Metamath Proof Explorer

Theorem sbthlem1

Description: Lemma for sbth . (Contributed by NM, 22-Mar-1998)

Ref Expression
Hypotheses sbthlem.1 
|- A e. _V
sbthlem.2 
|- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }
Assertion sbthlem1 
|- U. D C_ ( A \ ( g " ( B \ ( f " U. D ) ) ) )

Proof

Step Hyp Ref Expression
1 sbthlem.1 
 |-  A e. _V
2 sbthlem.2 
 |-  D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }
3 unissb 
 |-  ( U. D C_ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) <-> A. x e. D x C_ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) )
4 2 abeq2i 
 |-  ( x e. D <-> ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) )
5 difss2 
 |-  ( ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) -> ( g " ( B \ ( f " x ) ) ) C_ A )
6 ssconb 
 |-  ( ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ A ) -> ( x C_ ( A \ ( g " ( B \ ( f " x ) ) ) ) <-> ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) )
7 6 exbiri 
 |-  ( x C_ A -> ( ( g " ( B \ ( f " x ) ) ) C_ A -> ( ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) -> x C_ ( A \ ( g " ( B \ ( f " x ) ) ) ) ) ) )
8 5 7 syl5 
 |-  ( x C_ A -> ( ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) -> ( ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) -> x C_ ( A \ ( g " ( B \ ( f " x ) ) ) ) ) ) )
9 8 pm2.43d 
 |-  ( x C_ A -> ( ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) -> x C_ ( A \ ( g " ( B \ ( f " x ) ) ) ) ) )
10 9 imp 
 |-  ( ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) -> x C_ ( A \ ( g " ( B \ ( f " x ) ) ) ) )
11 4 10 sylbi 
 |-  ( x e. D -> x C_ ( A \ ( g " ( B \ ( f " x ) ) ) ) )
12 elssuni 
 |-  ( x e. D -> x C_ U. D )
13 imass2 
 |-  ( x C_ U. D -> ( f " x ) C_ ( f " U. D ) )
14 sscon 
 |-  ( ( f " x ) C_ ( f " U. D ) -> ( B \ ( f " U. D ) ) C_ ( B \ ( f " x ) ) )
15 12 13 14 3syl 
 |-  ( x e. D -> ( B \ ( f " U. D ) ) C_ ( B \ ( f " x ) ) )
16 imass2 
 |-  ( ( B \ ( f " U. D ) ) C_ ( B \ ( f " x ) ) -> ( g " ( B \ ( f " U. D ) ) ) C_ ( g " ( B \ ( f " x ) ) ) )
17 sscon 
 |-  ( ( g " ( B \ ( f " U. D ) ) ) C_ ( g " ( B \ ( f " x ) ) ) -> ( A \ ( g " ( B \ ( f " x ) ) ) ) C_ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) )
18 15 16 17 3syl 
 |-  ( x e. D -> ( A \ ( g " ( B \ ( f " x ) ) ) ) C_ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) )
19 11 18 sstrd 
 |-  ( x e. D -> x C_ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) )
20 3 19 mprgbir 
 |-  U. D C_ ( A \ ( g " ( B \ ( f " U. D ) ) ) )