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Metamath Proof Explorer

Theorem sizusglecusglem2

Description: Lemma 2 for sizusglecusg . (Contributed by Alexander van der Vekens, 13-Jan-2018) (Revised by AV, 13-Nov-2020)

Ref Expression
Hypotheses fusgrmaxsize.v 
|- V = ( Vtx ` G )
fusgrmaxsize.e 
|- E = ( Edg ` G )
usgrsscusgra.h 
|- V = ( Vtx ` H )
usgrsscusgra.f 
|- F = ( Edg ` H )
Assertion sizusglecusglem2 
|- ( ( G e. USGraph /\ H e. ComplUSGraph /\ F e. Fin ) -> E e. Fin )

Proof

Step Hyp Ref Expression
1 fusgrmaxsize.v 
 |-  V = ( Vtx ` G )
2 fusgrmaxsize.e 
 |-  E = ( Edg ` G )
3 usgrsscusgra.h 
 |-  V = ( Vtx ` H )
4 usgrsscusgra.f 
 |-  F = ( Edg ` H )
5 3 4 cusgrfi 
 |-  ( ( H e. ComplUSGraph /\ F e. Fin ) -> V e. Fin )
6 5 3adant1 
 |-  ( ( G e. USGraph /\ H e. ComplUSGraph /\ F e. Fin ) -> V e. Fin )
7 1 isfusgr 
 |-  ( G e. FinUSGraph <-> ( G e. USGraph /\ V e. Fin ) )
8 fusgrfis 
 |-  ( G e. FinUSGraph -> ( Edg ` G ) e. Fin )
9 7 8 sylbir 
 |-  ( ( G e. USGraph /\ V e. Fin ) -> ( Edg ` G ) e. Fin )
10 2 9 eqeltrid 
 |-  ( ( G e. USGraph /\ V e. Fin ) -> E e. Fin )
11 10 ex 
 |-  ( G e. USGraph -> ( V e. Fin -> E e. Fin ) )
12 11 3ad2ant1 
 |-  ( ( G e. USGraph /\ H e. ComplUSGraph /\ F e. Fin ) -> ( V e. Fin -> E e. Fin ) )
13 6 12 mpd 
 |-  ( ( G e. USGraph /\ H e. ComplUSGraph /\ F e. Fin ) -> E e. Fin )