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Metamath Proof Explorer

Theorem gricgrlic

Description: Isomorphic hypergraphs are locally isomorphic. (Contributed by AV, 12-Jun-2025) (Proof shortened by AV, 11-Jul-2025)

Ref Expression
Assertion gricgrlic 
|- ( ( G e. UHGraph /\ H e. UHGraph ) -> ( G ~=gr H -> G ~=lgr H ) )

Proof

Step Hyp Ref Expression
1 brgric 
 |-  ( G ~=gr H <-> ( G GraphIso H ) =/= (/) )
2 n0 
 |-  ( ( G GraphIso H ) =/= (/) <-> E. i i e. ( G GraphIso H ) )
3 uhgrimgrlim 
 |-  ( ( G e. UHGraph /\ H e. UHGraph /\ i e. ( G GraphIso H ) ) -> i e. ( G GraphLocIso H ) )
4 brgrilci 
 |-  ( i e. ( G GraphLocIso H ) -> G ~=lgr H )
5 3 4 syl 
 |-  ( ( G e. UHGraph /\ H e. UHGraph /\ i e. ( G GraphIso H ) ) -> G ~=lgr H )
6 5 3expa 
 |-  ( ( ( G e. UHGraph /\ H e. UHGraph ) /\ i e. ( G GraphIso H ) ) -> G ~=lgr H )
7 6 expcom 
 |-  ( i e. ( G GraphIso H ) -> ( ( G e. UHGraph /\ H e. UHGraph ) -> G ~=lgr H ) )
8 7 exlimiv 
 |-  ( E. i i e. ( G GraphIso H ) -> ( ( G e. UHGraph /\ H e. UHGraph ) -> G ~=lgr H ) )
9 2 8 sylbi 
 |-  ( ( G GraphIso H ) =/= (/) -> ( ( G e. UHGraph /\ H e. UHGraph ) -> G ~=lgr H ) )
10 1 9 sylbi 
 |-  ( G ~=gr H -> ( ( G e. UHGraph /\ H e. UHGraph ) -> G ~=lgr H ) )
11 10 com12 
 |-  ( ( G e. UHGraph /\ H e. UHGraph ) -> ( G ~=gr H -> G ~=lgr H ) )