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Theorem sbthlemi3 6854
Description: Lemma for isbth 6862. (Contributed by NM, 22-Mar-1998.)
Hypotheses
Ref Expression
sbthlem.1  |-  A  e. 
_V
sbthlem.2  |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f "
x ) ) ) 
C_  ( A  \  x ) ) }
Assertion
Ref Expression
sbthlemi3  |-  ( (EXMID  /\ 
ran  g  C_  A
)  ->  ( g " ( B  \ 
( f " U. D ) ) )  =  ( A  \  U. D ) )
Distinct variable groups:    x, A    x, B    x, D    x, f    x, g
Allowed substitution hints:    A( f, g)    B( f, g)    D( f, g)

Proof of Theorem sbthlemi3
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 sbthlem.1 . . . . . . 7  |-  A  e. 
_V
2 sbthlem.2 . . . . . . 7  |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f "
x ) ) ) 
C_  ( A  \  x ) ) }
31, 2sbthlem2 6853 . . . . . 6  |-  ( ran  g  C_  A  ->  ( A  \  ( g
" ( B  \ 
( f " U. D ) ) ) )  C_  U. D )
41, 2sbthlem1 6852 . . . . . 6  |-  U. D  C_  ( A  \  (
g " ( B 
\  ( f " U. D ) ) ) )
53, 4jctil 310 . . . . 5  |-  ( ran  g  C_  A  ->  ( U. D  C_  ( A  \  ( g "
( B  \  (
f " U. D
) ) ) )  /\  ( A  \ 
( g " ( B  \  ( f " U. D ) ) ) )  C_  U. D ) )
6 eqss 3116 . . . . 5  |-  ( U. D  =  ( A  \  ( g " ( B  \  ( f " U. D ) ) ) )  <->  ( U. D  C_  ( A  \  (
g " ( B 
\  ( f " U. D ) ) ) )  /\  ( A 
\  ( g "
( B  \  (
f " U. D
) ) ) ) 
C_  U. D ) )
75, 6sylibr 133 . . . 4  |-  ( ran  g  C_  A  ->  U. D  =  ( A 
\  ( g "
( B  \  (
f " U. D
) ) ) ) )
87difeq2d 3198 . . 3  |-  ( ran  g  C_  A  ->  ( A  \  U. D
)  =  ( A 
\  ( A  \ 
( g " ( B  \  ( f " U. D ) ) ) ) ) )
98adantl 275 . 2  |-  ( (EXMID  /\ 
ran  g  C_  A
)  ->  ( A  \ 
U. D )  =  ( A  \  ( A  \  ( g "
( B  \  (
f " U. D
) ) ) ) ) )
10 imassrn 4899 . . . . 5  |-  ( g
" ( B  \ 
( f " U. D ) ) ) 
C_  ran  g
11 sstr2 3108 . . . . 5  |-  ( ( g " ( B 
\  ( f " U. D ) ) ) 
C_  ran  g  ->  ( ran  g  C_  A  ->  ( g " ( B  \  ( f " U. D ) ) ) 
C_  A ) )
1210, 11ax-mp 5 . . . 4  |-  ( ran  g  C_  A  ->  ( g " ( B 
\  ( f " U. D ) ) ) 
C_  A )
13 exmidexmid 4127 . . . . . . 7  |-  (EXMID  -> DECID  y  e.  (
g " ( B 
\  ( f " U. D ) ) ) )
14 dcstab 830 . . . . . . 7  |-  (DECID  y  e.  ( g " ( B  \  ( f " U. D ) ) )  -> STAB  y  e.  ( g
" ( B  \ 
( f " U. D ) ) ) )
1513, 14syl 14 . . . . . 6  |-  (EXMID  -> STAB  y  e.  ( g " ( B  \  ( f " U. D ) ) ) )
1615alrimiv 1847 . . . . 5  |-  (EXMID  ->  A. ySTAB  y  e.  ( g " ( B  \  ( f " U. D ) ) ) )
17 dfss4st 3313 . . . . 5  |-  ( A. ySTAB  y  e.  ( g " ( B  \ 
( f " U. D ) ) )  ->  ( ( g
" ( B  \ 
( f " U. D ) ) ) 
C_  A  <->  ( A  \  ( A  \  (
g " ( B 
\  ( f " U. D ) ) ) ) )  =  ( g " ( B 
\  ( f " U. D ) ) ) ) )
1816, 17syl 14 . . . 4  |-  (EXMID  ->  (
( g " ( B  \  ( f " U. D ) ) ) 
C_  A  <->  ( A  \  ( A  \  (
g " ( B 
\  ( f " U. D ) ) ) ) )  =  ( g " ( B 
\  ( f " U. D ) ) ) ) )
1912, 18syl5ib 153 . . 3  |-  (EXMID  ->  ( ran  g  C_  A  -> 
( A  \  ( A  \  ( g "
( B  \  (
f " U. D
) ) ) ) )  =  ( g
" ( B  \ 
( f " U. D ) ) ) ) )
2019imp 123 . 2  |-  ( (EXMID  /\ 
ran  g  C_  A
)  ->  ( A  \  ( A  \  (
g " ( B 
\  ( f " U. D ) ) ) ) )  =  ( g " ( B 
\  ( f " U. D ) ) ) )
219, 20eqtr2d 2174 1  |-  ( (EXMID  /\ 
ran  g  C_  A
)  ->  ( g " ( B  \ 
( f " U. D ) ) )  =  ( A  \  U. D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104  STAB wstab 816  DECID wdc 820   A.wal 1330    = wceq 1332    e. wcel 1481   {cab 2126   _Vcvv 2689    \ cdif 3072    C_ wss 3075   U.cuni 3743  EXMIDwem 4125   ran crn 4547   "cima 4549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-exmid 4126  df-xp 4552  df-cnv 4554  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559
This theorem is referenced by:  sbthlemi4  6855  sbthlemi5  6856
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