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Theorem sbthlem6 6976
Description: Lemma for sbth 6981. (Contributed by NM, 27-Mar-1998.)
Hypotheses
Ref Expression
sbthlem.1  |-  A  e. 
_V
sbthlem.2  |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f "
x ) ) ) 
C_  ( A  \  x ) ) }
sbthlem.3  |-  H  =  ( ( f  |`  U. D )  u.  ( `' g  |`  ( A 
\  U. D ) ) )
Assertion
Ref Expression
sbthlem6  |-  ( ( ran  f  C_  B  /\  ( ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  ran  H  =  B )
Distinct variable groups:    x, A    x, B    x, D    x, f    x, g    x, H
Allowed substitution hints:    A( f, g)    B( f, g)    D( f, g)    H( f, g)

Proof of Theorem sbthlem6
StepHypRef Expression
1 df-ima 4702 . . . . 5  |-  ( `' g " ( A 
\  U. D ) )  =  ran  ( `' g  |`  ( A  \ 
U. D ) )
2 sbthlem.1 . . . . . 6  |-  A  e. 
_V
3 sbthlem.2 . . . . . 6  |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f "
x ) ) ) 
C_  ( A  \  x ) ) }
42, 3sbthlem4 6974 . . . . 5  |-  ( ( ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( `' g " ( A  \  U. D ) )  =  ( B 
\  ( f " U. D ) ) )
51, 4syl5reqr 2330 . . . 4  |-  ( ( ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( B  \  ( f " U. D ) )  =  ran  ( `' g  |`  ( A  \  U. D ) ) )
65uneq2d 3329 . . 3  |-  ( ( ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  (
( f " U. D )  u.  ( B  \  ( f " U. D ) ) )  =  ( ( f
" U. D )  u.  ran  ( `' g  |`  ( A  \ 
U. D ) ) ) )
7 rnun 5089 . . . 4  |-  ran  (
( f  |`  U. D
)  u.  ( `' g  |`  ( A  \ 
U. D ) ) )  =  ( ran  ( f  |`  U. D
)  u.  ran  ( `' g  |`  ( A 
\  U. D ) ) )
8 sbthlem.3 . . . . 5  |-  H  =  ( ( f  |`  U. D )  u.  ( `' g  |`  ( A 
\  U. D ) ) )
98rneqi 4905 . . . 4  |-  ran  H  =  ran  ( ( f  |`  U. D )  u.  ( `' g  |`  ( A  \  U. D
) ) )
10 df-ima 4702 . . . . 5  |-  ( f
" U. D )  =  ran  ( f  |`  U. D )
1110uneq1i 3325 . . . 4  |-  ( ( f " U. D
)  u.  ran  ( `' g  |`  ( A 
\  U. D ) ) )  =  ( ran  ( f  |`  U. D
)  u.  ran  ( `' g  |`  ( A 
\  U. D ) ) )
127, 9, 113eqtr4i 2313 . . 3  |-  ran  H  =  ( ( f
" U. D )  u.  ran  ( `' g  |`  ( A  \ 
U. D ) ) )
136, 12syl6reqr 2334 . 2  |-  ( ( ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ran  H  =  ( ( f
" U. D )  u.  ( B  \ 
( f " U. D ) ) ) )
14 imassrn 5025 . . . 4  |-  ( f
" U. D ) 
C_  ran  f
15 sstr2 3186 . . . 4  |-  ( ( f " U. D
)  C_  ran  f  -> 
( ran  f  C_  B  ->  ( f " U. D )  C_  B
) )
1614, 15ax-mp 8 . . 3  |-  ( ran  f  C_  B  ->  ( f " U. D
)  C_  B )
17 undif 3534 . . 3  |-  ( ( f " U. D
)  C_  B  <->  ( (
f " U. D
)  u.  ( B 
\  ( f " U. D ) ) )  =  B )
1816, 17sylib 188 . 2  |-  ( ran  f  C_  B  ->  ( ( f " U. D )  u.  ( B  \  ( f " U. D ) ) )  =  B )
1913, 18sylan9eqr 2337 1  |-  ( ( ran  f  C_  B  /\  ( ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  ran  H  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   _Vcvv 2788    \ cdif 3149    u. cun 3150    C_ wss 3152   U.cuni 3827   `'ccnv 4688   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692   Fun wfun 5249
This theorem is referenced by:  sbthlem9  6979
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5257
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