| 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
|
syl5req |
⊢ ( ( ( ( ( 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 ) ) ) ) ) |