢ ( 𝐹 ∈ ( 𝐺 GraphIso 𝐻 ) → ( 𝐻 ∈ UHGraph → ( 𝐺 ∈ UHGraph → ( 𝐾 ∈ 𝐼 ↔ ( ( 𝐹 “ 𝐾 ) ∈ 𝐸 ∧ 𝐾 ⊆ 𝑉 ) ) ) ) ) |
| 138 |
137
|
com13 |
⊢ ( 𝐺 ∈ UHGraph → ( 𝐻 ∈ UHGraph → ( 𝐹 ∈ ( 𝐺 GraphIso 𝐻 ) → ( 𝐾 ∈ 𝐼 ↔ ( ( 𝐹 “ 𝐾 ) ∈ 𝐸 ∧ 𝐾 ⊆ 𝑉 ) ) ) ) ) |
| 139 |
138
|
3imp |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ ( 𝐺 GraphIso 𝐻 ) ) → ( 𝐾 ∈ 𝐼 ↔ ( ( 𝐹 “ 𝐾 ) ∈ 𝐸 ∧ 𝐾 ⊆ 𝑉 ) ) ) |