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Theorem sbthlem1 6973
Description: Lemma for sbth 6983. (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 3859 . 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 2392 . . . 4  |-  ( x  e.  D  <->  ( x  C_  A  /\  ( g
" ( B  \ 
( f " x
) ) )  C_  ( A  \  x
) ) )
4 difss 3305 . . . . . . . 8  |-  ( A 
\  x )  C_  A
5 sstr2 3188 . . . . . . . 8  |-  ( ( g " ( B 
\  ( f "
x ) ) ) 
C_  ( A  \  x )  ->  (
( A  \  x
)  C_  A  ->  ( g " ( B 
\  ( f "
x ) ) ) 
C_  A ) )
64, 5mpi 16 . . . . . . 7  |-  ( ( g " ( B 
\  ( f "
x ) ) ) 
C_  ( A  \  x )  ->  (
g " ( B 
\  ( f "
x ) ) ) 
C_  A )
7 ssconb 3311 . . . . . . . 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 605 . . . . . . 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 28 . . . . . 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 44 . . . . 5  |-  ( x 
C_  A  ->  (
( g " ( B  \  ( f "
x ) ) ) 
C_  ( A  \  x )  ->  x  C_  ( A  \  (
g " ( B 
\  ( f "
x ) ) ) ) ) )
1110imp 418 . . . 4  |-  ( ( x  C_  A  /\  ( g " ( B  \  ( f "
x ) ) ) 
C_  ( A  \  x ) )  ->  x  C_  ( A  \ 
( g " ( B  \  ( f "
x ) ) ) ) )
123, 11sylbi 187 . . 3  |-  ( x  e.  D  ->  x  C_  ( A  \  (
g " ( B 
\  ( f "
x ) ) ) ) )
13 elssuni 3857 . . . . 5  |-  ( x  e.  D  ->  x  C_ 
U. D )
14 imass2 5051 . . . . 5  |-  ( x 
C_  U. D  ->  (
f " x ) 
C_  ( f " U. D ) )
15 sscon 3312 . . . . 5  |-  ( ( f " x ) 
C_  ( f " U. D )  ->  ( B  \  ( f " U. D ) )  C_  ( B  \  (
f " x ) ) )
1613, 14, 153syl 18 . . . 4  |-  ( x  e.  D  ->  ( B  \  ( f " U. D ) )  C_  ( B  \  (
f " x ) ) )
17 imass2 5051 . . . 4  |-  ( ( B  \  ( f
" U. D ) )  C_  ( B  \  ( f " x
) )  ->  (
g " ( B 
\  ( f " U. D ) ) ) 
C_  ( g "
( B  \  (
f " x ) ) ) )
18 sscon 3312 . . . 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 18 . . 3  |-  ( x  e.  D  ->  ( A  \  ( g "
( B  \  (
f " x ) ) ) )  C_  ( A  \  (
g " ( B 
\  ( f " U. D ) ) ) ) )
2012, 19sstrd 3191 . 2  |-  ( x  e.  D  ->  x  C_  ( A  \  (
g " ( B 
\  ( f " U. D ) ) ) ) )
211, 20mprgbir 2615 1  |-  U. D  C_  ( A  \  (
g " ( B 
\  ( f " U. D ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1625    e. wcel 1686   {cab 2271   _Vcvv 2790    \ cdif 3151    C_ wss 3154   U.cuni 3829   "cima 4694
This theorem is referenced by:  sbthlem2  6974  sbthlem3  6975  sbthlem5  6977
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ral 2550  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-xp 4697  df-cnv 4699  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704
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