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Metamath Proof Explorer

Theorem ssfin3ds

Description: A subset of a III-finite set is III-finite. (Contributed by Stefan O'Rear, 4-Nov-2014)

Ref Expression
Hypothesis isfin3ds.f 
|- F = { g | A. a e. ( ~P g ^m _om ) ( A. b e. _om ( a ` suc b ) C_ ( a ` b ) -> |^| ran a e. ran a ) }
Assertion ssfin3ds 
|- ( ( A e. F /\ B C_ A ) -> B e. F )

Proof

Step Hyp Ref Expression
1 isfin3ds.f 
 |-  F = { g | A. a e. ( ~P g ^m _om ) ( A. b e. _om ( a ` suc b ) C_ ( a ` b ) -> |^| ran a e. ran a ) }
2 pwexg 
 |-  ( A e. F -> ~P A e. _V )
3 simpr 
 |-  ( ( A e. F /\ B C_ A ) -> B C_ A )
4 3 sspwd 
 |-  ( ( A e. F /\ B C_ A ) -> ~P B C_ ~P A )
5 mapss 
 |-  ( ( ~P A e. _V /\ ~P B C_ ~P A ) -> ( ~P B ^m _om ) C_ ( ~P A ^m _om ) )
6 2 4 5 syl2an2r 
 |-  ( ( A e. F /\ B C_ A ) -> ( ~P B ^m _om ) C_ ( ~P A ^m _om ) )
7 1 isfin3ds 
 |-  ( A e. F -> ( A e. F <-> A. f e. ( ~P A ^m _om ) ( A. x e. _om ( f ` suc x ) C_ ( f ` x ) -> |^| ran f e. ran f ) ) )
8 7 ibi 
 |-  ( A e. F -> A. f e. ( ~P A ^m _om ) ( A. x e. _om ( f ` suc x ) C_ ( f ` x ) -> |^| ran f e. ran f ) )
9 8 adantr 
 |-  ( ( A e. F /\ B C_ A ) -> A. f e. ( ~P A ^m _om ) ( A. x e. _om ( f ` suc x ) C_ ( f ` x ) -> |^| ran f e. ran f ) )
10 ssralv 
 |-  ( ( ~P B ^m _om ) C_ ( ~P A ^m _om ) -> ( A. f e. ( ~P A ^m _om ) ( A. x e. _om ( f ` suc x ) C_ ( f ` x ) -> |^| ran f e. ran f ) -> A. f e. ( ~P B ^m _om ) ( A. x e. _om ( f ` suc x ) C_ ( f ` x ) -> |^| ran f e. ran f ) ) )
11 6 9 10 sylc 
 |-  ( ( A e. F /\ B C_ A ) -> A. f e. ( ~P B ^m _om ) ( A. x e. _om ( f ` suc x ) C_ ( f ` x ) -> |^| ran f e. ran f ) )
12 ssexg 
 |-  ( ( B C_ A /\ A e. F ) -> B e. _V )
13 12 ancoms 
 |-  ( ( A e. F /\ B C_ A ) -> B e. _V )
14 1 isfin3ds 
 |-  ( B e. _V -> ( B e. F <-> A. f e. ( ~P B ^m _om ) ( A. x e. _om ( f ` suc x ) C_ ( f ` x ) -> |^| ran f e. ran f ) ) )
15 13 14 syl 
 |-  ( ( A e. F /\ B C_ A ) -> ( B e. F <-> A. f e. ( ~P B ^m _om ) ( A. x e. _om ( f ` suc x ) C_ ( f ` x ) -> |^| ran f e. ran f ) ) )
16 11 15 mpbird 
 |-  ( ( A e. F /\ B C_ A ) -> B e. F )