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Theorem sbthlemi6 6802
Description: Lemma for isbth 6807. (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 501 . . 3  |-  ( ( (EXMID 
/\  ran  f  C_  B )  /\  (
( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g ) )  -> EXMID )
2 simprll 509 . . 3  |-  ( ( (EXMID 
/\  ran  f  C_  B )  /\  (
( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  dom  g  =  B
)
3 simprlr 510 . . 3  |-  ( ( (EXMID 
/\  ran  f  C_  B )  /\  (
( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  ran  g  C_  A )
4 simprr 504 . . 3  |-  ( ( (EXMID 
/\  ran  f  C_  B )  /\  (
( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  Fun  `' g )
5 df-ima 4512 . . . . . 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 6800 . . . . . 6  |-  ( (EXMID  /\  ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( `' g " ( A  \  U. D ) )  =  ( B 
\  ( f " U. D ) ) )
95, 8syl5reqr 2162 . . . . 5  |-  ( (EXMID  /\  ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( B  \  ( f " U. D ) )  =  ran  ( `' g  |`  ( A  \  U. D ) ) )
109uneq2d 3196 . . . 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 4905 . . . . 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 4727 . . . . 5  |-  ran  H  =  ran  ( ( f  |`  U. D )  u.  ( `' g  |`  ( A  \  U. D
) ) )
14 df-ima 4512 . . . . . 6  |-  ( f
" U. D )  =  ran  ( f  |`  U. D )
1514uneq1i 3192 . . . . 5  |-  ( ( f " U. D
)  u.  ran  ( `' g  |`  ( A 
\  U. D ) ) )  =  ( ran  ( f  |`  U. D
)  u.  ran  ( `' g  |`  ( A 
\  U. D ) ) )
1611, 13, 153eqtr4i 2145 . . . 4  |-  ran  H  =  ( ( f
" U. D )  u.  ran  ( `' g  |`  ( A  \ 
U. D ) ) )
1710, 16syl6reqr 2166 . . 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 1204 . 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 4850 . . . . . . 7  |-  ( f
" U. D ) 
C_  ran  f
20 sstr2 3070 . . . . . . 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 273 . . . . 5  |-  ( (EXMID  /\ 
ran  f  C_  B
)  ->  ( f " U. D )  C_  B )
23 exmidexmid 4080 . . . . . . . . 9  |-  (EXMID  -> DECID  y  e.  (
f " U. D
) )
2423ralrimivw 2480 . . . . . . . 8  |-  (EXMID  ->  A. y  e.  B DECID  y  e.  (
f " U. D
) )
2524biantrud 300 . . . . . . 7  |-  (EXMID  ->  (
( f " U. D )  C_  B  <->  ( ( f " U. D )  C_  B  /\  A. y  e.  B DECID  y  e.  ( f " U. D ) ) ) )
26 undifdcss 6764 . . . . . . 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 198 . . . . . 6  |-  (EXMID  ->  ( B  =  ( (
f " U. D
)  u.  ( B 
\  ( f " U. D ) ) )  <-> 
( f " U. D )  C_  B
) )
2827adantr 272 . . . . 5  |-  ( (EXMID  /\ 
ran  f  C_  B
)  ->  ( B  =  ( ( f
" U. D )  u.  ( B  \ 
( f " U. D ) ) )  <-> 
( f " U. D )  C_  B
) )
2922, 28mpbird 166 . . . 4  |-  ( (EXMID  /\ 
ran  f  C_  B
)  ->  B  =  ( ( f " U. D )  u.  ( B  \  ( f " U. D ) ) ) )
3029eqcomd 2120 . . 3  |-  ( (EXMID  /\ 
ran  f  C_  B
)  ->  ( (
f " U. D
)  u.  ( B 
\  ( f " U. D ) ) )  =  B )
3130adantr 272 . 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 2147 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 103    <-> wb 104  DECID wdc 802    /\ w3a 945    = wceq 1314    e. wcel 1463   {cab 2101   A.wral 2390   _Vcvv 2657    \ cdif 3034    u. cun 3035    C_ wss 3037   U.cuni 3702  EXMIDwem 4078   `'ccnv 4498   dom cdm 4499   ran crn 4500    |` cres 4501   "cima 4502   Fun wfun 5075
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091
This theorem depends on definitions:  df-bi 116  df-stab 799  df-dc 803  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-rab 2399  df-v 2659  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-exmid 4079  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-fun 5083
This theorem is referenced by:  sbthlemi9  6805
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