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Theorem sbthlemi6 6669
Description: Lemma for isbth 6674. (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
sbthlemi6  |-  ( ( (EXMID 
/\  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 sbthlemi6
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 simpll 496 . . 3  |-  ( ( (EXMID 
/\  ran  f  C_  B )  /\  (
( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g ) )  -> EXMID )
2 simprll 504 . . 3  |-  ( ( (EXMID 
/\  ran  f  C_  B )  /\  (
( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  dom  g  =  B
)
3 simprlr 505 . . 3  |-  ( ( (EXMID 
/\  ran  f  C_  B )  /\  (
( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  ran  g  C_  A )
4 simprr 499 . . 3  |-  ( ( (EXMID 
/\  ran  f  C_  B )  /\  (
( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  Fun  `' g )
5 df-ima 4451 . . . . . 6  |-  ( `' g " ( A 
\  U. D ) )  =  ran  ( `' g  |`  ( A  \ 
U. D ) )
6 sbthlem.1 . . . . . . 7  |-  A  e. 
_V
7 sbthlem.2 . . . . . . 7  |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f "
x ) ) ) 
C_  ( A  \  x ) ) }
86, 7sbthlemi4 6667 . . . . . 6  |-  ( (EXMID  /\  ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( `' g " ( A  \  U. D ) )  =  ( B 
\  ( f " U. D ) ) )
95, 8syl5reqr 2135 . . . . 5  |-  ( (EXMID  /\  ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( B  \  ( f " U. D ) )  =  ran  ( `' g  |`  ( A  \  U. D ) ) )
109uneq2d 3154 . . . 4  |-  ( (EXMID  /\  ( 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 ) ) ) )
11 rnun 4840 . . . . 5  |-  ran  (
( f  |`  U. D
)  u.  ( `' g  |`  ( A  \ 
U. D ) ) )  =  ( ran  ( f  |`  U. D
)  u.  ran  ( `' g  |`  ( A 
\  U. D ) ) )
12 sbthlem.3 . . . . . 6  |-  H  =  ( ( f  |`  U. D )  u.  ( `' g  |`  ( A 
\  U. D ) ) )
1312rneqi 4663 . . . . 5  |-  ran  H  =  ran  ( ( f  |`  U. D )  u.  ( `' g  |`  ( A  \  U. D
) ) )
14 df-ima 4451 . . . . . 6  |-  ( f
" U. D )  =  ran  ( f  |`  U. D )
1514uneq1i 3150 . . . . 5  |-  ( ( f " U. D
)  u.  ran  ( `' g  |`  ( A 
\  U. D ) ) )  =  ( ran  ( f  |`  U. D
)  u.  ran  ( `' g  |`  ( A 
\  U. D ) ) )
1611, 13, 153eqtr4i 2118 . . . 4  |-  ran  H  =  ( ( f
" U. D )  u.  ran  ( `' g  |`  ( A  \ 
U. D ) ) )
1710, 16syl6reqr 2139 . . 3  |-  ( (EXMID  /\  ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ran  H  =  ( ( f
" U. D )  u.  ( B  \ 
( f " U. D ) ) ) )
181, 2, 3, 4, 17syl121anc 1179 . 2  |-  ( ( (EXMID 
/\  ran  f  C_  B )  /\  (
( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  ran  H  =  ( ( f " U. D
)  u.  ( B 
\  ( f " U. D ) ) ) )
19 imassrn 4785 . . . . . . 7  |-  ( f
" U. D ) 
C_  ran  f
20 sstr2 3032 . . . . . . 7  |-  ( ( f " U. D
)  C_  ran  f  -> 
( ran  f  C_  B  ->  ( f " U. D )  C_  B
) )
2119, 20ax-mp 7 . . . . . 6  |-  ( ran  f  C_  B  ->  ( f " U. D
)  C_  B )
2221adantl 271 . . . . 5  |-  ( (EXMID  /\ 
ran  f  C_  B
)  ->  ( f " U. D )  C_  B )
23 exmidexmid 4031 . . . . . . . . 9  |-  (EXMID  -> DECID  y  e.  (
f " U. D
) )
2423ralrimivw 2447 . . . . . . . 8  |-  (EXMID  ->  A. y  e.  B DECID  y  e.  (
f " U. D
) )
2524biantrud 298 . . . . . . 7  |-  (EXMID  ->  (
( f " U. D )  C_  B  <->  ( ( f " U. D )  C_  B  /\  A. y  e.  B DECID  y  e.  ( f " U. D ) ) ) )
26 undifdcss 6631 . . . . . . 7  |-  ( B  =  ( ( f
" U. D )  u.  ( B  \ 
( f " U. D ) ) )  <-> 
( ( f " U. D )  C_  B  /\  A. y  e.  B DECID  y  e.  ( f " U. D ) ) )
2725, 26syl6rbbr 197 . . . . . 6  |-  (EXMID  ->  ( B  =  ( (
f " U. D
)  u.  ( B 
\  ( f " U. D ) ) )  <-> 
( f " U. D )  C_  B
) )
2827adantr 270 . . . . 5  |-  ( (EXMID  /\ 
ran  f  C_  B
)  ->  ( B  =  ( ( f
" U. D )  u.  ( B  \ 
( f " U. D ) ) )  <-> 
( f " U. D )  C_  B
) )
2922, 28mpbird 165 . . . 4  |-  ( (EXMID  /\ 
ran  f  C_  B
)  ->  B  =  ( ( f " U. D )  u.  ( B  \  ( f " U. D ) ) ) )
3029eqcomd 2093 . . 3  |-  ( (EXMID  /\ 
ran  f  C_  B
)  ->  ( (
f " U. D
)  u.  ( B 
\  ( f " U. D ) ) )  =  B )
3130adantr 270 . 2  |-  ( ( (EXMID 
/\  ran  f  C_  B )  /\  (
( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g ) )  -> 
( ( f " U. D )  u.  ( B  \  ( f " U. D ) ) )  =  B )
3218, 31eqtrd 2120 1  |-  ( ( (EXMID 
/\  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 102    <-> wb 103  DECID wdc 780    /\ w3a 924    = wceq 1289    e. wcel 1438   {cab 2074   A.wral 2359   _Vcvv 2619    \ cdif 2996    u. cun 2997    C_ wss 2999   U.cuni 3653  EXMIDwem 4029   `'ccnv 4437   dom cdm 4438   ran crn 4439    |` cres 4440   "cima 4441   Fun wfun 5009
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-stab 776  df-dc 781  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-exmid 4030  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-fun 5017
This theorem is referenced by:  sbthlemi9  6672
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