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Theorem sbthlem3 7628
Description: Lemma for sbth 7636. (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 7627 . . . . 5  |-  ( ran  g  C_  A  ->  ( A  \  ( g
" ( B  \ 
( f " U. D ) ) ) )  C_  U. D )
41, 2sbthlem1 7626 . . . . 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 3502 . . . 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 3605 . 2  |-  ( ran  g  C_  A  ->  ( A  \  U. D
)  =  ( A 
\  ( A  \ 
( g " ( B  \  ( f " U. D ) ) ) ) ) )
9 imassrn 5335 . . . 4  |-  ( g
" ( B  \ 
( f " U. D ) ) ) 
C_  ran  g
10 sstr2 3494 . . . 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 3715 . . 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 2483 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 1381    e. wcel 1802   {cab 2426   _Vcvv 3093    \ cdif 3456    C_ wss 3459   U.cuni 4231   ran crn 4987   "cima 4989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4555  ax-nul 4563  ax-pr 4673
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-nul 3769  df-if 3924  df-sn 4012  df-pr 4014  df-op 4018  df-uni 4232  df-br 4435  df-opab 4493  df-xp 4992  df-cnv 4994  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999
This theorem is referenced by:  sbthlem4  7629  sbthlem5  7630
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