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Theorem sbthlem3 7641
Description: Lemma for sbth 7649. (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
sbthlem3  |-  ( 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 sbthlem3
StepHypRef Expression
1 sbthlem.1 . . . . . 6  |-  A  e. 
_V
2 sbthlem.2 . . . . . 6  |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f "
x ) ) ) 
C_  ( A  \  x ) ) }
31, 2sbthlem2 7640 . . . . 5  |-  ( ran  g  C_  A  ->  ( A  \  ( g
" ( B  \ 
( f " U. D ) ) ) )  C_  U. D )
41, 2sbthlem1 7639 . . . . 5  |-  U. D  C_  ( A  \  (
g " ( B 
\  ( f " U. D ) ) ) )
53, 4jctil 537 . . . 4  |-  ( ran  g  C_  A  ->  ( U. D  C_  ( A  \  ( g "
( B  \  (
f " U. D
) ) ) )  /\  ( A  \ 
( g " ( B  \  ( f " U. D ) ) ) )  C_  U. D ) )
6 eqss 3524 . . . 4  |-  ( 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 212 . . 3  |-  ( ran  g  C_  A  ->  U. D  =  ( A 
\  ( g "
( B  \  (
f " U. D
) ) ) ) )
87difeq2d 3627 . 2  |-  ( ran  g  C_  A  ->  ( A  \  U. D
)  =  ( A 
\  ( A  \ 
( g " ( B  \  ( f " U. D ) ) ) ) ) )
9 imassrn 5354 . . . 4  |-  ( g
" ( B  \ 
( f " U. D ) ) ) 
C_  ran  g
10 sstr2 3516 . . . 4  |-  ( ( g " ( B 
\  ( f " U. D ) ) ) 
C_  ran  g  ->  ( ran  g  C_  A  ->  ( g " ( B  \  ( f " U. D ) ) ) 
C_  A ) )
119, 10ax-mp 5 . . 3  |-  ( ran  g  C_  A  ->  ( g " ( B 
\  ( f " U. D ) ) ) 
C_  A )
12 dfss4 3737 . . 3  |-  ( ( g " ( B 
\  ( f " U. D ) ) ) 
C_  A  <->  ( A  \  ( A  \  (
g " ( B 
\  ( f " U. D ) ) ) ) )  =  ( g " ( B 
\  ( f " U. D ) ) ) )
1311, 12sylib 196 . 2  |-  ( ran  g  C_  A  ->  ( A  \  ( A 
\  ( g "
( B  \  (
f " U. D
) ) ) ) )  =  ( g
" ( B  \ 
( f " U. D ) ) ) )
148, 13eqtr2d 2509 1  |-  ( ran  g  C_  A  ->  ( g " ( B 
\  ( f " U. D ) ) )  =  ( A  \  U. D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452   _Vcvv 3118    \ cdif 3478    C_ wss 3481   U.cuni 4251   ran crn 5006   "cima 5008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-xp 5011  df-cnv 5013  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018
This theorem is referenced by:  sbthlem4  7642  sbthlem5  7643
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