MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbthlem3 Structured version   Unicode version

Theorem sbthlem3 7548
Description: Lemma for sbth 7556. (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 7547 . . . . 5  |-  ( ran  g  C_  A  ->  ( A  \  ( g
" ( B  \ 
( f " U. D ) ) ) )  C_  U. D )
41, 2sbthlem1 7546 . . . . 5  |-  U. D  C_  ( A  \  (
g " ( B 
\  ( f " U. D ) ) ) )
53, 4jctil 535 . . . 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 3432 . . . 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 3536 . 2  |-  ( ran  g  C_  A  ->  ( A  \  U. D
)  =  ( A 
\  ( A  \ 
( g " ( B  \  ( f " U. D ) ) ) ) ) )
9 imassrn 5260 . . . 4  |-  ( g
" ( B  \ 
( f " U. D ) ) ) 
C_  ran  g
10 sstr2 3424 . . . 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 3657 . . 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 2424 1  |-  ( ran  g  C_  A  ->  ( g " ( B 
\  ( f " U. D ) ) )  =  ( A  \  U. D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1826   {cab 2367   _Vcvv 3034    \ cdif 3386    C_ wss 3389   U.cuni 4163   ran crn 4914   "cima 4916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-xp 4919  df-cnv 4921  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926
This theorem is referenced by:  sbthlem4  7549  sbthlem5  7550
  Copyright terms: Public domain W3C validator