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 ) ) ) |