Metamath Proof Explorer

Theorem sbthlem2

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 sbthlem2 
|- ( ran g C_ A -> ( A \ ( g " ( B \ ( f " U. D ) ) ) ) C_ 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 1 2 sbthlem1 
 |-  U. D C_ ( A \ ( g " ( B \ ( f " U. D ) ) ) )
4 imass2 
 |-  ( U. D C_ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) -> ( f " U. D ) C_ ( f " ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) )
5 sscon 
 |-  ( ( f " U. D ) C_ ( f " ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) -> ( B \ ( f " ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) ) C_ ( B \ ( f " U. D ) ) )
6 3 4 5 mp2b 
 |-  ( B \ ( f " ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) ) C_ ( B \ ( f " U. D ) )
7 imass2 
 |-  ( ( B \ ( f " ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) ) C_ ( B \ ( f " U. D ) ) -> ( g " ( B \ ( f " ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) ) ) C_ ( g " ( B \ ( f " U. D ) ) ) )
8 sscon 
 |-  ( ( g " ( B \ ( f " ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) ) ) C_ ( g " ( B \ ( f " U. D ) ) ) -> ( A \ ( g " ( B \ ( f " U. D ) ) ) ) C_ ( A \ ( g " ( B \ ( f " ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) ) ) ) )
9 6 7 8 mp2b 
 |-  ( A \ ( g " ( B \ ( f " U. D ) ) ) ) C_ ( A \ ( g " ( B \ ( f " ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) ) ) )
10 imassrn 
 |-  ( g " ( B \ ( f " ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) ) ) C_ ran g
11 sstr2 
 |-  ( ( g " ( B \ ( f " ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) ) ) C_ ran g -> ( ran g C_ A -> ( g " ( B \ ( f " ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) ) ) C_ A ) )
12 10 11 ax-mp 
 |-  ( ran g C_ A -> ( g " ( B \ ( f " ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) ) ) C_ A )
13 difss 
 |-  ( A \ ( g " ( B \ ( f " U. D ) ) ) ) C_ A
14 ssconb 
 |-  ( ( ( g " ( B \ ( f " ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) ) ) C_ A /\ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) C_ A ) -> ( ( g " ( B \ ( f " ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) ) ) C_ ( A \ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) <-> ( A \ ( g " ( B \ ( f " U. D ) ) ) ) C_ ( A \ ( g " ( B \ ( f " ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) ) ) ) ) )
15 12 13 14 sylancl 
 |-  ( ran g C_ A -> ( ( g " ( B \ ( f " ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) ) ) C_ ( A \ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) <-> ( A \ ( g " ( B \ ( f " U. D ) ) ) ) C_ ( A \ ( g " ( B \ ( f " ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) ) ) ) ) )
16 9 15 mpbiri 
 |-  ( ran g C_ A -> ( g " ( B \ ( f " ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) ) ) C_ ( A \ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) )
17 16 13 jctil 
 |-  ( ran g C_ A -> ( ( A \ ( g " ( B \ ( f " U. D ) ) ) ) C_ A /\ ( g " ( B \ ( f " ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) ) ) C_ ( A \ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) ) )
18 1 difexi 
 |-  ( A \ ( g " ( B \ ( f " U. D ) ) ) ) e. _V
19 sseq1 
 |-  ( x = ( A \ ( g " ( B \ ( f " U. D ) ) ) ) -> ( x C_ A <-> ( A \ ( g " ( B \ ( f " U. D ) ) ) ) C_ A ) )
20 imaeq2 
 |-  ( x = ( A \ ( g " ( B \ ( f " U. D ) ) ) ) -> ( f " x ) = ( f " ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) )
21 20 difeq2d 
 |-  ( x = ( A \ ( g " ( B \ ( f " U. D ) ) ) ) -> ( B \ ( f " x ) ) = ( B \ ( f " ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) ) )
22 21 imaeq2d 
 |-  ( x = ( A \ ( g " ( B \ ( f " U. D ) ) ) ) -> ( g " ( B \ ( f " x ) ) ) = ( g " ( B \ ( f " ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) ) ) )
23 difeq2 
 |-  ( x = ( A \ ( g " ( B \ ( f " U. D ) ) ) ) -> ( A \ x ) = ( A \ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) )
24 22 23 sseq12d 
 |-  ( x = ( A \ ( g " ( B \ ( f " U. D ) ) ) ) -> ( ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) <-> ( g " ( B \ ( f " ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) ) ) C_ ( A \ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) ) )
25 19 24 anbi12d 
 |-  ( x = ( A \ ( g " ( B \ ( f " U. D ) ) ) ) -> ( ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) <-> ( ( A \ ( g " ( B \ ( f " U. D ) ) ) ) C_ A /\ ( g " ( B \ ( f " ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) ) ) C_ ( A \ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) ) ) )
26 18 25 elab 
 |-  ( ( A \ ( g " ( B \ ( f " U. D ) ) ) ) e. { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) } <-> ( ( A \ ( g " ( B \ ( f " U. D ) ) ) ) C_ A /\ ( g " ( B \ ( f " ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) ) ) C_ ( A \ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) ) ) )
27 17 26 sylibr 
 |-  ( ran g C_ A -> ( A \ ( g " ( B \ ( f " U. D ) ) ) ) e. { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) } )
28 27 2 eleqtrrdi 
 |-  ( ran g C_ A -> ( A \ ( g " ( B \ ( f " U. D ) ) ) ) e. D )
29 elssuni 
 |-  ( ( A \ ( g " ( B \ ( f " U. D ) ) ) ) e. D -> ( A \ ( g " ( B \ ( f " U. D ) ) ) ) C_ U. D )
30 28 29 syl 
 |-  ( ran g C_ A -> ( A \ ( g " ( B \ ( f " U. D ) ) ) ) C_ U. D )