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Theorem sbthlem1 6939
Description: Lemma for sbth 6949. (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
sbthlem1  |-  U. D  C_  ( A  \  (
g " ( B 
\  ( f " 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 sbthlem1
StepHypRef Expression
1 unissb 3831 . 2  |-  ( U. D  C_  ( A  \ 
( g " ( B  \  ( f " U. D ) ) ) )  <->  A. x  e.  D  x  C_  ( A  \ 
( g " ( B  \  ( f " U. D ) ) ) ) )
2 sbthlem.2 . . . . 5  |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f "
x ) ) ) 
C_  ( A  \  x ) ) }
32abeq2i 2365 . . . 4  |-  ( x  e.  D  <->  ( x  C_  A  /\  ( g
" ( B  \ 
( f " x
) ) )  C_  ( A  \  x
) ) )
4 difss 3278 . . . . . . . 8  |-  ( A 
\  x )  C_  A
5 sstr2 3161 . . . . . . . 8  |-  ( ( g " ( B 
\  ( f "
x ) ) ) 
C_  ( A  \  x )  ->  (
( A  \  x
)  C_  A  ->  ( g " ( B 
\  ( f "
x ) ) ) 
C_  A ) )
64, 5mpi 18 . . . . . . 7  |-  ( ( g " ( B 
\  ( f "
x ) ) ) 
C_  ( A  \  x )  ->  (
g " ( B 
\  ( f "
x ) ) ) 
C_  A )
7 ssconb 3284 . . . . . . . 8  |-  ( ( x  C_  A  /\  ( g " ( B  \  ( f "
x ) ) ) 
C_  A )  -> 
( x  C_  ( A  \  ( g "
( B  \  (
f " x ) ) ) )  <->  ( g " ( B  \ 
( f " x
) ) )  C_  ( A  \  x
) ) )
87exbiri 608 . . . . . . 7  |-  ( x 
C_  A  ->  (
( g " ( B  \  ( f "
x ) ) ) 
C_  A  ->  (
( g " ( B  \  ( f "
x ) ) ) 
C_  ( A  \  x )  ->  x  C_  ( A  \  (
g " ( B 
\  ( f "
x ) ) ) ) ) ) )
96, 8syl5 30 . . . . . 6  |-  ( x 
C_  A  ->  (
( g " ( B  \  ( f "
x ) ) ) 
C_  ( A  \  x )  ->  (
( g " ( B  \  ( f "
x ) ) ) 
C_  ( A  \  x )  ->  x  C_  ( A  \  (
g " ( B 
\  ( f "
x ) ) ) ) ) ) )
109pm2.43d 46 . . . . 5  |-  ( x 
C_  A  ->  (
( g " ( B  \  ( f "
x ) ) ) 
C_  ( A  \  x )  ->  x  C_  ( A  \  (
g " ( B 
\  ( f "
x ) ) ) ) ) )
1110imp 420 . . . 4  |-  ( ( x  C_  A  /\  ( g " ( B  \  ( f "
x ) ) ) 
C_  ( A  \  x ) )  ->  x  C_  ( A  \ 
( g " ( B  \  ( f "
x ) ) ) ) )
123, 11sylbi 189 . . 3  |-  ( x  e.  D  ->  x  C_  ( A  \  (
g " ( B 
\  ( f "
x ) ) ) ) )
13 elssuni 3829 . . . . 5  |-  ( x  e.  D  ->  x  C_ 
U. D )
14 imass2 5037 . . . . 5  |-  ( x 
C_  U. D  ->  (
f " x ) 
C_  ( f " U. D ) )
15 sscon 3285 . . . . 5  |-  ( ( f " x ) 
C_  ( f " U. D )  ->  ( B  \  ( f " U. D ) )  C_  ( B  \  (
f " x ) ) )
1613, 14, 153syl 20 . . . 4  |-  ( x  e.  D  ->  ( B  \  ( f " U. D ) )  C_  ( B  \  (
f " x ) ) )
17 imass2 5037 . . . 4  |-  ( ( B  \  ( f
" U. D ) )  C_  ( B  \  ( f " x
) )  ->  (
g " ( B 
\  ( f " U. D ) ) ) 
C_  ( g "
( B  \  (
f " x ) ) ) )
18 sscon 3285 . . . 4  |-  ( ( g " ( B 
\  ( f " U. D ) ) ) 
C_  ( g "
( B  \  (
f " x ) ) )  ->  ( A  \  ( g "
( B  \  (
f " x ) ) ) )  C_  ( A  \  (
g " ( B 
\  ( f " U. D ) ) ) ) )
1916, 17, 183syl 20 . . 3  |-  ( x  e.  D  ->  ( A  \  ( g "
( B  \  (
f " x ) ) ) )  C_  ( A  \  (
g " ( B 
\  ( f " U. D ) ) ) ) )
2012, 19sstrd 3164 . 2  |-  ( x  e.  D  ->  x  C_  ( A  \  (
g " ( B 
\  ( f " U. D ) ) ) ) )
211, 20mprgbir 2588 1  |-  U. D  C_  ( A  \  (
g " ( B 
\  ( f " U. D ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   {cab 2244   _Vcvv 2763    \ cdif 3124    C_ wss 3127   U.cuni 3801   "cima 4664
This theorem is referenced by:  sbthlem2  6940  sbthlem3  6941  sbthlem5  6943
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ral 2523  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-xp 4675  df-cnv 4677  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682
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