| Step |
Hyp |
Ref |
Expression |
| 1 |
|
isubgredg.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
| 2 |
|
isubgredg.e |
⊢ 𝐸 = ( Edg ‘ 𝐺 ) |
| 3 |
|
isubgredg.h |
⊢ 𝐻 = ( 𝐺 ISubGr 𝑆 ) |
| 4 |
|
isubgredg.i |
⊢ 𝐼 = ( Edg ‘ 𝐻 ) |
| 5 |
3
|
fveq2i |
⊢ ( iEdg ‘ 𝐻 ) = ( iEdg ‘ ( 𝐺 ISubGr 𝑆 ) ) |
| 6 |
|
eqid |
⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 ) |
| 7 |
1 6
|
isubgriedg |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → ( iEdg ‘ ( 𝐺 ISubGr 𝑆 ) ) = ( ( iEdg ‘ 𝐺 ) ↾ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) ) |
| 8 |
5 7
|
eqtrid |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → ( iEdg ‘ 𝐻 ) = ( ( iEdg ‘ 𝐺 ) ↾ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) ) |
| 9 |
8
|
rneqd |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → ran ( iEdg ‘ 𝐻 ) = ran ( ( iEdg ‘ 𝐺 ) ↾ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) ) |
| 10 |
9
|
eleq2d |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → ( 𝐾 ∈ ran ( iEdg ‘ 𝐻 ) ↔ 𝐾 ∈ ran ( ( iEdg ‘ 𝐺 ) ↾ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) ) ) |
| 11 |
1 6
|
uhgrf |
⊢ ( 𝐺 ∈ UHGraph → ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) ⟶ ( 𝒫 𝑉 ∖ { ∅ } ) ) |
| 12 |
11
|
adantr |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) ⟶ ( 𝒫 𝑉 ∖ { ∅ } ) ) |
| 13 |
12
|
ffnd |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → ( iEdg ‘ 𝐺 ) Fn dom ( iEdg ‘ 𝐺 ) ) |
| 14 |
|
ssrab2 |
⊢ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ⊆ dom ( iEdg ‘ 𝐺 ) |
| 15 |
14
|
a1i |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ⊆ dom ( iEdg ‘ 𝐺 ) ) |
| 16 |
13 15
|
fnssresd |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → ( ( iEdg ‘ 𝐺 ) ↾ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) Fn { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) |
| 17 |
|
fvelrnb |
⊢ ( ( ( iEdg ‘ 𝐺 ) ↾ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) Fn { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } → ( 𝐾 ∈ ran ( ( iEdg ‘ 𝐺 ) ↾ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) ↔ ∃ 𝑥 ∈ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ( ( ( iEdg ‘ 𝐺 ) ↾ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) ‘ 𝑥 ) = 𝐾 ) ) |
| 18 |
16 17
|
syl |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → ( 𝐾 ∈ ran ( ( iEdg ‘ 𝐺 ) ↾ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) ↔ ∃ 𝑥 ∈ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ( ( ( iEdg ‘ 𝐺 ) ↾ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) ‘ 𝑥 ) = 𝐾 ) ) |
| 19 |
|
fvres |
⊢ ( 𝑥 ∈ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } → ( ( ( iEdg ‘ 𝐺 ) ↾ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) ‘ 𝑥 ) = ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ) |
| 20 |
19
|
adantl |
⊢ ( ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) ∧ 𝑥 ∈ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) → ( ( ( iEdg ‘ 𝐺 ) ↾ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) ‘ 𝑥 ) = ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ) |
| 21 |
20
|
eqeq1d |
⊢ ( ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) ∧ 𝑥 ∈ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) → ( ( ( ( iEdg ‘ 𝐺 ) ↾ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) ‘ 𝑥 ) = 𝐾 ↔ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) = 𝐾 ) ) |
| 22 |
|
fveq2 |
⊢ ( 𝑖 = 𝑥 → ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) = ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ) |
| 23 |
22
|
sseq1d |
⊢ ( 𝑖 = 𝑥 → ( ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 ↔ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 ) ) |
| 24 |
23
|
elrab |
⊢ ( 𝑥 ∈ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ↔ ( 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∧ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 ) ) |
| 25 |
6
|
uhgrfun |
⊢ ( 𝐺 ∈ UHGraph → Fun ( iEdg ‘ 𝐺 ) ) |
| 26 |
25
|
adantr |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → Fun ( iEdg ‘ 𝐺 ) ) |
| 27 |
|
simpl |
⊢ ( ( 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∧ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 ) → 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ) |
| 28 |
|
fvelrn |
⊢ ( ( Fun ( iEdg ‘ 𝐺 ) ∧ 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ∈ ran ( iEdg ‘ 𝐺 ) ) |
| 29 |
26 27 28
|
syl2anr |
⊢ ( ( ( 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∧ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 ) ∧ ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ∈ ran ( iEdg ‘ 𝐺 ) ) |
| 30 |
|
simpr |
⊢ ( ( 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∧ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 ) |
| 31 |
30
|
adantr |
⊢ ( ( ( 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∧ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 ) ∧ ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 ) |
| 32 |
29 31
|
jca |
⊢ ( ( ( 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∧ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 ) ∧ ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) ) → ( ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ∈ ran ( iEdg ‘ 𝐺 ) ∧ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 ) ) |
| 33 |
32
|
ex |
⊢ ( ( 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∧ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 ) → ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → ( ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ∈ ran ( iEdg ‘ 𝐺 ) ∧ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 ) ) ) |
| 34 |
24 33
|
sylbi |
⊢ ( 𝑥 ∈ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } → ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → ( ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ∈ ran ( iEdg ‘ 𝐺 ) ∧ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 ) ) ) |
| 35 |
34
|
impcom |
⊢ ( ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) ∧ 𝑥 ∈ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) → ( ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ∈ ran ( iEdg ‘ 𝐺 ) ∧ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 ) ) |
| 36 |
|
eleq1 |
⊢ ( ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) = 𝐾 → ( ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ∈ ran ( iEdg ‘ 𝐺 ) ↔ 𝐾 ∈ ran ( iEdg ‘ 𝐺 ) ) ) |
| 37 |
|
sseq1 |
⊢ ( ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) = 𝐾 → ( ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 ↔ 𝐾 ⊆ 𝑆 ) ) |
| 38 |
36 37
|
anbi12d |
⊢ ( ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) = 𝐾 → ( ( ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ∈ ran ( iEdg ‘ 𝐺 ) ∧ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 ) ↔ ( 𝐾 ∈ ran ( iEdg ‘ 𝐺 ) ∧ 𝐾 ⊆ 𝑆 ) ) ) |
| 39 |
35 38
|
syl5ibcom |
⊢ ( ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) ∧ 𝑥 ∈ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) → ( ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) = 𝐾 → ( 𝐾 ∈ ran ( iEdg ‘ 𝐺 ) ∧ 𝐾 ⊆ 𝑆 ) ) ) |
| 40 |
21 39
|
sylbid |
⊢ ( ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) ∧ 𝑥 ∈ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) → ( ( ( ( iEdg ‘ 𝐺 ) ↾ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) ‘ 𝑥 ) = 𝐾 → ( 𝐾 ∈ ran ( iEdg ‘ 𝐺 ) ∧ 𝐾 ⊆ 𝑆 ) ) ) |
| 41 |
40
|
rexlimdva |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → ( ∃ 𝑥 ∈ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ( ( ( iEdg ‘ 𝐺 ) ↾ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) ‘ 𝑥 ) = 𝐾 → ( 𝐾 ∈ ran ( iEdg ‘ 𝐺 ) ∧ 𝐾 ⊆ 𝑆 ) ) ) |
| 42 |
|
edgval |
⊢ ( Edg ‘ 𝐺 ) = ran ( iEdg ‘ 𝐺 ) |
| 43 |
42
|
eqcomi |
⊢ ran ( iEdg ‘ 𝐺 ) = ( Edg ‘ 𝐺 ) |
| 44 |
43
|
eleq2i |
⊢ ( 𝐾 ∈ ran ( iEdg ‘ 𝐺 ) ↔ 𝐾 ∈ ( Edg ‘ 𝐺 ) ) |
| 45 |
6
|
edgiedgb |
⊢ ( Fun ( iEdg ‘ 𝐺 ) → ( 𝐾 ∈ ( Edg ‘ 𝐺 ) ↔ ∃ 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ) ) |
| 46 |
44 45
|
bitrid |
⊢ ( Fun ( iEdg ‘ 𝐺 ) → ( 𝐾 ∈ ran ( iEdg ‘ 𝐺 ) ↔ ∃ 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ) ) |
| 47 |
25 46
|
syl |
⊢ ( 𝐺 ∈ UHGraph → ( 𝐾 ∈ ran ( iEdg ‘ 𝐺 ) ↔ ∃ 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ) ) |
| 48 |
47
|
adantr |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → ( 𝐾 ∈ ran ( iEdg ‘ 𝐺 ) ↔ ∃ 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ) ) |
| 49 |
|
simprl |
⊢ ( ( ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) ∧ 𝐾 ⊆ 𝑆 ) ∧ ( 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ) ) → 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ) |
| 50 |
|
simpr |
⊢ ( ( 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ) → 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ) |
| 51 |
50
|
sseq1d |
⊢ ( ( 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ) → ( 𝐾 ⊆ 𝑆 ↔ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 ) ) |
| 52 |
51
|
biimpcd |
⊢ ( 𝐾 ⊆ 𝑆 → ( ( 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 ) ) |
| 53 |
52
|
adantl |
⊢ ( ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) ∧ 𝐾 ⊆ 𝑆 ) → ( ( 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 ) ) |
| 54 |
53
|
imp |
⊢ ( ( ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) ∧ 𝐾 ⊆ 𝑆 ) ∧ ( 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ) ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 ) |
| 55 |
49 54 24
|
sylanbrc |
⊢ ( ( ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) ∧ 𝐾 ⊆ 𝑆 ) ∧ ( 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ) ) → 𝑥 ∈ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) |
| 56 |
|
simpr |
⊢ ( ( ( ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) ∧ 𝐾 ⊆ 𝑆 ) ∧ ( 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ) ) ∧ 𝑥 ∈ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) → 𝑥 ∈ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) |
| 57 |
50
|
eqcomd |
⊢ ( ( 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) = 𝐾 ) |
| 58 |
57
|
adantl |
⊢ ( ( ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) ∧ 𝐾 ⊆ 𝑆 ) ∧ ( 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ) ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) = 𝐾 ) |
| 59 |
19 58
|
sylan9eqr |
⊢ ( ( ( ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) ∧ 𝐾 ⊆ 𝑆 ) ∧ ( 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ) ) ∧ 𝑥 ∈ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) → ( ( ( iEdg ‘ 𝐺 ) ↾ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) ‘ 𝑥 ) = 𝐾 ) |
| 60 |
56 59
|
jca |
⊢ ( ( ( ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) ∧ 𝐾 ⊆ 𝑆 ) ∧ ( 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ) ) ∧ 𝑥 ∈ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) → ( 𝑥 ∈ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ∧ ( ( ( iEdg ‘ 𝐺 ) ↾ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) ‘ 𝑥 ) = 𝐾 ) ) |
| 61 |
55 60
|
mpdan |
⊢ ( ( ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) ∧ 𝐾 ⊆ 𝑆 ) ∧ ( 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ) ) → ( 𝑥 ∈ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ∧ ( ( ( iEdg ‘ 𝐺 ) ↾ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) ‘ 𝑥 ) = 𝐾 ) ) |
| 62 |
61
|
ex |
⊢ ( ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) ∧ 𝐾 ⊆ 𝑆 ) → ( ( 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ) → ( 𝑥 ∈ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ∧ ( ( ( iEdg ‘ 𝐺 ) ↾ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) ‘ 𝑥 ) = 𝐾 ) ) ) |
| 63 |
62
|
eximdv |
⊢ ( ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) ∧ 𝐾 ⊆ 𝑆 ) → ( ∃ 𝑥 ( 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ) → ∃ 𝑥 ( 𝑥 ∈ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ∧ ( ( ( iEdg ‘ 𝐺 ) ↾ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) ‘ 𝑥 ) = 𝐾 ) ) ) |
| 64 |
|
df-rex |
⊢ ( ∃ 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ↔ ∃ 𝑥 ( 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∧ 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ) ) |
| 65 |
|
df-rex |
⊢ ( ∃ 𝑥 ∈ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ( ( ( iEdg ‘ 𝐺 ) ↾ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) ‘ 𝑥 ) = 𝐾 ↔ ∃ 𝑥 ( 𝑥 ∈ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ∧ ( ( ( iEdg ‘ 𝐺 ) ↾ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) ‘ 𝑥 ) = 𝐾 ) ) |
| 66 |
63 64 65
|
3imtr4g |
⊢ ( ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) ∧ 𝐾 ⊆ 𝑆 ) → ( ∃ 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) → ∃ 𝑥 ∈ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ( ( ( iEdg ‘ 𝐺 ) ↾ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) ‘ 𝑥 ) = 𝐾 ) ) |
| 67 |
66
|
ex |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → ( 𝐾 ⊆ 𝑆 → ( ∃ 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) → ∃ 𝑥 ∈ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ( ( ( iEdg ‘ 𝐺 ) ↾ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) ‘ 𝑥 ) = 𝐾 ) ) ) |
| 68 |
67
|
com23 |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → ( ∃ 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) 𝐾 = ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) → ( 𝐾 ⊆ 𝑆 → ∃ 𝑥 ∈ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ( ( ( iEdg ‘ 𝐺 ) ↾ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) ‘ 𝑥 ) = 𝐾 ) ) ) |
| 69 |
48 68
|
sylbid |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → ( 𝐾 ∈ ran ( iEdg ‘ 𝐺 ) → ( 𝐾 ⊆ 𝑆 → ∃ 𝑥 ∈ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ( ( ( iEdg ‘ 𝐺 ) ↾ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) ‘ 𝑥 ) = 𝐾 ) ) ) |
| 70 |
69
|
impd |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → ( ( 𝐾 ∈ ran ( iEdg ‘ 𝐺 ) ∧ 𝐾 ⊆ 𝑆 ) → ∃ 𝑥 ∈ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ( ( ( iEdg ‘ 𝐺 ) ↾ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) ‘ 𝑥 ) = 𝐾 ) ) |
| 71 |
41 70
|
impbid |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → ( ∃ 𝑥 ∈ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ( ( ( iEdg ‘ 𝐺 ) ↾ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) ⊆ 𝑆 } ) ‘ 𝑥 ) = 𝐾 ↔ ( 𝐾 ∈ ran ( iEdg ‘ 𝐺 ) ∧ 𝐾 ⊆ 𝑆 ) ) ) |
| 72 |
10 18 71
|
3bitrd |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → ( 𝐾 ∈ ran ( iEdg ‘ 𝐻 ) ↔ ( 𝐾 ∈ ran ( iEdg ‘ 𝐺 ) ∧ 𝐾 ⊆ 𝑆 ) ) ) |
| 73 |
|
edgval |
⊢ ( Edg ‘ 𝐻 ) = ran ( iEdg ‘ 𝐻 ) |
| 74 |
4 73
|
eqtri |
⊢ 𝐼 = ran ( iEdg ‘ 𝐻 ) |
| 75 |
74
|
eleq2i |
⊢ ( 𝐾 ∈ 𝐼 ↔ 𝐾 ∈ ran ( iEdg ‘ 𝐻 ) ) |
| 76 |
2 42
|
eqtri |
⊢ 𝐸 = ran ( iEdg ‘ 𝐺 ) |
| 77 |
76
|
eleq2i |
⊢ ( 𝐾 ∈ 𝐸 ↔ 𝐾 ∈ ran ( iEdg ‘ 𝐺 ) ) |
| 78 |
77
|
anbi1i |
⊢ ( ( 𝐾 ∈ 𝐸 ∧ 𝐾 ⊆ 𝑆 ) ↔ ( 𝐾 ∈ ran ( iEdg ‘ 𝐺 ) ∧ 𝐾 ⊆ 𝑆 ) ) |
| 79 |
72 75 78
|
3bitr4g |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → ( 𝐾 ∈ 𝐼 ↔ ( 𝐾 ∈ 𝐸 ∧ 𝐾 ⊆ 𝑆 ) ) ) |