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Theorem sbthlemi4 6973
Description: Lemma for isbth 6980. (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 ) ) }
Assertion
Ref Expression
sbthlemi4  |-  ( (EXMID  /\  ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( `' g " ( A  \  U. D ) )  =  ( B 
\  ( f " U. D ) ) )
Distinct variable groups:    x, A    x, B    x, D    x, f    x, g
Allowed substitution hints:    A( f, g)    B( f, g)    D( f, g)

Proof of Theorem sbthlemi4
StepHypRef Expression
1 df-ima 4651 . 2  |-  ( `' g " ( A 
\  U. D ) )  =  ran  ( `' g  |`  ( A  \ 
U. D ) )
2 difss 3273 . . . . . . . 8  |-  ( B 
\  ( f " U. D ) )  C_  B
3 sseq2 3191 . . . . . . . 8  |-  ( dom  g  =  B  -> 
( ( B  \ 
( f " U. D ) )  C_  dom  g  <->  ( B  \ 
( f " U. D ) )  C_  B ) )
42, 3mpbiri 168 . . . . . . 7  |-  ( dom  g  =  B  -> 
( B  \  (
f " U. D
) )  C_  dom  g )
5 ssdmres 4941 . . . . . . 7  |-  ( ( B  \  ( f
" U. D ) )  C_  dom  g  <->  dom  ( g  |`  ( B  \  (
f " U. D
) ) )  =  ( B  \  (
f " U. D
) ) )
64, 5sylib 122 . . . . . 6  |-  ( dom  g  =  B  ->  dom  ( g  |`  ( B  \  ( f " U. D ) ) )  =  ( B  \ 
( f " U. D ) ) )
7 dfdm4 4831 . . . . . 6  |-  dom  (
g  |`  ( B  \ 
( f " U. D ) ) )  =  ran  `' ( g  |`  ( B  \  ( f " U. D ) ) )
86, 7eqtr3di 2235 . . . . 5  |-  ( dom  g  =  B  -> 
( B  \  (
f " U. D
) )  =  ran  `' ( g  |`  ( B  \  ( f " U. D ) ) ) )
98adantr 276 . . . 4  |-  ( ( dom  g  =  B  /\  ran  g  C_  A )  ->  ( B  \  ( f " U. D ) )  =  ran  `' ( g  |`  ( B  \  (
f " U. D
) ) ) )
1093ad2ant2 1020 . . 3  |-  ( (EXMID  /\  ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( B  \  ( f " U. D ) )  =  ran  `' ( g  |`  ( B  \  (
f " U. D
) ) ) )
11 funcnvres 5301 . . . . . . 7  |-  ( Fun  `' g  ->  `' ( g  |`  ( B  \  ( f " U. D ) ) )  =  ( `' g  |`  ( g " ( B  \  ( f " U. D ) ) ) ) )
12113ad2ant3 1021 . . . . . 6  |-  ( (EXMID  /\ 
ran  g  C_  A  /\  Fun  `' g )  ->  `' ( g  |`  ( B  \  (
f " U. D
) ) )  =  ( `' g  |`  ( g " ( B  \  ( f " U. D ) ) ) ) )
13 sbthlem.1 . . . . . . . . 9  |-  A  e. 
_V
14 sbthlem.2 . . . . . . . . 9  |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f "
x ) ) ) 
C_  ( A  \  x ) ) }
1513, 14sbthlemi3 6972 . . . . . . . 8  |-  ( (EXMID  /\ 
ran  g  C_  A
)  ->  ( g " ( B  \ 
( f " U. D ) ) )  =  ( A  \  U. D ) )
1615reseq2d 4919 . . . . . . 7  |-  ( (EXMID  /\ 
ran  g  C_  A
)  ->  ( `' g  |`  ( g "
( B  \  (
f " U. D
) ) ) )  =  ( `' g  |`  ( A  \  U. D ) ) )
17163adant3 1018 . . . . . 6  |-  ( (EXMID  /\ 
ran  g  C_  A  /\  Fun  `' g )  ->  ( `' g  |`  ( g " ( B  \  ( f " U. D ) ) ) )  =  ( `' g  |`  ( A  \ 
U. D ) ) )
1812, 17eqtrd 2220 . . . . 5  |-  ( (EXMID  /\ 
ran  g  C_  A  /\  Fun  `' g )  ->  `' ( g  |`  ( B  \  (
f " U. D
) ) )  =  ( `' g  |`  ( A  \  U. D
) ) )
1918rneqd 4868 . . . 4  |-  ( (EXMID  /\ 
ran  g  C_  A  /\  Fun  `' g )  ->  ran  `' (
g  |`  ( B  \ 
( f " U. D ) ) )  =  ran  ( `' g  |`  ( A  \ 
U. D ) ) )
20193adant2l 1233 . . 3  |-  ( (EXMID  /\  ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ran  `' ( g  |`  ( B  \  ( f " U. D ) ) )  =  ran  ( `' g  |`  ( A  \ 
U. D ) ) )
2110, 20eqtrd 2220 . 2  |-  ( (EXMID  /\  ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( B  \  ( f " U. D ) )  =  ran  ( `' g  |`  ( A  \  U. D ) ) )
221, 21eqtr4id 2239 1  |-  ( (EXMID  /\  ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( `' g " ( A  \  U. D ) )  =  ( B 
\  ( f " U. D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 979    = wceq 1363    e. wcel 2158   {cab 2173   _Vcvv 2749    \ cdif 3138    C_ wss 3141   U.cuni 3821  EXMIDwem 4206   `'ccnv 4637   dom cdm 4638   ran crn 4639    |` cres 4640   "cima 4641   Fun wfun 5222
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-exmid 4207  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-fun 5230
This theorem is referenced by:  sbthlemi6  6975  sbthlemi8  6977
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