| 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 |
|
simpl |
⊢ ( ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) → 𝑔 : 𝐸 –1-1-onto→ 𝐾 ) |
| 6 |
5
|
ad2antlr |
⊢ ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) → 𝑔 : 𝐸 –1-1-onto→ 𝐾 ) |
| 7 |
|
f1ocnvdm |
⊢ ( ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) → ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 ) |
| 8 |
6 7
|
sylan |
⊢ ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) → ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 ) |
| 9 |
|
uspgrupgr |
⊢ ( 𝐴 ∈ USPGraph → 𝐴 ∈ UPGraph ) |
| 10 |
9
|
adantr |
⊢ ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) → 𝐴 ∈ UPGraph ) |
| 11 |
10
|
ad4antr |
⊢ ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) → 𝐴 ∈ UPGraph ) |
| 12 |
1 3
|
upgredg |
⊢ ( ( 𝐴 ∈ UPGraph ∧ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 ) → ∃ 𝑥 ∈ 𝑉 ∃ 𝑦 ∈ 𝑉 ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) = { 𝑥 , 𝑦 } ) |
| 13 |
11 12
|
sylan |
⊢ ( ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) ∧ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 ) → ∃ 𝑥 ∈ 𝑉 ∃ 𝑦 ∈ 𝑉 ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) = { 𝑥 , 𝑦 } ) |
| 14 |
|
eleq1 |
⊢ ( ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) = { 𝑥 , 𝑦 } → ( ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 ↔ { 𝑥 , 𝑦 } ∈ 𝐸 ) ) |
| 15 |
14
|
biimpd |
⊢ ( ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) = { 𝑥 , 𝑦 } → ( ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 → { 𝑥 , 𝑦 } ∈ 𝐸 ) ) |
| 16 |
15
|
com12 |
⊢ ( ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 → ( ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) = { 𝑥 , 𝑦 } → { 𝑥 , 𝑦 } ∈ 𝐸 ) ) |
| 17 |
16
|
ad2antlr |
⊢ ( ( ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) ∧ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) → ( ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) = { 𝑥 , 𝑦 } → { 𝑥 , 𝑦 } ∈ 𝐸 ) ) |
| 18 |
17
|
imp |
⊢ ( ( ( ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) ∧ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) = { 𝑥 , 𝑦 } ) → { 𝑥 , 𝑦 } ∈ 𝐸 ) |
| 19 |
5
|
ad6antlr |
⊢ ( ( ( ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) ∧ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ { 𝑥 , 𝑦 } ∈ 𝐸 ) → 𝑔 : 𝐸 –1-1-onto→ 𝐾 ) |
| 20 |
|
simpr |
⊢ ( ( ( ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) ∧ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ { 𝑥 , 𝑦 } ∈ 𝐸 ) → { 𝑥 , 𝑦 } ∈ 𝐸 ) |
| 21 |
|
simpr |
⊢ ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) → { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) |
| 22 |
21
|
ad5ant12 |
⊢ ( ( ( ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) ∧ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ { 𝑥 , 𝑦 } ∈ 𝐸 ) → { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) |
| 23 |
19 20 22
|
3jca |
⊢ ( ( ( ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) ∧ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ { 𝑥 , 𝑦 } ∈ 𝐸 ) → ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ { 𝑥 , 𝑦 } ∈ 𝐸 ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) ) |
| 24 |
|
f1ocnvfvb |
⊢ ( ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ { 𝑥 , 𝑦 } ∈ 𝐸 ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) → ( ( 𝑔 ‘ { 𝑥 , 𝑦 } ) = { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ↔ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) = { 𝑥 , 𝑦 } ) ) |
| 25 |
23 24
|
syl |
⊢ ( ( ( ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) ∧ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ { 𝑥 , 𝑦 } ∈ 𝐸 ) → ( ( 𝑔 ‘ { 𝑥 , 𝑦 } ) = { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ↔ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) = { 𝑥 , 𝑦 } ) ) |
| 26 |
|
imaeq2 |
⊢ ( 𝑒 = { 𝑥 , 𝑦 } → ( 𝑓 “ 𝑒 ) = ( 𝑓 “ { 𝑥 , 𝑦 } ) ) |
| 27 |
|
fveq2 |
⊢ ( 𝑒 = { 𝑥 , 𝑦 } → ( 𝑔 ‘ 𝑒 ) = ( 𝑔 ‘ { 𝑥 , 𝑦 } ) ) |
| 28 |
26 27
|
eqeq12d |
⊢ ( 𝑒 = { 𝑥 , 𝑦 } → ( ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ↔ ( 𝑓 “ { 𝑥 , 𝑦 } ) = ( 𝑔 ‘ { 𝑥 , 𝑦 } ) ) ) |
| 29 |
28
|
rspccv |
⊢ ( ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) → ( { 𝑥 , 𝑦 } ∈ 𝐸 → ( 𝑓 “ { 𝑥 , 𝑦 } ) = ( 𝑔 ‘ { 𝑥 , 𝑦 } ) ) ) |
| 30 |
29
|
adantl |
⊢ ( ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) → ( { 𝑥 , 𝑦 } ∈ 𝐸 → ( 𝑓 “ { 𝑥 , 𝑦 } ) = ( 𝑔 ‘ { 𝑥 , 𝑦 } ) ) ) |
| 31 |
30
|
adantl |
⊢ ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) → ( { 𝑥 , 𝑦 } ∈ 𝐸 → ( 𝑓 “ { 𝑥 , 𝑦 } ) = ( 𝑔 ‘ { 𝑥 , 𝑦 } ) ) ) |
| 32 |
31
|
ad4antr |
⊢ ( ( ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) ∧ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) → ( { 𝑥 , 𝑦 } ∈ 𝐸 → ( 𝑓 “ { 𝑥 , 𝑦 } ) = ( 𝑔 ‘ { 𝑥 , 𝑦 } ) ) ) |
| 33 |
32
|
imp |
⊢ ( ( ( ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) ∧ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ { 𝑥 , 𝑦 } ∈ 𝐸 ) → ( 𝑓 “ { 𝑥 , 𝑦 } ) = ( 𝑔 ‘ { 𝑥 , 𝑦 } ) ) |
| 34 |
|
eqeq1 |
⊢ ( ( 𝑔 ‘ { 𝑥 , 𝑦 } ) = ( 𝑓 “ { 𝑥 , 𝑦 } ) → ( ( 𝑔 ‘ { 𝑥 , 𝑦 } ) = { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ↔ ( 𝑓 “ { 𝑥 , 𝑦 } ) = { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ) |
| 35 |
34
|
eqcoms |
⊢ ( ( 𝑓 “ { 𝑥 , 𝑦 } ) = ( 𝑔 ‘ { 𝑥 , 𝑦 } ) → ( ( 𝑔 ‘ { 𝑥 , 𝑦 } ) = { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ↔ ( 𝑓 “ { 𝑥 , 𝑦 } ) = { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ) |
| 36 |
35
|
adantl |
⊢ ( ( ( ( ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) ∧ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ { 𝑥 , 𝑦 } ∈ 𝐸 ) ∧ ( 𝑓 “ { 𝑥 , 𝑦 } ) = ( 𝑔 ‘ { 𝑥 , 𝑦 } ) ) → ( ( 𝑔 ‘ { 𝑥 , 𝑦 } ) = { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ↔ ( 𝑓 “ { 𝑥 , 𝑦 } ) = { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ) |
| 37 |
|
f1ofn |
⊢ ( 𝑓 : 𝑉 –1-1-onto→ 𝑊 → 𝑓 Fn 𝑉 ) |
| 38 |
37
|
ad6antlr |
⊢ ( ( ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) ∧ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) → 𝑓 Fn 𝑉 ) |
| 39 |
|
simpl |
⊢ ( ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) → 𝑥 ∈ 𝑉 ) |
| 40 |
39
|
adantl |
⊢ ( ( ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) ∧ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) → 𝑥 ∈ 𝑉 ) |
| 41 |
|
simpr |
⊢ ( ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) → 𝑦 ∈ 𝑉 ) |
| 42 |
41
|
adantl |
⊢ ( ( ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) ∧ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) → 𝑦 ∈ 𝑉 ) |
| 43 |
38 40 42
|
3jca |
⊢ ( ( ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) ∧ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) → ( 𝑓 Fn 𝑉 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) |
| 44 |
43
|
adantr |
⊢ ( ( ( ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) ∧ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ { 𝑥 , 𝑦 } ∈ 𝐸 ) → ( 𝑓 Fn 𝑉 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) |
| 45 |
|
fnimapr |
⊢ ( ( 𝑓 Fn 𝑉 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) → ( 𝑓 “ { 𝑥 , 𝑦 } ) = { ( 𝑓 ‘ 𝑥 ) , ( 𝑓 ‘ 𝑦 ) } ) |
| 46 |
44 45
|
syl |
⊢ ( ( ( ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) ∧ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ { 𝑥 , 𝑦 } ∈ 𝐸 ) → ( 𝑓 “ { 𝑥 , 𝑦 } ) = { ( 𝑓 ‘ 𝑥 ) , ( 𝑓 ‘ 𝑦 ) } ) |
| 47 |
46
|
eqeq1d |
⊢ ( ( ( ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) ∧ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ { 𝑥 , 𝑦 } ∈ 𝐸 ) → ( ( 𝑓 “ { 𝑥 , 𝑦 } ) = { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ↔ { ( 𝑓 ‘ 𝑥 ) , ( 𝑓 ‘ 𝑦 ) } = { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ) |
| 48 |
|
fvex |
⊢ ( 𝑓 ‘ 𝑥 ) ∈ V |
| 49 |
|
fvex |
⊢ ( 𝑓 ‘ 𝑦 ) ∈ V |
| 50 |
|
fvex |
⊢ ( 𝑓 ‘ 𝑎 ) ∈ V |
| 51 |
|
fvex |
⊢ ( 𝑓 ‘ 𝑏 ) ∈ V |
| 52 |
48 49 50 51
|
preq12b |
⊢ ( { ( 𝑓 ‘ 𝑥 ) , ( 𝑓 ‘ 𝑦 ) } = { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ↔ ( ( ( 𝑓 ‘ 𝑥 ) = ( 𝑓 ‘ 𝑎 ) ∧ ( 𝑓 ‘ 𝑦 ) = ( 𝑓 ‘ 𝑏 ) ) ∨ ( ( 𝑓 ‘ 𝑥 ) = ( 𝑓 ‘ 𝑏 ) ∧ ( 𝑓 ‘ 𝑦 ) = ( 𝑓 ‘ 𝑎 ) ) ) ) |
| 53 |
|
f1of1 |
⊢ ( 𝑓 : 𝑉 –1-1-onto→ 𝑊 → 𝑓 : 𝑉 –1-1→ 𝑊 ) |
| 54 |
|
simpl |
⊢ ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) → 𝑎 ∈ 𝑉 ) |
| 55 |
54 39
|
anim12ci |
⊢ ( ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) → ( 𝑥 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉 ) ) |
| 56 |
|
f1veqaeq |
⊢ ( ( 𝑓 : 𝑉 –1-1→ 𝑊 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉 ) ) → ( ( 𝑓 ‘ 𝑥 ) = ( 𝑓 ‘ 𝑎 ) → 𝑥 = 𝑎 ) ) |
| 57 |
53 55 56
|
syl2anr |
⊢ ( ( ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) → ( ( 𝑓 ‘ 𝑥 ) = ( 𝑓 ‘ 𝑎 ) → 𝑥 = 𝑎 ) ) |
| 58 |
|
simpr |
⊢ ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) → 𝑏 ∈ 𝑉 ) |
| 59 |
58 41
|
anim12ci |
⊢ ( ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) → ( 𝑦 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) |
| 60 |
|
f1veqaeq |
⊢ ( ( 𝑓 : 𝑉 –1-1→ 𝑊 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) → ( ( 𝑓 ‘ 𝑦 ) = ( 𝑓 ‘ 𝑏 ) → 𝑦 = 𝑏 ) ) |
| 61 |
53 59 60
|
syl2anr |
⊢ ( ( ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) → ( ( 𝑓 ‘ 𝑦 ) = ( 𝑓 ‘ 𝑏 ) → 𝑦 = 𝑏 ) ) |
| 62 |
57 61
|
anim12d |
⊢ ( ( ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) → ( ( ( 𝑓 ‘ 𝑥 ) = ( 𝑓 ‘ 𝑎 ) ∧ ( 𝑓 ‘ 𝑦 ) = ( 𝑓 ‘ 𝑏 ) ) → ( 𝑥 = 𝑎 ∧ 𝑦 = 𝑏 ) ) ) |
| 63 |
62
|
impcom |
⊢ ( ( ( ( 𝑓 ‘ 𝑥 ) = ( 𝑓 ‘ 𝑎 ) ∧ ( 𝑓 ‘ 𝑦 ) = ( 𝑓 ‘ 𝑏 ) ) ∧ ( ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ) → ( 𝑥 = 𝑎 ∧ 𝑦 = 𝑏 ) ) |
| 64 |
63
|
orcd |
⊢ ( ( ( ( 𝑓 ‘ 𝑥 ) = ( 𝑓 ‘ 𝑎 ) ∧ ( 𝑓 ‘ 𝑦 ) = ( 𝑓 ‘ 𝑏 ) ) ∧ ( ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ) → ( ( 𝑥 = 𝑎 ∧ 𝑦 = 𝑏 ) ∨ ( 𝑥 = 𝑏 ∧ 𝑦 = 𝑎 ) ) ) |
| 65 |
64
|
ex |
⊢ ( ( ( 𝑓 ‘ 𝑥 ) = ( 𝑓 ‘ 𝑎 ) ∧ ( 𝑓 ‘ 𝑦 ) = ( 𝑓 ‘ 𝑏 ) ) → ( ( ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) → ( ( 𝑥 = 𝑎 ∧ 𝑦 = 𝑏 ) ∨ ( 𝑥 = 𝑏 ∧ 𝑦 = 𝑎 ) ) ) ) |
| 66 |
58 39
|
anim12ci |
⊢ ( ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) → ( 𝑥 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) |
| 67 |
|
f1veqaeq |
⊢ ( ( 𝑓 : 𝑉 –1-1→ 𝑊 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) → ( ( 𝑓 ‘ 𝑥 ) = ( 𝑓 ‘ 𝑏 ) → 𝑥 = 𝑏 ) ) |
| 68 |
53 66 67
|
syl2anr |
⊢ ( ( ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) → ( ( 𝑓 ‘ 𝑥 ) = ( 𝑓 ‘ 𝑏 ) → 𝑥 = 𝑏 ) ) |
| 69 |
54 41
|
anim12ci |
⊢ ( ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) → ( 𝑦 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉 ) ) |
| 70 |
|
f1veqaeq |
⊢ ( ( 𝑓 : 𝑉 –1-1→ 𝑊 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉 ) ) → ( ( 𝑓 ‘ 𝑦 ) = ( 𝑓 ‘ 𝑎 ) → 𝑦 = 𝑎 ) ) |
| 71 |
53 69 70
|
syl2anr |
⊢ ( ( ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) → ( ( 𝑓 ‘ 𝑦 ) = ( 𝑓 ‘ 𝑎 ) → 𝑦 = 𝑎 ) ) |
| 72 |
68 71
|
anim12d |
⊢ ( ( ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) → ( ( ( 𝑓 ‘ 𝑥 ) = ( 𝑓 ‘ 𝑏 ) ∧ ( 𝑓 ‘ 𝑦 ) = ( 𝑓 ‘ 𝑎 ) ) → ( 𝑥 = 𝑏 ∧ 𝑦 = 𝑎 ) ) ) |
| 73 |
72
|
impcom |
⊢ ( ( ( ( 𝑓 ‘ 𝑥 ) = ( 𝑓 ‘ 𝑏 ) ∧ ( 𝑓 ‘ 𝑦 ) = ( 𝑓 ‘ 𝑎 ) ) ∧ ( ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ) → ( 𝑥 = 𝑏 ∧ 𝑦 = 𝑎 ) ) |
| 74 |
73
|
olcd |
⊢ ( ( ( ( 𝑓 ‘ 𝑥 ) = ( 𝑓 ‘ 𝑏 ) ∧ ( 𝑓 ‘ 𝑦 ) = ( 𝑓 ‘ 𝑎 ) ) ∧ ( ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ) → ( ( 𝑥 = 𝑎 ∧ 𝑦 = 𝑏 ) ∨ ( 𝑥 = 𝑏 ∧ 𝑦 = 𝑎 ) ) ) |
| 75 |
74
|
ex |
⊢ ( ( ( 𝑓 ‘ 𝑥 ) = ( 𝑓 ‘ 𝑏 ) ∧ ( 𝑓 ‘ 𝑦 ) = ( 𝑓 ‘ 𝑎 ) ) → ( ( ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) → ( ( 𝑥 = 𝑎 ∧ 𝑦 = 𝑏 ) ∨ ( 𝑥 = 𝑏 ∧ 𝑦 = 𝑎 ) ) ) ) |
| 76 |
65 75
|
jaoi |
⊢ ( ( ( ( 𝑓 ‘ 𝑥 ) = ( 𝑓 ‘ 𝑎 ) ∧ ( 𝑓 ‘ 𝑦 ) = ( 𝑓 ‘ 𝑏 ) ) ∨ ( ( 𝑓 ‘ 𝑥 ) = ( 𝑓 ‘ 𝑏 ) ∧ ( 𝑓 ‘ 𝑦 ) = ( 𝑓 ‘ 𝑎 ) ) ) → ( ( ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) → ( ( 𝑥 = 𝑎 ∧ 𝑦 = 𝑏 ) ∨ ( 𝑥 = 𝑏 ∧ 𝑦 = 𝑎 ) ) ) ) |
| 77 |
|
vex |
⊢ 𝑥 ∈ V |
| 78 |
|
vex |
⊢ 𝑦 ∈ V |
| 79 |
|
vex |
⊢ 𝑎 ∈ V |
| 80 |
|
vex |
⊢ 𝑏 ∈ V |
| 81 |
77 78 79 80
|
preq12b |
⊢ ( { 𝑥 , 𝑦 } = { 𝑎 , 𝑏 } ↔ ( ( 𝑥 = 𝑎 ∧ 𝑦 = 𝑏 ) ∨ ( 𝑥 = 𝑏 ∧ 𝑦 = 𝑎 ) ) ) |
| 82 |
76 81
|
syl6ibr |
⊢ ( ( ( ( 𝑓 ‘ 𝑥 ) = ( 𝑓 ‘ 𝑎 ) ∧ ( 𝑓 ‘ 𝑦 ) = ( 𝑓 ‘ 𝑏 ) ) ∨ ( ( 𝑓 ‘ 𝑥 ) = ( 𝑓 ‘ 𝑏 ) ∧ ( 𝑓 ‘ 𝑦 ) = ( 𝑓 ‘ 𝑎 ) ) ) → ( ( ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) → { 𝑥 , 𝑦 } = { 𝑎 , 𝑏 } ) ) |
| 83 |
52 82
|
sylbi |
⊢ ( { ( 𝑓 ‘ 𝑥 ) , ( 𝑓 ‘ 𝑦 ) } = { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } → ( ( ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) → { 𝑥 , 𝑦 } = { 𝑎 , 𝑏 } ) ) |
| 84 |
83
|
com12 |
⊢ ( ( ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) → ( { ( 𝑓 ‘ 𝑥 ) , ( 𝑓 ‘ 𝑦 ) } = { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } → { 𝑥 , 𝑦 } = { 𝑎 , 𝑏 } ) ) |
| 85 |
84
|
expcom |
⊢ ( 𝑓 : 𝑉 –1-1-onto→ 𝑊 → ( ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) → ( { ( 𝑓 ‘ 𝑥 ) , ( 𝑓 ‘ 𝑦 ) } = { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } → { 𝑥 , 𝑦 } = { 𝑎 , 𝑏 } ) ) ) |
| 86 |
85
|
expd |
⊢ ( 𝑓 : 𝑉 –1-1-onto→ 𝑊 → ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) → ( ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) → ( { ( 𝑓 ‘ 𝑥 ) , ( 𝑓 ‘ 𝑦 ) } = { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } → { 𝑥 , 𝑦 } = { 𝑎 , 𝑏 } ) ) ) ) |
| 87 |
86
|
ad2antlr |
⊢ ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) → ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) → ( ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) → ( { ( 𝑓 ‘ 𝑥 ) , ( 𝑓 ‘ 𝑦 ) } = { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } → { 𝑥 , 𝑦 } = { 𝑎 , 𝑏 } ) ) ) ) |
| 88 |
87
|
imp |
⊢ ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) → ( ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) → ( { ( 𝑓 ‘ 𝑥 ) , ( 𝑓 ‘ 𝑦 ) } = { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } → { 𝑥 , 𝑦 } = { 𝑎 , 𝑏 } ) ) ) |
| 89 |
88
|
adantr |
⊢ ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) → ( ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) → ( { ( 𝑓 ‘ 𝑥 ) , ( 𝑓 ‘ 𝑦 ) } = { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } → { 𝑥 , 𝑦 } = { 𝑎 , 𝑏 } ) ) ) |
| 90 |
89
|
adantr |
⊢ ( ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) ∧ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 ) → ( ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) → ( { ( 𝑓 ‘ 𝑥 ) , ( 𝑓 ‘ 𝑦 ) } = { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } → { 𝑥 , 𝑦 } = { 𝑎 , 𝑏 } ) ) ) |
| 91 |
90
|
imp31 |
⊢ ( ( ( ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) ∧ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑥 ) , ( 𝑓 ‘ 𝑦 ) } = { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) → { 𝑥 , 𝑦 } = { 𝑎 , 𝑏 } ) |
| 92 |
91
|
eleq1d |
⊢ ( ( ( ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) ∧ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑥 ) , ( 𝑓 ‘ 𝑦 ) } = { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) → ( { 𝑥 , 𝑦 } ∈ 𝐸 ↔ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) |
| 93 |
92
|
biimpd |
⊢ ( ( ( ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) ∧ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑥 ) , ( 𝑓 ‘ 𝑦 ) } = { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) → ( { 𝑥 , 𝑦 } ∈ 𝐸 → { 𝑎 , 𝑏 } ∈ 𝐸 ) ) |
| 94 |
93
|
impancom |
⊢ ( ( ( ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) ∧ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ { 𝑥 , 𝑦 } ∈ 𝐸 ) → ( { ( 𝑓 ‘ 𝑥 ) , ( 𝑓 ‘ 𝑦 ) } = { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } → { 𝑎 , 𝑏 } ∈ 𝐸 ) ) |
| 95 |
47 94
|
sylbid |
⊢ ( ( ( ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) ∧ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ { 𝑥 , 𝑦 } ∈ 𝐸 ) → ( ( 𝑓 “ { 𝑥 , 𝑦 } ) = { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } → { 𝑎 , 𝑏 } ∈ 𝐸 ) ) |
| 96 |
95
|
adantr |
⊢ ( ( ( ( ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) ∧ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ { 𝑥 , 𝑦 } ∈ 𝐸 ) ∧ ( 𝑓 “ { 𝑥 , 𝑦 } ) = ( 𝑔 ‘ { 𝑥 , 𝑦 } ) ) → ( ( 𝑓 “ { 𝑥 , 𝑦 } ) = { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } → { 𝑎 , 𝑏 } ∈ 𝐸 ) ) |
| 97 |
36 96
|
sylbid |
⊢ ( ( ( ( ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) ∧ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ { 𝑥 , 𝑦 } ∈ 𝐸 ) ∧ ( 𝑓 “ { 𝑥 , 𝑦 } ) = ( 𝑔 ‘ { 𝑥 , 𝑦 } ) ) → ( ( 𝑔 ‘ { 𝑥 , 𝑦 } ) = { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } → { 𝑎 , 𝑏 } ∈ 𝐸 ) ) |
| 98 |
33 97
|
mpdan |
⊢ ( ( ( ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) ∧ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ { 𝑥 , 𝑦 } ∈ 𝐸 ) → ( ( 𝑔 ‘ { 𝑥 , 𝑦 } ) = { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } → { 𝑎 , 𝑏 } ∈ 𝐸 ) ) |
| 99 |
25 98
|
sylbird |
⊢ ( ( ( ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) ∧ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ { 𝑥 , 𝑦 } ∈ 𝐸 ) → ( ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) = { 𝑥 , 𝑦 } → { 𝑎 , 𝑏 } ∈ 𝐸 ) ) |
| 100 |
99
|
impancom |
⊢ ( ( ( ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) ∧ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) = { 𝑥 , 𝑦 } ) → ( { 𝑥 , 𝑦 } ∈ 𝐸 → { 𝑎 , 𝑏 } ∈ 𝐸 ) ) |
| 101 |
18 100
|
mpd |
⊢ ( ( ( ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) ∧ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) ∧ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) = { 𝑥 , 𝑦 } ) → { 𝑎 , 𝑏 } ∈ 𝐸 ) |
| 102 |
101
|
ex |
⊢ ( ( ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) ∧ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 ) ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) → ( ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) = { 𝑥 , 𝑦 } → { 𝑎 , 𝑏 } ∈ 𝐸 ) ) |
| 103 |
102
|
rexlimdvva |
⊢ ( ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) ∧ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 ) → ( ∃ 𝑥 ∈ 𝑉 ∃ 𝑦 ∈ 𝑉 ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) = { 𝑥 , 𝑦 } → { 𝑎 , 𝑏 } ∈ 𝐸 ) ) |
| 104 |
13 103
|
mpd |
⊢ ( ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) ∧ ( ◡ 𝑔 ‘ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ) ∈ 𝐸 ) → { 𝑎 , 𝑏 } ∈ 𝐸 ) |
| 105 |
8 104
|
mpdan |
⊢ ( ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 ) → { 𝑎 , 𝑏 } ∈ 𝐸 ) |
| 106 |
105
|
ex |
⊢ ( ( ( ( ( 𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph ) ∧ 𝑓 : 𝑉 –1-1-onto→ 𝑊 ) ∧ ( 𝑔 : 𝐸 –1-1-onto→ 𝐾 ∧ ∀ 𝑒 ∈ 𝐸 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) → ( { ( 𝑓 ‘ 𝑎 ) , ( 𝑓 ‘ 𝑏 ) } ∈ 𝐾 → { 𝑎 , 𝑏 } ∈ 𝐸 ) ) |