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Theorem sbthlem1 7209
Description: Lemma for sbth 7219. (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 4037 . 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 2542 . . . 4  |-  ( x  e.  D  <->  ( x  C_  A  /\  ( g
" ( B  \ 
( f " x
) ) )  C_  ( A  \  x
) ) )
4 difss2 3468 . . . . . . 7  |-  ( ( g " ( B 
\  ( f "
x ) ) ) 
C_  ( A  \  x )  ->  (
g " ( B 
\  ( f "
x ) ) ) 
C_  A )
5 ssconb 3472 . . . . . . . 8  |-  ( ( x  C_  A  /\  ( g " ( B  \  ( f "
x ) ) ) 
C_  A )  -> 
( x  C_  ( A  \  ( g "
( B  \  (
f " x ) ) ) )  <->  ( g " ( B  \ 
( f " x
) ) )  C_  ( A  \  x
) ) )
65exbiri 606 . . . . . . 7  |-  ( x 
C_  A  ->  (
( g " ( B  \  ( f "
x ) ) ) 
C_  A  ->  (
( g " ( B  \  ( f "
x ) ) ) 
C_  ( A  \  x )  ->  x  C_  ( A  \  (
g " ( B 
\  ( f "
x ) ) ) ) ) ) )
74, 6syl5 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 ) ) ) ) ) ) )
87pm2.43d 46 . . . . 5  |-  ( x 
C_  A  ->  (
( g " ( B  \  ( f "
x ) ) ) 
C_  ( A  \  x )  ->  x  C_  ( A  \  (
g " ( B 
\  ( f "
x ) ) ) ) ) )
98imp 419 . . . 4  |-  ( ( x  C_  A  /\  ( g " ( B  \  ( f "
x ) ) ) 
C_  ( A  \  x ) )  ->  x  C_  ( A  \ 
( g " ( B  \  ( f "
x ) ) ) ) )
103, 9sylbi 188 . . 3  |-  ( x  e.  D  ->  x  C_  ( A  \  (
g " ( B 
\  ( f "
x ) ) ) ) )
11 elssuni 4035 . . . . 5  |-  ( x  e.  D  ->  x  C_ 
U. D )
12 imass2 5232 . . . . 5  |-  ( x 
C_  U. D  ->  (
f " x ) 
C_  ( f " U. D ) )
13 sscon 3473 . . . . 5  |-  ( ( f " x ) 
C_  ( f " U. D )  ->  ( B  \  ( f " U. D ) )  C_  ( B  \  (
f " x ) ) )
1411, 12, 133syl 19 . . . 4  |-  ( x  e.  D  ->  ( B  \  ( f " U. D ) )  C_  ( B  \  (
f " x ) ) )
15 imass2 5232 . . . 4  |-  ( ( B  \  ( f
" U. D ) )  C_  ( B  \  ( f " x
) )  ->  (
g " ( B 
\  ( f " U. D ) ) ) 
C_  ( g "
( B  \  (
f " x ) ) ) )
16 sscon 3473 . . . 4  |-  ( ( g " ( B 
\  ( f " U. D ) ) ) 
C_  ( g "
( B  \  (
f " x ) ) )  ->  ( A  \  ( g "
( B  \  (
f " x ) ) ) )  C_  ( A  \  (
g " ( B 
\  ( f " U. D ) ) ) ) )
1714, 15, 163syl 19 . . 3  |-  ( x  e.  D  ->  ( A  \  ( g "
( B  \  (
f " x ) ) ) )  C_  ( A  \  (
g " ( B 
\  ( f " U. D ) ) ) ) )
1810, 17sstrd 3350 . 2  |-  ( x  e.  D  ->  x  C_  ( A  \  (
g " ( B 
\  ( f " U. D ) ) ) ) )
191, 18mprgbir 2768 1  |-  U. D  C_  ( A  \  (
g " ( B 
\  ( f " U. D ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2421   _Vcvv 2948    \ cdif 3309    C_ wss 3312   U.cuni 4007   "cima 4873
This theorem is referenced by:  sbthlem2  7210  sbthlem3  7211  sbthlem5  7213
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-xp 4876  df-cnv 4878  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883
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