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Theorem sbthlemi6 6951
Description: Lemma for isbth 6956. (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 527 . . 3  |-  ( ( (EXMID 
/\  ran  f  C_  B )  /\  (
( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g ) )  -> EXMID )
2 simprll 537 . . 3  |-  ( ( (EXMID 
/\  ran  f  C_  B )  /\  (
( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  dom  g  =  B
)
3 simprlr 538 . . 3  |-  ( ( (EXMID 
/\  ran  f  C_  B )  /\  (
( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  ran  g  C_  A )
4 simprr 531 . . 3  |-  ( ( (EXMID 
/\  ran  f  C_  B )  /\  (
( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  Fun  `' g )
5 rnun 5029 . . . . 5  |-  ran  (
( f  |`  U. D
)  u.  ( `' g  |`  ( A  \ 
U. D ) ) )  =  ( ran  ( f  |`  U. D
)  u.  ran  ( `' g  |`  ( A 
\  U. D ) ) )
6 sbthlem.3 . . . . . 6  |-  H  =  ( ( f  |`  U. D )  u.  ( `' g  |`  ( A 
\  U. D ) ) )
76rneqi 4848 . . . . 5  |-  ran  H  =  ran  ( ( f  |`  U. D )  u.  ( `' g  |`  ( A  \  U. D
) ) )
8 df-ima 4633 . . . . . 6  |-  ( f
" U. D )  =  ran  ( f  |`  U. D )
98uneq1i 3283 . . . . 5  |-  ( ( f " U. D
)  u.  ran  ( `' g  |`  ( A 
\  U. D ) ) )  =  ( ran  ( f  |`  U. D
)  u.  ran  ( `' g  |`  ( A 
\  U. D ) ) )
105, 7, 93eqtr4i 2206 . . . 4  |-  ran  H  =  ( ( f
" U. D )  u.  ran  ( `' g  |`  ( A  \ 
U. D ) ) )
11 sbthlem.1 . . . . . . 7  |-  A  e. 
_V
12 sbthlem.2 . . . . . . 7  |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f "
x ) ) ) 
C_  ( A  \  x ) ) }
1311, 12sbthlemi4 6949 . . . . . 6  |-  ( (EXMID  /\  ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( `' g " ( A  \  U. D ) )  =  ( B 
\  ( f " U. D ) ) )
14 df-ima 4633 . . . . . 6  |-  ( `' g " ( A 
\  U. D ) )  =  ran  ( `' g  |`  ( A  \ 
U. D ) )
1513, 14eqtr3di 2223 . . . . 5  |-  ( (EXMID  /\  ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( B  \  ( f " U. D ) )  =  ran  ( `' g  |`  ( A  \  U. D ) ) )
1615uneq2d 3287 . . . 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 ) ) ) )
1710, 16eqtr4id 2227 . . 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 1243 . 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 4974 . . . . . . 7  |-  ( f
" U. D ) 
C_  ran  f
20 sstr2 3160 . . . . . . 7  |-  ( ( f " U. D
)  C_  ran  f  -> 
( ran  f  C_  B  ->  ( f " U. D )  C_  B
) )
2119, 20ax-mp 5 . . . . . 6  |-  ( ran  f  C_  B  ->  ( f " U. D
)  C_  B )
2221adantl 277 . . . . 5  |-  ( (EXMID  /\ 
ran  f  C_  B
)  ->  ( f " U. D )  C_  B )
23 undifdcss 6912 . . . . . . 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 ) ) )
24 exmidexmid 4191 . . . . . . . . 9  |-  (EXMID  -> DECID  y  e.  (
f " U. D
) )
2524ralrimivw 2549 . . . . . . . 8  |-  (EXMID  ->  A. y  e.  B DECID  y  e.  (
f " U. D
) )
2625biantrud 304 . . . . . . 7  |-  (EXMID  ->  (
( f " U. D )  C_  B  <->  ( ( f " U. D )  C_  B  /\  A. y  e.  B DECID  y  e.  ( f " U. D ) ) ) )
2723, 26bitr4id 199 . . . . . 6  |-  (EXMID  ->  ( B  =  ( (
f " U. D
)  u.  ( B 
\  ( f " U. D ) ) )  <-> 
( f " U. D )  C_  B
) )
2827adantr 276 . . . . 5  |-  ( (EXMID  /\ 
ran  f  C_  B
)  ->  ( B  =  ( ( f
" U. D )  u.  ( B  \ 
( f " U. D ) ) )  <-> 
( f " U. D )  C_  B
) )
2922, 28mpbird 167 . . . 4  |-  ( (EXMID  /\ 
ran  f  C_  B
)  ->  B  =  ( ( f " U. D )  u.  ( B  \  ( f " U. D ) ) ) )
3029eqcomd 2181 . . 3  |-  ( (EXMID  /\ 
ran  f  C_  B
)  ->  ( (
f " U. D
)  u.  ( B 
\  ( f " U. D ) ) )  =  B )
3130adantr 276 . 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 2208 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 104    <-> wb 105  DECID wdc 834    /\ w3a 978    = wceq 1353    e. wcel 2146   {cab 2161   A.wral 2453   _Vcvv 2735    \ cdif 3124    u. cun 3125    C_ wss 3127   U.cuni 3805  EXMIDwem 4189   `'ccnv 4619   dom cdm 4620   ran crn 4621    |` cres 4622   "cima 4623   Fun wfun 5202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-exmid 4190  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-fun 5210
This theorem is referenced by:  sbthlemi9  6954
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