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Theorem sbthlem3 7428
Description: Lemma for sbth 7436. (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 7427 . . . . 5  |-  ( ran  g  C_  A  ->  ( A  \  ( g
" ( B  \ 
( f " U. D ) ) ) )  C_  U. D )
41, 2sbthlem1 7426 . . . . 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 3376 . . . 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 3479 . 2  |-  ( ran  g  C_  A  ->  ( A  \  U. D
)  =  ( A 
\  ( A  \ 
( g " ( B  \  ( f " U. D ) ) ) ) ) )
9 imassrn 5185 . . . 4  |-  ( g
" ( B  \ 
( f " U. D ) ) ) 
C_  ran  g
10 sstr2 3368 . . . 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 3589 . . 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 2476 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 1369    e. wcel 1756   {cab 2429   _Vcvv 2977    \ cdif 3330    C_ wss 3333   U.cuni 4096   ran crn 4846   "cima 4848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-xp 4851  df-cnv 4853  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858
This theorem is referenced by:  sbthlem4  7429  sbthlem5  7430
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