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Theorem sbthlemi6 6955
Description: Lemma for isbth 6960. (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 5033 . . . . 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 4851 . . . . 5  |-  ran  H  =  ran  ( ( f  |`  U. D )  u.  ( `' g  |`  ( A  \  U. D
) ) )
8 df-ima 4636 . . . . . 6  |-  ( f
" U. D )  =  ran  ( f  |`  U. D )
98uneq1i 3285 . . . . 5  |-  ( ( f " U. D
)  u.  ran  ( `' g  |`  ( A 
\  U. D ) ) )  =  ( ran  ( f  |`  U. D
)  u.  ran  ( `' g  |`  ( A 
\  U. D ) ) )
105, 7, 93eqtr4i 2208 . . . 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 6953 . . . . . 6  |-  ( (EXMID  /\  ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( `' g " ( A  \  U. D ) )  =  ( B 
\  ( f " U. D ) ) )
14 df-ima 4636 . . . . . 6  |-  ( `' g " ( A 
\  U. D ) )  =  ran  ( `' g  |`  ( A  \ 
U. D ) )
1513, 14eqtr3di 2225 . . . . 5  |-  ( (EXMID  /\  ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( B  \  ( f " U. D ) )  =  ran  ( `' g  |`  ( A  \  U. D ) ) )
1615uneq2d 3289 . . . 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 2229 . . 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 4977 . . . . . . 7  |-  ( f
" U. D ) 
C_  ran  f
20 sstr2 3162 . . . . . . 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 6916 . . . . . . 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 4193 . . . . . . . . 9  |-  (EXMID  -> DECID  y  e.  (
f " U. D
) )
2524ralrimivw 2551 . . . . . . . 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 2183 . . 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 2210 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 2148   {cab 2163   A.wral 2455   _Vcvv 2737    \ cdif 3126    u. cun 3127    C_ wss 3129   U.cuni 3807  EXMIDwem 4191   `'ccnv 4622   dom cdm 4623   ran crn 4624    |` cres 4625   "cima 4626   Fun wfun 5206
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-exmid 4192  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-fun 5214
This theorem is referenced by:  sbthlemi9  6958
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