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Metamath Proof Explorer

Theorem sbthlem6

Description: Lemma for sbth . (Contributed by NM, 27-Mar-1998)

Ref Expression
Hypotheses sbthlem.1 
|- A e. _V
sbthlem.2 
|- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }
sbthlem.3 
|- H = ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )
Assertion sbthlem6 
|- ( ( ran f C_ B /\ ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ) -> ran H = B )

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 sbthlem.3 
 |-  H = ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )
4 rnun 
 |-  ran ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) ) = ( ran ( f |` U. D ) u. ran ( `' g |` ( A \ U. D ) ) )
5 3 rneqi 
 |-  ran H = ran ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )
6 df-ima 
 |-  ( f " U. D ) = ran ( f |` U. D )
7 6 uneq1i 
 |-  ( ( f " U. D ) u. ran ( `' g |` ( A \ U. D ) ) ) = ( ran ( f |` U. D ) u. ran ( `' g |` ( A \ U. D ) ) )
8 4 5 7 3eqtr4i 
 |-  ran H = ( ( f " U. D ) u. ran ( `' g |` ( A \ U. D ) ) )
9 1 2 sbthlem4 
 |-  ( ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( `' g " ( A \ U. D ) ) = ( B \ ( f " U. D ) ) )
10 df-ima 
 |-  ( `' g " ( A \ U. D ) ) = ran ( `' g |` ( A \ U. D ) )
11 9 10 eqtr3di 
 |-  ( ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( B \ ( f " U. D ) ) = ran ( `' g |` ( A \ U. D ) ) )
12 11 uneq2d 
 |-  ( ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( ( f " U. D ) u. ( B \ ( f " U. D ) ) ) = ( ( f " U. D ) u. ran ( `' g |` ( A \ U. D ) ) ) )
13 8 12 eqtr4id 
 |-  ( ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ran H = ( ( f " U. D ) u. ( B \ ( f " U. D ) ) ) )
14 imassrn 
 |-  ( f " U. D ) C_ ran f
15 sstr2 
 |-  ( ( f " U. D ) C_ ran f -> ( ran f C_ B -> ( f " U. D ) C_ B ) )
16 14 15 ax-mp 
 |-  ( ran f C_ B -> ( f " U. D ) C_ B )
17 undif 
 |-  ( ( f " U. D ) C_ B <-> ( ( f " U. D ) u. ( B \ ( f " U. D ) ) ) = B )
18 16 17 sylib 
 |-  ( ran f C_ B -> ( ( f " U. D ) u. ( B \ ( f " U. D ) ) ) = B )
19 13 18 sylan9eqr 
 |-  ( ( ran f C_ B /\ ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ) -> ran H = B )