Metamath Proof Explorer

Theorem uhgrimedg

Description: An isomorphism between graphs preserves edges, i.e. there is an edge in one graph connecting vertices iff there is an edge in the other graph connecting the corresponding vertices. (Contributed by AV, 25-Oct-2025)

Ref Expression
Hypotheses uhgrimedgi.e 
|- E = ( Edg ` G )
uhgrimedgi.d 
|- D = ( Edg ` H )
Assertion uhgrimedg 
|- ( ( ( G e. UHGraph /\ H e. UHGraph ) /\ F e. ( G GraphIso H ) /\ K C_ ( Vtx ` G ) ) -> ( K e. E <-> ( F " K ) e. D ) )

Proof

Step Hyp Ref Expression
1 uhgrimedgi.e 
 |-  E = ( Edg ` G )
2 uhgrimedgi.d 
 |-  D = ( Edg ` H )
3 simp1 
 |-  ( ( ( G e. UHGraph /\ H e. UHGraph ) /\ F e. ( G GraphIso H ) /\ K C_ ( Vtx ` G ) ) -> ( G e. UHGraph /\ H e. UHGraph ) )
4 simp2 
 |-  ( ( ( G e. UHGraph /\ H e. UHGraph ) /\ F e. ( G GraphIso H ) /\ K C_ ( Vtx ` G ) ) -> F e. ( G GraphIso H ) )
5 4 anim1i 
 |-  ( ( ( ( G e. UHGraph /\ H e. UHGraph ) /\ F e. ( G GraphIso H ) /\ K C_ ( Vtx ` G ) ) /\ K e. E ) -> ( F e. ( G GraphIso H ) /\ K e. E ) )
6 1 2 uhgrimedgi 
 |-  ( ( ( G e. UHGraph /\ H e. UHGraph ) /\ ( F e. ( G GraphIso H ) /\ K e. E ) ) -> ( F " K ) e. D )
7 3 5 6 syl2an2r 
 |-  ( ( ( ( G e. UHGraph /\ H e. UHGraph ) /\ F e. ( G GraphIso H ) /\ K C_ ( Vtx ` G ) ) /\ K e. E ) -> ( F " K ) e. D )
8 eqid 
 |-  ( Vtx ` G ) = ( Vtx ` G )
9 eqid 
 |-  ( Vtx ` H ) = ( Vtx ` H )
10 8 9 grimf1o 
 |-  ( F e. ( G GraphIso H ) -> F : ( Vtx ` G ) -1-1-onto-> ( Vtx ` H ) )
11 f1of1 
 |-  ( F : ( Vtx ` G ) -1-1-onto-> ( Vtx ` H ) -> F : ( Vtx ` G ) -1-1-> ( Vtx ` H ) )
12 10 11 syl 
 |-  ( F e. ( G GraphIso H ) -> F : ( Vtx ` G ) -1-1-> ( Vtx ` H ) )
13 12 3ad2ant2 
 |-  ( ( ( G e. UHGraph /\ H e. UHGraph ) /\ F e. ( G GraphIso H ) /\ K C_ ( Vtx ` G ) ) -> F : ( Vtx ` G ) -1-1-> ( Vtx ` H ) )
14 simp3 
 |-  ( ( ( G e. UHGraph /\ H e. UHGraph ) /\ F e. ( G GraphIso H ) /\ K C_ ( Vtx ` G ) ) -> K C_ ( Vtx ` G ) )
15 13 14 jca 
 |-  ( ( ( G e. UHGraph /\ H e. UHGraph ) /\ F e. ( G GraphIso H ) /\ K C_ ( Vtx ` G ) ) -> ( F : ( Vtx ` G ) -1-1-> ( Vtx ` H ) /\ K C_ ( Vtx ` G ) ) )
16 15 adantr 
 |-  ( ( ( ( G e. UHGraph /\ H e. UHGraph ) /\ F e. ( G GraphIso H ) /\ K C_ ( Vtx ` G ) ) /\ ( F " K ) e. D ) -> ( F : ( Vtx ` G ) -1-1-> ( Vtx ` H ) /\ K C_ ( Vtx ` G ) ) )
17 f1imacnv 
 |-  ( ( F : ( Vtx ` G ) -1-1-> ( Vtx ` H ) /\ K C_ ( Vtx ` G ) ) -> ( `' F " ( F " K ) ) = K )
18 16 17 syl 
 |-  ( ( ( ( G e. UHGraph /\ H e. UHGraph ) /\ F e. ( G GraphIso H ) /\ K C_ ( Vtx ` G ) ) /\ ( F " K ) e. D ) -> ( `' F " ( F " K ) ) = K )
19 pm3.22 
 |-  ( ( G e. UHGraph /\ H e. UHGraph ) -> ( H e. UHGraph /\ G e. UHGraph ) )
20 19 3ad2ant1 
 |-  ( ( ( G e. UHGraph /\ H e. UHGraph ) /\ F e. ( G GraphIso H ) /\ K C_ ( Vtx ` G ) ) -> ( H e. UHGraph /\ G e. UHGraph ) )
21 simpl 
 |-  ( ( G e. UHGraph /\ H e. UHGraph ) -> G e. UHGraph )
22 21 anim1i 
 |-  ( ( ( G e. UHGraph /\ H e. UHGraph ) /\ F e. ( G GraphIso H ) ) -> ( G e. UHGraph /\ F e. ( G GraphIso H ) ) )
23 22 3adant3 
 |-  ( ( ( G e. UHGraph /\ H e. UHGraph ) /\ F e. ( G GraphIso H ) /\ K C_ ( Vtx ` G ) ) -> ( G e. UHGraph /\ F e. ( G GraphIso H ) ) )
24 grimcnv 
 |-  ( G e. UHGraph -> ( F e. ( G GraphIso H ) -> `' F e. ( H GraphIso G ) ) )
25 24 imp 
 |-  ( ( G e. UHGraph /\ F e. ( G GraphIso H ) ) -> `' F e. ( H GraphIso G ) )
26 23 25 syl 
 |-  ( ( ( G e. UHGraph /\ H e. UHGraph ) /\ F e. ( G GraphIso H ) /\ K C_ ( Vtx ` G ) ) -> `' F e. ( H GraphIso G ) )
27 26 anim1i 
 |-  ( ( ( ( G e. UHGraph /\ H e. UHGraph ) /\ F e. ( G GraphIso H ) /\ K C_ ( Vtx ` G ) ) /\ ( F " K ) e. D ) -> ( `' F e. ( H GraphIso G ) /\ ( F " K ) e. D ) )
28 2 1 uhgrimedgi 
 |-  ( ( ( H e. UHGraph /\ G e. UHGraph ) /\ ( `' F e. ( H GraphIso G ) /\ ( F " K ) e. D ) ) -> ( `' F " ( F " K ) ) e. E )
29 20 27 28 syl2an2r 
 |-  ( ( ( ( G e. UHGraph /\ H e. UHGraph ) /\ F e. ( G GraphIso H ) /\ K C_ ( Vtx ` G ) ) /\ ( F " K ) e. D ) -> ( `' F " ( F " K ) ) e. E )
30 18 29 eqeltrrd 
 |-  ( ( ( ( G e. UHGraph /\ H e. UHGraph ) /\ F e. ( G GraphIso H ) /\ K C_ ( Vtx ` G ) ) /\ ( F " K ) e. D ) -> K e. E )
31 7 30 impbida 
 |-  ( ( ( G e. UHGraph /\ H e. UHGraph ) /\ F e. ( G GraphIso H ) /\ K C_ ( Vtx ` G ) ) -> ( K e. E <-> ( F " K ) e. D ) )