Step |
Hyp |
Ref |
Expression |
1 |
|
isomushgr.v |
⊢ 𝑉 = ( Vtx ‘ 𝐴 ) |
2 |
|
isomushgr.w |
⊢ 𝑊 = ( Vtx ‘ 𝐵 ) |
3 |
|
isomushgr.e |
⊢ 𝐸 = ( Edg ‘ 𝐴 ) |
4 |
|
isomushgr.k |
⊢ 𝐾 = ( Edg ‘ 𝐵 ) |
5 |
|
eqid |
⊢ ( iEdg ‘ 𝐴 ) = ( iEdg ‘ 𝐴 ) |
6 |
|
eqid |
⊢ ( iEdg ‘ 𝐵 ) = ( iEdg ‘ 𝐵 ) |
7 |
1 2 5 6
|
isomgr |
⊢ ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) → ( 𝐴 IsomGr 𝐵 ↔ ∃ 𝑓 ( 𝑓 : 𝑉 –1-1-onto→ 𝑊 ∧ ∃ ℎ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) ) ) |
8 |
|
fvex |
⊢ ( iEdg ‘ 𝐵 ) ∈ V |
9 |
|
vex |
⊢ ℎ ∈ V |
10 |
|
fvex |
⊢ ( iEdg ‘ 𝐴 ) ∈ V |
11 |
10
|
cnvex |
⊢ ◡ ( iEdg ‘ 𝐴 ) ∈ V |
12 |
9 11
|
coex |
⊢ ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) ∈ V |
13 |
8 12
|
coex |
⊢ ( ( iEdg ‘ 𝐵 ) ∘ ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) ) ∈ V |
14 |
13
|
a1i |
⊢ ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) → ( ( iEdg ‘ 𝐵 ) ∘ ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) ) ∈ V ) |
15 |
2 6
|
ushgrf |
⊢ ( 𝐵 ∈ USHGraph → ( iEdg ‘ 𝐵 ) : dom ( iEdg ‘ 𝐵 ) –1-1→ ( 𝒫 𝑊 ∖ { ∅ } ) ) |
16 |
|
f1f1orn |
⊢ ( ( iEdg ‘ 𝐵 ) : dom ( iEdg ‘ 𝐵 ) –1-1→ ( 𝒫 𝑊 ∖ { ∅ } ) → ( iEdg ‘ 𝐵 ) : dom ( iEdg ‘ 𝐵 ) –1-1-onto→ ran ( iEdg ‘ 𝐵 ) ) |
17 |
15 16
|
syl |
⊢ ( 𝐵 ∈ USHGraph → ( iEdg ‘ 𝐵 ) : dom ( iEdg ‘ 𝐵 ) –1-1-onto→ ran ( iEdg ‘ 𝐵 ) ) |
18 |
|
edgval |
⊢ ( Edg ‘ 𝐵 ) = ran ( iEdg ‘ 𝐵 ) |
19 |
4 18
|
eqtri |
⊢ 𝐾 = ran ( iEdg ‘ 𝐵 ) |
20 |
|
f1oeq3 |
⊢ ( 𝐾 = ran ( iEdg ‘ 𝐵 ) → ( ( iEdg ‘ 𝐵 ) : dom ( iEdg ‘ 𝐵 ) –1-1-onto→ 𝐾 ↔ ( iEdg ‘ 𝐵 ) : dom ( iEdg ‘ 𝐵 ) –1-1-onto→ ran ( iEdg ‘ 𝐵 ) ) ) |
21 |
19 20
|
ax-mp |
⊢ ( ( iEdg ‘ 𝐵 ) : dom ( iEdg ‘ 𝐵 ) –1-1-onto→ 𝐾 ↔ ( iEdg ‘ 𝐵 ) : dom ( iEdg ‘ 𝐵 ) –1-1-onto→ ran ( iEdg ‘ 𝐵 ) ) |
22 |
17 21
|
sylibr |
⊢ ( 𝐵 ∈ USHGraph → ( iEdg ‘ 𝐵 ) : dom ( iEdg ‘ 𝐵 ) –1-1-onto→ 𝐾 ) |
23 |
22
|
ad3antlr |
⊢ ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) → ( iEdg ‘ 𝐵 ) : dom ( iEdg ‘ 𝐵 ) –1-1-onto→ 𝐾 ) |
24 |
|
simprl |
⊢ ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) → ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ) |
25 |
1 5
|
ushgrf |
⊢ ( 𝐴 ∈ USHGraph → ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) –1-1→ ( 𝒫 𝑉 ∖ { ∅ } ) ) |
26 |
|
f1f1orn |
⊢ ( ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) –1-1→ ( 𝒫 𝑉 ∖ { ∅ } ) → ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ ran ( iEdg ‘ 𝐴 ) ) |
27 |
25 26
|
syl |
⊢ ( 𝐴 ∈ USHGraph → ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ ran ( iEdg ‘ 𝐴 ) ) |
28 |
|
edgval |
⊢ ( Edg ‘ 𝐴 ) = ran ( iEdg ‘ 𝐴 ) |
29 |
3 28
|
eqtri |
⊢ 𝐸 = ran ( iEdg ‘ 𝐴 ) |
30 |
|
f1oeq3 |
⊢ ( 𝐸 = ran ( iEdg ‘ 𝐴 ) → ( ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ 𝐸 ↔ ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ ran ( iEdg ‘ 𝐴 ) ) ) |
31 |
29 30
|
ax-mp |
⊢ ( ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ 𝐸 ↔ ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ ran ( iEdg ‘ 𝐴 ) ) |
32 |
27 31
|
sylibr |
⊢ ( 𝐴 ∈ USHGraph → ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ 𝐸 ) |
33 |
|
f1ocnv |
⊢ ( ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ 𝐸 → ◡ ( iEdg ‘ 𝐴 ) : 𝐸 –1-1-onto→ dom ( iEdg ‘ 𝐴 ) ) |
34 |
32 33
|
syl |
⊢ ( 𝐴 ∈ USHGraph → ◡ ( iEdg ‘ 𝐴 ) : 𝐸 –1-1-onto→ dom ( iEdg ‘ 𝐴 ) ) |
35 |
34
|
ad3antrrr |
⊢ ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) → ◡ ( iEdg ‘ 𝐴 ) : 𝐸 –1-1-onto→ dom ( iEdg ‘ 𝐴 ) ) |
36 |
|
f1oco |
⊢ ( ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ◡ ( iEdg ‘ 𝐴 ) : 𝐸 –1-1-onto→ dom ( iEdg ‘ 𝐴 ) ) → ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) : 𝐸 –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ) |
37 |
24 35 36
|
syl2anc |
⊢ ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) → ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) : 𝐸 –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ) |
38 |
|
f1oco |
⊢ ( ( ( iEdg ‘ 𝐵 ) : dom ( iEdg ‘ 𝐵 ) –1-1-onto→ 𝐾 ∧ ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) : 𝐸 –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ) → ( ( iEdg ‘ 𝐵 ) ∘ ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) ) : 𝐸 –1-1-onto→ 𝐾 ) |
39 |
23 37 38
|
syl2anc |
⊢ ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) → ( ( iEdg ‘ 𝐵 ) ∘ ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) ) : 𝐸 –1-1-onto→ 𝐾 ) |
40 |
29
|
eleq2i |
⊢ ( 𝑒 ∈ 𝐸 ↔ 𝑒 ∈ ran ( iEdg ‘ 𝐴 ) ) |
41 |
|
f1fn |
⊢ ( ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) –1-1→ ( 𝒫 𝑉 ∖ { ∅ } ) → ( iEdg ‘ 𝐴 ) Fn dom ( iEdg ‘ 𝐴 ) ) |
42 |
25 41
|
syl |
⊢ ( 𝐴 ∈ USHGraph → ( iEdg ‘ 𝐴 ) Fn dom ( iEdg ‘ 𝐴 ) ) |
43 |
|
fvelrnb |
⊢ ( ( iEdg ‘ 𝐴 ) Fn dom ( iEdg ‘ 𝐴 ) → ( 𝑒 ∈ ran ( iEdg ‘ 𝐴 ) ↔ ∃ 𝑗 ∈ dom ( iEdg ‘ 𝐴 ) ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) = 𝑒 ) ) |
44 |
42 43
|
syl |
⊢ ( 𝐴 ∈ USHGraph → ( 𝑒 ∈ ran ( iEdg ‘ 𝐴 ) ↔ ∃ 𝑗 ∈ dom ( iEdg ‘ 𝐴 ) ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) = 𝑒 ) ) |
45 |
44
|
ad3antrrr |
⊢ ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) → ( 𝑒 ∈ ran ( iEdg ‘ 𝐴 ) ↔ ∃ 𝑗 ∈ dom ( iEdg ‘ 𝐴 ) ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) = 𝑒 ) ) |
46 |
|
fveq2 |
⊢ ( 𝑖 = 𝑗 → ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) = ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) ) |
47 |
46
|
imaeq2d |
⊢ ( 𝑖 = 𝑗 → ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) ) ) |
48 |
|
2fveq3 |
⊢ ( 𝑖 = 𝑗 → ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑗 ) ) ) |
49 |
47 48
|
eqeq12d |
⊢ ( 𝑖 = 𝑗 → ( ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ↔ ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑗 ) ) ) ) |
50 |
49
|
rspccv |
⊢ ( ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) → ( 𝑗 ∈ dom ( iEdg ‘ 𝐴 ) → ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑗 ) ) ) ) |
51 |
50
|
ad2antll |
⊢ ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) → ( 𝑗 ∈ dom ( iEdg ‘ 𝐴 ) → ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑗 ) ) ) ) |
52 |
51
|
imp |
⊢ ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) ∧ 𝑗 ∈ dom ( iEdg ‘ 𝐴 ) ) → ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑗 ) ) ) |
53 |
|
coass |
⊢ ( ( ( iEdg ‘ 𝐵 ) ∘ ℎ ) ∘ ◡ ( iEdg ‘ 𝐴 ) ) = ( ( iEdg ‘ 𝐵 ) ∘ ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) ) |
54 |
53
|
eqcomi |
⊢ ( ( iEdg ‘ 𝐵 ) ∘ ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) ) = ( ( ( iEdg ‘ 𝐵 ) ∘ ℎ ) ∘ ◡ ( iEdg ‘ 𝐴 ) ) |
55 |
54
|
fveq1i |
⊢ ( ( ( iEdg ‘ 𝐵 ) ∘ ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) ) ‘ ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) ) = ( ( ( ( iEdg ‘ 𝐵 ) ∘ ℎ ) ∘ ◡ ( iEdg ‘ 𝐴 ) ) ‘ ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) ) |
56 |
|
dff1o4 |
⊢ ( ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ ran ( iEdg ‘ 𝐴 ) ↔ ( ( iEdg ‘ 𝐴 ) Fn dom ( iEdg ‘ 𝐴 ) ∧ ◡ ( iEdg ‘ 𝐴 ) Fn ran ( iEdg ‘ 𝐴 ) ) ) |
57 |
27 56
|
sylib |
⊢ ( 𝐴 ∈ USHGraph → ( ( iEdg ‘ 𝐴 ) Fn dom ( iEdg ‘ 𝐴 ) ∧ ◡ ( iEdg ‘ 𝐴 ) Fn ran ( iEdg ‘ 𝐴 ) ) ) |
58 |
57
|
simprd |
⊢ ( 𝐴 ∈ USHGraph → ◡ ( iEdg ‘ 𝐴 ) Fn ran ( iEdg ‘ 𝐴 ) ) |
59 |
58
|
ad4antr |
⊢ ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) ∧ 𝑗 ∈ dom ( iEdg ‘ 𝐴 ) ) → ◡ ( iEdg ‘ 𝐴 ) Fn ran ( iEdg ‘ 𝐴 ) ) |
60 |
|
f1of |
⊢ ( ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ ran ( iEdg ‘ 𝐴 ) → ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) ⟶ ran ( iEdg ‘ 𝐴 ) ) |
61 |
27 60
|
syl |
⊢ ( 𝐴 ∈ USHGraph → ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) ⟶ ran ( iEdg ‘ 𝐴 ) ) |
62 |
61
|
ad3antrrr |
⊢ ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) → ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) ⟶ ran ( iEdg ‘ 𝐴 ) ) |
63 |
62
|
ffvelrnda |
⊢ ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) ∧ 𝑗 ∈ dom ( iEdg ‘ 𝐴 ) ) → ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) ∈ ran ( iEdg ‘ 𝐴 ) ) |
64 |
|
fvco2 |
⊢ ( ( ◡ ( iEdg ‘ 𝐴 ) Fn ran ( iEdg ‘ 𝐴 ) ∧ ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) ∈ ran ( iEdg ‘ 𝐴 ) ) → ( ( ( ( iEdg ‘ 𝐵 ) ∘ ℎ ) ∘ ◡ ( iEdg ‘ 𝐴 ) ) ‘ ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) ) = ( ( ( iEdg ‘ 𝐵 ) ∘ ℎ ) ‘ ( ◡ ( iEdg ‘ 𝐴 ) ‘ ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) ) ) ) |
65 |
59 63 64
|
syl2anc |
⊢ ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) ∧ 𝑗 ∈ dom ( iEdg ‘ 𝐴 ) ) → ( ( ( ( iEdg ‘ 𝐵 ) ∘ ℎ ) ∘ ◡ ( iEdg ‘ 𝐴 ) ) ‘ ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) ) = ( ( ( iEdg ‘ 𝐵 ) ∘ ℎ ) ‘ ( ◡ ( iEdg ‘ 𝐴 ) ‘ ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) ) ) ) |
66 |
32
|
ad3antrrr |
⊢ ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) → ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ 𝐸 ) |
67 |
|
f1ocnvfv1 |
⊢ ( ( ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ 𝐸 ∧ 𝑗 ∈ dom ( iEdg ‘ 𝐴 ) ) → ( ◡ ( iEdg ‘ 𝐴 ) ‘ ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) ) = 𝑗 ) |
68 |
66 67
|
sylan |
⊢ ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) ∧ 𝑗 ∈ dom ( iEdg ‘ 𝐴 ) ) → ( ◡ ( iEdg ‘ 𝐴 ) ‘ ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) ) = 𝑗 ) |
69 |
68
|
fveq2d |
⊢ ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) ∧ 𝑗 ∈ dom ( iEdg ‘ 𝐴 ) ) → ( ( ( iEdg ‘ 𝐵 ) ∘ ℎ ) ‘ ( ◡ ( iEdg ‘ 𝐴 ) ‘ ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) ) ) = ( ( ( iEdg ‘ 𝐵 ) ∘ ℎ ) ‘ 𝑗 ) ) |
70 |
|
f1ofn |
⊢ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) → ℎ Fn dom ( iEdg ‘ 𝐴 ) ) |
71 |
70
|
ad2antrl |
⊢ ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) → ℎ Fn dom ( iEdg ‘ 𝐴 ) ) |
72 |
|
fvco2 |
⊢ ( ( ℎ Fn dom ( iEdg ‘ 𝐴 ) ∧ 𝑗 ∈ dom ( iEdg ‘ 𝐴 ) ) → ( ( ( iEdg ‘ 𝐵 ) ∘ ℎ ) ‘ 𝑗 ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑗 ) ) ) |
73 |
71 72
|
sylan |
⊢ ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) ∧ 𝑗 ∈ dom ( iEdg ‘ 𝐴 ) ) → ( ( ( iEdg ‘ 𝐵 ) ∘ ℎ ) ‘ 𝑗 ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑗 ) ) ) |
74 |
65 69 73
|
3eqtrd |
⊢ ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) ∧ 𝑗 ∈ dom ( iEdg ‘ 𝐴 ) ) → ( ( ( ( iEdg ‘ 𝐵 ) ∘ ℎ ) ∘ ◡ ( iEdg ‘ 𝐴 ) ) ‘ ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑗 ) ) ) |
75 |
55 74
|
eqtr2id |
⊢ ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) ∧ 𝑗 ∈ dom ( iEdg ‘ 𝐴 ) ) → ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑗 ) ) = ( ( ( iEdg ‘ 𝐵 ) ∘ ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) ) ‘ ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) ) ) |
76 |
75
|
ad2antrr |
⊢ ( ( ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) ∧ 𝑗 ∈ dom ( iEdg ‘ 𝐴 ) ) ∧ ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑗 ) ) ) ∧ ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) = 𝑒 ) → ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑗 ) ) = ( ( ( iEdg ‘ 𝐵 ) ∘ ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) ) ‘ ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) ) ) |
77 |
|
imaeq2 |
⊢ ( 𝑒 = ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) → ( 𝑓 “ 𝑒 ) = ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) ) ) |
78 |
77
|
eqcoms |
⊢ ( ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) = 𝑒 → ( 𝑓 “ 𝑒 ) = ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) ) ) |
79 |
|
simpr |
⊢ ( ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) ∧ 𝑗 ∈ dom ( iEdg ‘ 𝐴 ) ) ∧ ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑗 ) ) ) → ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑗 ) ) ) |
80 |
78 79
|
sylan9eqr |
⊢ ( ( ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) ∧ 𝑗 ∈ dom ( iEdg ‘ 𝐴 ) ) ∧ ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑗 ) ) ) ∧ ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) = 𝑒 ) → ( 𝑓 “ 𝑒 ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑗 ) ) ) |
81 |
|
fveq2 |
⊢ ( 𝑒 = ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) → ( ( ( iEdg ‘ 𝐵 ) ∘ ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) ) ‘ 𝑒 ) = ( ( ( iEdg ‘ 𝐵 ) ∘ ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) ) ‘ ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) ) ) |
82 |
81
|
eqcoms |
⊢ ( ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) = 𝑒 → ( ( ( iEdg ‘ 𝐵 ) ∘ ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) ) ‘ 𝑒 ) = ( ( ( iEdg ‘ 𝐵 ) ∘ ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) ) ‘ ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) ) ) |
83 |
82
|
adantl |
⊢ ( ( ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) ∧ 𝑗 ∈ dom ( iEdg ‘ 𝐴 ) ) ∧ ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑗 ) ) ) ∧ ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) = 𝑒 ) → ( ( ( iEdg ‘ 𝐵 ) ∘ ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) ) ‘ 𝑒 ) = ( ( ( iEdg ‘ 𝐵 ) ∘ ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) ) ‘ ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) ) ) |
84 |
76 80 83
|
3eqtr4d |
⊢ ( ( ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) ∧ 𝑗 ∈ dom ( iEdg ‘ 𝐴 ) ) ∧ ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑗 ) ) ) ∧ ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) = 𝑒 ) → ( 𝑓 “ 𝑒 ) = ( ( ( iEdg ‘ 𝐵 ) ∘ ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) ) ‘ 𝑒 ) ) |
85 |
84
|
ex |
⊢ ( ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) ∧ 𝑗 ∈ dom ( iEdg ‘ 𝐴 ) ) ∧ ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑗 ) ) ) → ( ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) = 𝑒 → ( 𝑓 “ 𝑒 ) = ( ( ( iEdg ‘ 𝐵 ) ∘ ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) ) ‘ 𝑒 ) ) ) |
86 |
52 85
|
mpdan |
⊢ ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) ∧ 𝑗 ∈ dom ( iEdg ‘ 𝐴 ) ) → ( ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) = 𝑒 → ( 𝑓 “ 𝑒 ) = ( ( ( iEdg ‘ 𝐵 ) ∘ ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) ) ‘ 𝑒 ) ) ) |
87 |
86
|
rexlimdva |
⊢ ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) → ( ∃ 𝑗 ∈ dom ( iEdg ‘ 𝐴 ) ( ( iEdg ‘ 𝐴 ) ‘ 𝑗 ) = 𝑒 → ( 𝑓 “ 𝑒 ) = ( ( ( iEdg ‘ 𝐵 ) ∘ ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) ) ‘ 𝑒 ) ) ) |
88 |
45 87
|
sylbid |
⊢ ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) → ( 𝑒 ∈ ran ( iEdg ‘ 𝐴 ) → ( 𝑓 “ 𝑒 ) = ( ( ( iEdg ‘ 𝐵 ) ∘ ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) ) ‘ 𝑒 ) ) ) |
89 |
40 88
|
syl5bi |
⊢ ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) → ( 𝑒 ∈ 𝐸 → ( 𝑓 “ 𝑒 ) = ( ( ( iEdg ‘ 𝐵 ) ∘ ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) ) ‘ 𝑒 ) ) ) |
90 |
89
|
ralrimiv |
⊢ ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) → ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( ( ( iEdg ‘ 𝐵 ) ∘ ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) ) ‘ 𝑒 ) ) |
91 |
39 90
|
jca |
⊢ ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) → ( ( ( iEdg ‘ 𝐵 ) ∘ ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) ) : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( ( ( iEdg ‘ 𝐵 ) ∘ ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) ) ‘ 𝑒 ) ) ) |
92 |
|
f1oeq1 |
⊢ ( 𝑔 = ( ( iEdg ‘ 𝐵 ) ∘ ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) ) → ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ↔ ( ( iEdg ‘ 𝐵 ) ∘ ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) ) : 𝐸 –1-1-onto→ 𝐾 ) ) |
93 |
|
fveq1 |
⊢ ( 𝑔 = ( ( iEdg ‘ 𝐵 ) ∘ ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) ) → ( 𝑔 ‘ 𝑒 ) = ( ( ( iEdg ‘ 𝐵 ) ∘ ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) ) ‘ 𝑒 ) ) |
94 |
93
|
eqeq2d |
⊢ ( 𝑔 = ( ( iEdg ‘ 𝐵 ) ∘ ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) ) → ( ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ↔ ( 𝑓 “ 𝑒 ) = ( ( ( iEdg ‘ 𝐵 ) ∘ ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) ) ‘ 𝑒 ) ) ) |
95 |
94
|
ralbidv |
⊢ ( 𝑔 = ( ( iEdg ‘ 𝐵 ) ∘ ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) ) → ( ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ↔ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( ( ( iEdg ‘ 𝐵 ) ∘ ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) ) ‘ 𝑒 ) ) ) |
96 |
92 95
|
anbi12d |
⊢ ( 𝑔 = ( ( iEdg ‘ 𝐵 ) ∘ ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) ) → ( ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ↔ ( ( ( iEdg ‘ 𝐵 ) ∘ ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) ) : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( ( ( iEdg ‘ 𝐵 ) ∘ ( ℎ ∘ ◡ ( iEdg ‘ 𝐴 ) ) ) ‘ 𝑒 ) ) ) ) |
97 |
14 91 96
|
spcedv |
⊢ ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) → ∃ 𝑔 ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) |
98 |
97
|
ex |
⊢ ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) → ( ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) → ∃ 𝑔 ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ) |
99 |
98
|
exlimdv |
⊢ ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) → ( ∃ ℎ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) → ∃ 𝑔 ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ) |
100 |
8
|
cnvex |
⊢ ◡ ( iEdg ‘ 𝐵 ) ∈ V |
101 |
|
vex |
⊢ 𝑔 ∈ V |
102 |
101 10
|
coex |
⊢ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ∈ V |
103 |
100 102
|
coex |
⊢ ( ◡ ( iEdg ‘ 𝐵 ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) ∈ V |
104 |
103
|
a1i |
⊢ ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) → ( ◡ ( iEdg ‘ 𝐵 ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) ∈ V ) |
105 |
17
|
ad3antlr |
⊢ ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) → ( iEdg ‘ 𝐵 ) : dom ( iEdg ‘ 𝐵 ) –1-1-onto→ ran ( iEdg ‘ 𝐵 ) ) |
106 |
|
f1ocnv |
⊢ ( ( iEdg ‘ 𝐵 ) : dom ( iEdg ‘ 𝐵 ) –1-1-onto→ ran ( iEdg ‘ 𝐵 ) → ◡ ( iEdg ‘ 𝐵 ) : ran ( iEdg ‘ 𝐵 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ) |
107 |
105 106
|
syl |
⊢ ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) → ◡ ( iEdg ‘ 𝐵 ) : ran ( iEdg ‘ 𝐵 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ) |
108 |
|
f1oeq23 |
⊢ ( ( 𝐸 = ran ( iEdg ‘ 𝐴 ) ∧ 𝐾 = ran ( iEdg ‘ 𝐵 ) ) → ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ↔ 𝑔 : ran ( iEdg ‘ 𝐴 ) –1-1-onto→ ran ( iEdg ‘ 𝐵 ) ) ) |
109 |
29 19 108
|
mp2an |
⊢ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ↔ 𝑔 : ran ( iEdg ‘ 𝐴 ) –1-1-onto→ ran ( iEdg ‘ 𝐵 ) ) |
110 |
109
|
biimpi |
⊢ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 → 𝑔 : ran ( iEdg ‘ 𝐴 ) –1-1-onto→ ran ( iEdg ‘ 𝐵 ) ) |
111 |
110
|
ad2antrl |
⊢ ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) → 𝑔 : ran ( iEdg ‘ 𝐴 ) –1-1-onto→ ran ( iEdg ‘ 𝐵 ) ) |
112 |
27
|
ad3antrrr |
⊢ ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) → ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ ran ( iEdg ‘ 𝐴 ) ) |
113 |
|
f1oco |
⊢ ( ( 𝑔 : ran ( iEdg ‘ 𝐴 ) –1-1-onto→ ran ( iEdg ‘ 𝐵 ) ∧ ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ ran ( iEdg ‘ 𝐴 ) ) → ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ ran ( iEdg ‘ 𝐵 ) ) |
114 |
111 112 113
|
syl2anc |
⊢ ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) → ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ ran ( iEdg ‘ 𝐵 ) ) |
115 |
|
f1oco |
⊢ ( ( ◡ ( iEdg ‘ 𝐵 ) : ran ( iEdg ‘ 𝐵 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ ran ( iEdg ‘ 𝐵 ) ) → ( ◡ ( iEdg ‘ 𝐵 ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ) |
116 |
107 114 115
|
syl2anc |
⊢ ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) → ( ◡ ( iEdg ‘ 𝐵 ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ) |
117 |
61
|
ad2antrr |
⊢ ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) → ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) ⟶ ran ( iEdg ‘ 𝐴 ) ) |
118 |
117
|
ffund |
⊢ ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) → Fun ( iEdg ‘ 𝐴 ) ) |
119 |
118
|
adantr |
⊢ ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) → Fun ( iEdg ‘ 𝐴 ) ) |
120 |
|
fvelrn |
⊢ ( ( Fun ( iEdg ‘ 𝐴 ) ∧ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ) → ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ∈ ran ( iEdg ‘ 𝐴 ) ) |
121 |
119 120
|
sylan |
⊢ ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ) → ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ∈ ran ( iEdg ‘ 𝐴 ) ) |
122 |
29
|
raleqi |
⊢ ( ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ↔ ∀ 𝑒 ∈ ran ( iEdg ‘ 𝐴 ) ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) |
123 |
122
|
biimpi |
⊢ ( ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) → ∀ 𝑒 ∈ ran ( iEdg ‘ 𝐴 ) ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) |
124 |
123
|
ad2antll |
⊢ ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) → ∀ 𝑒 ∈ ran ( iEdg ‘ 𝐴 ) ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) |
125 |
124
|
adantr |
⊢ ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ) → ∀ 𝑒 ∈ ran ( iEdg ‘ 𝐴 ) ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) |
126 |
|
imaeq2 |
⊢ ( 𝑒 = ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) → ( 𝑓 “ 𝑒 ) = ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) ) |
127 |
|
fveq2 |
⊢ ( 𝑒 = ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) → ( 𝑔 ‘ 𝑒 ) = ( 𝑔 ‘ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) ) |
128 |
126 127
|
eqeq12d |
⊢ ( 𝑒 = ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) → ( ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ↔ ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( 𝑔 ‘ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) ) ) |
129 |
128
|
rspccva |
⊢ ( ( ∀ 𝑒 ∈ ran ( iEdg ‘ 𝐴 ) ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ∧ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ∈ ran ( iEdg ‘ 𝐴 ) ) → ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( 𝑔 ‘ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) ) |
130 |
125 129
|
sylan |
⊢ ( ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ) ∧ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ∈ ran ( iEdg ‘ 𝐴 ) ) → ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( 𝑔 ‘ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) ) |
131 |
|
feq3 |
⊢ ( 𝐸 = ran ( iEdg ‘ 𝐴 ) → ( ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) ⟶ 𝐸 ↔ ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) ⟶ ran ( iEdg ‘ 𝐴 ) ) ) |
132 |
29 131
|
ax-mp |
⊢ ( ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) ⟶ 𝐸 ↔ ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) ⟶ ran ( iEdg ‘ 𝐴 ) ) |
133 |
61 132
|
sylibr |
⊢ ( 𝐴 ∈ USHGraph → ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) ⟶ 𝐸 ) |
134 |
133
|
ad2antrr |
⊢ ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) → ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) ⟶ 𝐸 ) |
135 |
|
f1ofn |
⊢ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 → 𝑔 Fn 𝐸 ) |
136 |
135
|
adantr |
⊢ ( ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) → 𝑔 Fn 𝐸 ) |
137 |
134 136
|
anim12ci |
⊢ ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) → ( 𝑔 Fn 𝐸 ∧ ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) ⟶ 𝐸 ) ) |
138 |
137
|
ad2antrr |
⊢ ( ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ) ∧ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ∈ ran ( iEdg ‘ 𝐴 ) ) → ( 𝑔 Fn 𝐸 ∧ ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) ⟶ 𝐸 ) ) |
139 |
|
fnfco |
⊢ ( ( 𝑔 Fn 𝐸 ∧ ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) ⟶ 𝐸 ) → ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) Fn dom ( iEdg ‘ 𝐴 ) ) |
140 |
138 139
|
syl |
⊢ ( ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ) ∧ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ∈ ran ( iEdg ‘ 𝐴 ) ) → ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) Fn dom ( iEdg ‘ 𝐴 ) ) |
141 |
|
simplr |
⊢ ( ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ) ∧ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ∈ ran ( iEdg ‘ 𝐴 ) ) → 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ) |
142 |
|
fvco2 |
⊢ ( ( ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) Fn dom ( iEdg ‘ 𝐴 ) ∧ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ) → ( ( ( I ↾ ran ( iEdg ‘ 𝐵 ) ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) ‘ 𝑖 ) = ( ( I ↾ ran ( iEdg ‘ 𝐵 ) ) ‘ ( ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ‘ 𝑖 ) ) ) |
143 |
140 141 142
|
syl2anc |
⊢ ( ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ) ∧ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ∈ ran ( iEdg ‘ 𝐴 ) ) → ( ( ( I ↾ ran ( iEdg ‘ 𝐵 ) ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) ‘ 𝑖 ) = ( ( I ↾ ran ( iEdg ‘ 𝐵 ) ) ‘ ( ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ‘ 𝑖 ) ) ) |
144 |
|
f1of |
⊢ ( ( iEdg ‘ 𝐵 ) : dom ( iEdg ‘ 𝐵 ) –1-1-onto→ 𝐾 → ( iEdg ‘ 𝐵 ) : dom ( iEdg ‘ 𝐵 ) ⟶ 𝐾 ) |
145 |
22 144
|
syl |
⊢ ( 𝐵 ∈ USHGraph → ( iEdg ‘ 𝐵 ) : dom ( iEdg ‘ 𝐵 ) ⟶ 𝐾 ) |
146 |
145
|
ffund |
⊢ ( 𝐵 ∈ USHGraph → Fun ( iEdg ‘ 𝐵 ) ) |
147 |
|
funcocnv2 |
⊢ ( Fun ( iEdg ‘ 𝐵 ) → ( ( iEdg ‘ 𝐵 ) ∘ ◡ ( iEdg ‘ 𝐵 ) ) = ( I ↾ ran ( iEdg ‘ 𝐵 ) ) ) |
148 |
146 147
|
syl |
⊢ ( 𝐵 ∈ USHGraph → ( ( iEdg ‘ 𝐵 ) ∘ ◡ ( iEdg ‘ 𝐵 ) ) = ( I ↾ ran ( iEdg ‘ 𝐵 ) ) ) |
149 |
148
|
eqcomd |
⊢ ( 𝐵 ∈ USHGraph → ( I ↾ ran ( iEdg ‘ 𝐵 ) ) = ( ( iEdg ‘ 𝐵 ) ∘ ◡ ( iEdg ‘ 𝐵 ) ) ) |
150 |
149
|
ad5antlr |
⊢ ( ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ) ∧ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ∈ ran ( iEdg ‘ 𝐴 ) ) → ( I ↾ ran ( iEdg ‘ 𝐵 ) ) = ( ( iEdg ‘ 𝐵 ) ∘ ◡ ( iEdg ‘ 𝐵 ) ) ) |
151 |
150
|
coeq1d |
⊢ ( ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ) ∧ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ∈ ran ( iEdg ‘ 𝐴 ) ) → ( ( I ↾ ran ( iEdg ‘ 𝐵 ) ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) = ( ( ( iEdg ‘ 𝐵 ) ∘ ◡ ( iEdg ‘ 𝐵 ) ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) ) |
152 |
151
|
fveq1d |
⊢ ( ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ) ∧ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ∈ ran ( iEdg ‘ 𝐴 ) ) → ( ( ( I ↾ ran ( iEdg ‘ 𝐵 ) ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) ‘ 𝑖 ) = ( ( ( ( iEdg ‘ 𝐵 ) ∘ ◡ ( iEdg ‘ 𝐵 ) ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) ‘ 𝑖 ) ) |
153 |
|
coass |
⊢ ( ( ( iEdg ‘ 𝐵 ) ∘ ◡ ( iEdg ‘ 𝐵 ) ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) = ( ( iEdg ‘ 𝐵 ) ∘ ( ◡ ( iEdg ‘ 𝐵 ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) ) |
154 |
153
|
fveq1i |
⊢ ( ( ( ( iEdg ‘ 𝐵 ) ∘ ◡ ( iEdg ‘ 𝐵 ) ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) ‘ 𝑖 ) = ( ( ( iEdg ‘ 𝐵 ) ∘ ( ◡ ( iEdg ‘ 𝐵 ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) ) ‘ 𝑖 ) |
155 |
152 154
|
eqtrdi |
⊢ ( ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ) ∧ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ∈ ran ( iEdg ‘ 𝐴 ) ) → ( ( ( I ↾ ran ( iEdg ‘ 𝐵 ) ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) ‘ 𝑖 ) = ( ( ( iEdg ‘ 𝐵 ) ∘ ( ◡ ( iEdg ‘ 𝐵 ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) ) ‘ 𝑖 ) ) |
156 |
|
f1of |
⊢ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 → 𝑔 : 𝐸 ⟶ 𝐾 ) |
157 |
|
eqid |
⊢ 𝐸 = 𝐸 |
158 |
157 19
|
feq23i |
⊢ ( 𝑔 : 𝐸 ⟶ 𝐾 ↔ 𝑔 : 𝐸 ⟶ ran ( iEdg ‘ 𝐵 ) ) |
159 |
156 158
|
sylib |
⊢ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 → 𝑔 : 𝐸 ⟶ ran ( iEdg ‘ 𝐵 ) ) |
160 |
159
|
adantr |
⊢ ( ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) → 𝑔 : 𝐸 ⟶ ran ( iEdg ‘ 𝐵 ) ) |
161 |
|
f1of |
⊢ ( ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ 𝐸 → ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) ⟶ 𝐸 ) |
162 |
32 161
|
syl |
⊢ ( 𝐴 ∈ USHGraph → ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) ⟶ 𝐸 ) |
163 |
162
|
ad2antrr |
⊢ ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) → ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) ⟶ 𝐸 ) |
164 |
|
fco |
⊢ ( ( 𝑔 : 𝐸 ⟶ ran ( iEdg ‘ 𝐵 ) ∧ ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) ⟶ 𝐸 ) → ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) : dom ( iEdg ‘ 𝐴 ) ⟶ ran ( iEdg ‘ 𝐵 ) ) |
165 |
160 163 164
|
syl2anr |
⊢ ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) → ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) : dom ( iEdg ‘ 𝐴 ) ⟶ ran ( iEdg ‘ 𝐵 ) ) |
166 |
165
|
anim1i |
⊢ ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ) → ( ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) : dom ( iEdg ‘ 𝐴 ) ⟶ ran ( iEdg ‘ 𝐵 ) ∧ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ) ) |
167 |
166
|
adantr |
⊢ ( ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ) ∧ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ∈ ran ( iEdg ‘ 𝐴 ) ) → ( ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) : dom ( iEdg ‘ 𝐴 ) ⟶ ran ( iEdg ‘ 𝐵 ) ∧ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ) ) |
168 |
|
ffvelrn |
⊢ ( ( ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) : dom ( iEdg ‘ 𝐴 ) ⟶ ran ( iEdg ‘ 𝐵 ) ∧ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ) → ( ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ‘ 𝑖 ) ∈ ran ( iEdg ‘ 𝐵 ) ) |
169 |
167 168
|
syl |
⊢ ( ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ) ∧ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ∈ ran ( iEdg ‘ 𝐴 ) ) → ( ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ‘ 𝑖 ) ∈ ran ( iEdg ‘ 𝐵 ) ) |
170 |
|
fvresi |
⊢ ( ( ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ‘ 𝑖 ) ∈ ran ( iEdg ‘ 𝐵 ) → ( ( I ↾ ran ( iEdg ‘ 𝐵 ) ) ‘ ( ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ‘ 𝑖 ) ) = ( ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ‘ 𝑖 ) ) |
171 |
169 170
|
syl |
⊢ ( ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ) ∧ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ∈ ran ( iEdg ‘ 𝐴 ) ) → ( ( I ↾ ran ( iEdg ‘ 𝐵 ) ) ‘ ( ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ‘ 𝑖 ) ) = ( ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ‘ 𝑖 ) ) |
172 |
162
|
ad3antrrr |
⊢ ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) → ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) ⟶ 𝐸 ) |
173 |
172
|
anim1i |
⊢ ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ) → ( ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) ⟶ 𝐸 ∧ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ) ) |
174 |
173
|
adantr |
⊢ ( ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ) ∧ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ∈ ran ( iEdg ‘ 𝐴 ) ) → ( ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) ⟶ 𝐸 ∧ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ) ) |
175 |
|
fvco3 |
⊢ ( ( ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) ⟶ 𝐸 ∧ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ) → ( ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ‘ 𝑖 ) = ( 𝑔 ‘ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) ) |
176 |
174 175
|
syl |
⊢ ( ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ) ∧ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ∈ ran ( iEdg ‘ 𝐴 ) ) → ( ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ‘ 𝑖 ) = ( 𝑔 ‘ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) ) |
177 |
171 176
|
eqtrd |
⊢ ( ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ) ∧ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ∈ ran ( iEdg ‘ 𝐴 ) ) → ( ( I ↾ ran ( iEdg ‘ 𝐵 ) ) ‘ ( ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ‘ 𝑖 ) ) = ( 𝑔 ‘ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) ) |
178 |
143 155 177
|
3eqtr3rd |
⊢ ( ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ) ∧ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ∈ ran ( iEdg ‘ 𝐴 ) ) → ( 𝑔 ‘ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( ( iEdg ‘ 𝐵 ) ∘ ( ◡ ( iEdg ‘ 𝐵 ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) ) ‘ 𝑖 ) ) |
179 |
|
dff1o4 |
⊢ ( ( iEdg ‘ 𝐵 ) : dom ( iEdg ‘ 𝐵 ) –1-1-onto→ 𝐾 ↔ ( ( iEdg ‘ 𝐵 ) Fn dom ( iEdg ‘ 𝐵 ) ∧ ◡ ( iEdg ‘ 𝐵 ) Fn 𝐾 ) ) |
180 |
22 179
|
sylib |
⊢ ( 𝐵 ∈ USHGraph → ( ( iEdg ‘ 𝐵 ) Fn dom ( iEdg ‘ 𝐵 ) ∧ ◡ ( iEdg ‘ 𝐵 ) Fn 𝐾 ) ) |
181 |
180
|
simprd |
⊢ ( 𝐵 ∈ USHGraph → ◡ ( iEdg ‘ 𝐵 ) Fn 𝐾 ) |
182 |
181
|
ad5antlr |
⊢ ( ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ) ∧ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ∈ ran ( iEdg ‘ 𝐴 ) ) → ◡ ( iEdg ‘ 𝐵 ) Fn 𝐾 ) |
183 |
156
|
adantr |
⊢ ( ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) → 𝑔 : 𝐸 ⟶ 𝐾 ) |
184 |
134 183
|
anim12ci |
⊢ ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) → ( 𝑔 : 𝐸 ⟶ 𝐾 ∧ ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) ⟶ 𝐸 ) ) |
185 |
184
|
ad2antrr |
⊢ ( ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ) ∧ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ∈ ran ( iEdg ‘ 𝐴 ) ) → ( 𝑔 : 𝐸 ⟶ 𝐾 ∧ ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) ⟶ 𝐸 ) ) |
186 |
|
fco |
⊢ ( ( 𝑔 : 𝐸 ⟶ 𝐾 ∧ ( iEdg ‘ 𝐴 ) : dom ( iEdg ‘ 𝐴 ) ⟶ 𝐸 ) → ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) : dom ( iEdg ‘ 𝐴 ) ⟶ 𝐾 ) |
187 |
185 186
|
syl |
⊢ ( ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ) ∧ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ∈ ran ( iEdg ‘ 𝐴 ) ) → ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) : dom ( iEdg ‘ 𝐴 ) ⟶ 𝐾 ) |
188 |
|
fnfco |
⊢ ( ( ◡ ( iEdg ‘ 𝐵 ) Fn 𝐾 ∧ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) : dom ( iEdg ‘ 𝐴 ) ⟶ 𝐾 ) → ( ◡ ( iEdg ‘ 𝐵 ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) Fn dom ( iEdg ‘ 𝐴 ) ) |
189 |
182 187 188
|
syl2anc |
⊢ ( ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ) ∧ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ∈ ran ( iEdg ‘ 𝐴 ) ) → ( ◡ ( iEdg ‘ 𝐵 ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) Fn dom ( iEdg ‘ 𝐴 ) ) |
190 |
|
fvco2 |
⊢ ( ( ( ◡ ( iEdg ‘ 𝐵 ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) Fn dom ( iEdg ‘ 𝐴 ) ∧ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ) → ( ( ( iEdg ‘ 𝐵 ) ∘ ( ◡ ( iEdg ‘ 𝐵 ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) ) ‘ 𝑖 ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ( ◡ ( iEdg ‘ 𝐵 ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) ‘ 𝑖 ) ) ) |
191 |
189 141 190
|
syl2anc |
⊢ ( ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ) ∧ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ∈ ran ( iEdg ‘ 𝐴 ) ) → ( ( ( iEdg ‘ 𝐵 ) ∘ ( ◡ ( iEdg ‘ 𝐵 ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) ) ‘ 𝑖 ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ( ◡ ( iEdg ‘ 𝐵 ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) ‘ 𝑖 ) ) ) |
192 |
130 178 191
|
3eqtrd |
⊢ ( ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ) ∧ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ∈ ran ( iEdg ‘ 𝐴 ) ) → ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ( ◡ ( iEdg ‘ 𝐵 ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) ‘ 𝑖 ) ) ) |
193 |
121 192
|
mpdan |
⊢ ( ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ) → ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ( ◡ ( iEdg ‘ 𝐵 ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) ‘ 𝑖 ) ) ) |
194 |
193
|
ralrimiva |
⊢ ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) → ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ( ◡ ( iEdg ‘ 𝐵 ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) ‘ 𝑖 ) ) ) |
195 |
116 194
|
jca |
⊢ ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) → ( ( ◡ ( iEdg ‘ 𝐵 ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ( ◡ ( iEdg ‘ 𝐵 ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) ‘ 𝑖 ) ) ) ) |
196 |
|
f1oeq1 |
⊢ ( ℎ = ( ◡ ( iEdg ‘ 𝐵 ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) → ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ↔ ( ◡ ( iEdg ‘ 𝐵 ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ) ) |
197 |
|
fveq1 |
⊢ ( ℎ = ( ◡ ( iEdg ‘ 𝐵 ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) → ( ℎ ‘ 𝑖 ) = ( ( ◡ ( iEdg ‘ 𝐵 ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) ‘ 𝑖 ) ) |
198 |
197
|
fveq2d |
⊢ ( ℎ = ( ◡ ( iEdg ‘ 𝐵 ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) → ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ( ◡ ( iEdg ‘ 𝐵 ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) ‘ 𝑖 ) ) ) |
199 |
198
|
eqeq2d |
⊢ ( ℎ = ( ◡ ( iEdg ‘ 𝐵 ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) → ( ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ↔ ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ( ◡ ( iEdg ‘ 𝐵 ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) ‘ 𝑖 ) ) ) ) |
200 |
199
|
ralbidv |
⊢ ( ℎ = ( ◡ ( iEdg ‘ 𝐵 ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) → ( ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ↔ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ( ◡ ( iEdg ‘ 𝐵 ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) ‘ 𝑖 ) ) ) ) |
201 |
196 200
|
anbi12d |
⊢ ( ℎ = ( ◡ ( iEdg ‘ 𝐵 ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) → ( ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ↔ ( ( ◡ ( iEdg ‘ 𝐵 ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ( ◡ ( iEdg ‘ 𝐵 ) ∘ ( 𝑔 ∘ ( iEdg ‘ 𝐴 ) ) ) ‘ 𝑖 ) ) ) ) ) |
202 |
104 195 201
|
spcedv |
⊢ ( ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) → ∃ ℎ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) |
203 |
202
|
ex |
⊢ ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) → ( ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) → ∃ ℎ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) ) |
204 |
203
|
exlimdv |
⊢ ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) → ( ∃ 𝑔 ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) → ∃ ℎ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) ) |
205 |
99 204
|
impbid |
⊢ ( ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) → ( ∃ ℎ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ↔ ∃ 𝑔 ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ) |
206 |
205
|
pm5.32da |
⊢ ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) → ( ( 𝑓 : 𝑉 –1-1-onto→ 𝑊 ∧ ∃ ℎ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) ↔ ( 𝑓 : 𝑉 –1-1-onto→ 𝑊 ∧ ∃ 𝑔 ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ) ) |
207 |
206
|
exbidv |
⊢ ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) → ( ∃ 𝑓 ( 𝑓 : 𝑉 –1-1-onto→ 𝑊 ∧ ∃ ℎ ( ℎ : dom ( iEdg ‘ 𝐴 ) –1-1-onto→ dom ( iEdg ‘ 𝐵 ) ∧ ∀ 𝑖 ∈ dom ( iEdg ‘ 𝐴 ) ( 𝑓 “ ( ( iEdg ‘ 𝐴 ) ‘ 𝑖 ) ) = ( ( iEdg ‘ 𝐵 ) ‘ ( ℎ ‘ 𝑖 ) ) ) ) ↔ ∃ 𝑓 ( 𝑓 : 𝑉 –1-1-onto→ 𝑊 ∧ ∃ 𝑔 ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ) ) |
208 |
7 207
|
bitrd |
⊢ ( ( 𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph ) → ( 𝐴 IsomGr 𝐵 ↔ ∃ 𝑓 ( 𝑓 : 𝑉 –1-1-onto→ 𝑊 ∧ ∃ 𝑔 ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ) ) |