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Metamath Proof Explorer

Theorem isf32lem12

Description: Lemma for isfin3-2 . (Contributed by Stefan O'Rear, 6-Nov-2014) (Revised by Mario Carneiro, 17-May-2015)

Ref Expression
Hypothesis isf32lem40.f 
|- F = { g | A. a e. ( ~P g ^m _om ) ( A. x e. _om ( a ` suc x ) C_ ( a ` x ) -> |^| ran a e. ran a ) }
Assertion isf32lem12 
|- ( G e. V -> ( -. _om ~<_* G -> G e. F ) )

Proof

Step Hyp Ref Expression
1 isf32lem40.f 
 |-  F = { g | A. a e. ( ~P g ^m _om ) ( A. x e. _om ( a ` suc x ) C_ ( a ` x ) -> |^| ran a e. ran a ) }
2 elmapi 
 |-  ( f e. ( ~P G ^m _om ) -> f : _om --> ~P G )
3 isf32lem11 
 |-  ( ( G e. V /\ ( f : _om --> ~P G /\ A. b e. _om ( f ` suc b ) C_ ( f ` b ) /\ -. |^| ran f e. ran f ) ) -> _om ~<_* G )
4 3 expcom 
 |-  ( ( f : _om --> ~P G /\ A. b e. _om ( f ` suc b ) C_ ( f ` b ) /\ -. |^| ran f e. ran f ) -> ( G e. V -> _om ~<_* G ) )
5 4 3expa 
 |-  ( ( ( f : _om --> ~P G /\ A. b e. _om ( f ` suc b ) C_ ( f ` b ) ) /\ -. |^| ran f e. ran f ) -> ( G e. V -> _om ~<_* G ) )
6 5 impancom 
 |-  ( ( ( f : _om --> ~P G /\ A. b e. _om ( f ` suc b ) C_ ( f ` b ) ) /\ G e. V ) -> ( -. |^| ran f e. ran f -> _om ~<_* G ) )
7 6 con1d 
 |-  ( ( ( f : _om --> ~P G /\ A. b e. _om ( f ` suc b ) C_ ( f ` b ) ) /\ G e. V ) -> ( -. _om ~<_* G -> |^| ran f e. ran f ) )
8 7 exp31 
 |-  ( f : _om --> ~P G -> ( A. b e. _om ( f ` suc b ) C_ ( f ` b ) -> ( G e. V -> ( -. _om ~<_* G -> |^| ran f e. ran f ) ) ) )
9 2 8 syl 
 |-  ( f e. ( ~P G ^m _om ) -> ( A. b e. _om ( f ` suc b ) C_ ( f ` b ) -> ( G e. V -> ( -. _om ~<_* G -> |^| ran f e. ran f ) ) ) )
10 9 com4t 
 |-  ( G e. V -> ( -. _om ~<_* G -> ( f e. ( ~P G ^m _om ) -> ( A. b e. _om ( f ` suc b ) C_ ( f ` b ) -> |^| ran f e. ran f ) ) ) )
11 10 ralrimdv 
 |-  ( G e. V -> ( -. _om ~<_* G -> A. f e. ( ~P G ^m _om ) ( A. b e. _om ( f ` suc b ) C_ ( f ` b ) -> |^| ran f e. ran f ) ) )
12 1 isfin3ds 
 |-  ( G e. V -> ( G e. F <-> A. f e. ( ~P G ^m _om ) ( A. b e. _om ( f ` suc b ) C_ ( f ` b ) -> |^| ran f e. ran f ) ) )
13 11 12 sylibrd 
 |-  ( G e. V -> ( -. _om ~<_* G -> G e. F ) )