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Theorem sbthlemi4 6776
Description: Lemma for isbth 6783. (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 dfdm4 4669 . . . . . 6  |-  dom  (
g  |`  ( B  \ 
( f " U. D ) ) )  =  ran  `' ( g  |`  ( B  \  ( f " U. D ) ) )
2 difss 3149 . . . . . . . 8  |-  ( B 
\  ( f " U. D ) )  C_  B
3 sseq2 3071 . . . . . . . 8  |-  ( dom  g  =  B  -> 
( ( B  \ 
( f " U. D ) )  C_  dom  g  <->  ( B  \ 
( f " U. D ) )  C_  B ) )
42, 3mpbiri 167 . . . . . . 7  |-  ( dom  g  =  B  -> 
( B  \  (
f " U. D
) )  C_  dom  g )
5 ssdmres 4777 . . . . . . 7  |-  ( ( B  \  ( f
" U. D ) )  C_  dom  g  <->  dom  ( g  |`  ( B  \  (
f " U. D
) ) )  =  ( B  \  (
f " U. D
) ) )
64, 5sylib 121 . . . . . 6  |-  ( dom  g  =  B  ->  dom  ( g  |`  ( B  \  ( f " U. D ) ) )  =  ( B  \ 
( f " U. D ) ) )
71, 6syl5reqr 2147 . . . . 5  |-  ( dom  g  =  B  -> 
( B  \  (
f " U. D
) )  =  ran  `' ( g  |`  ( B  \  ( f " U. D ) ) ) )
87adantr 272 . . . 4  |-  ( ( dom  g  =  B  /\  ran  g  C_  A )  ->  ( B  \  ( f " U. D ) )  =  ran  `' ( g  |`  ( B  \  (
f " U. D
) ) ) )
983ad2ant2 971 . . 3  |-  ( (EXMID  /\  ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( B  \  ( f " U. D ) )  =  ran  `' ( g  |`  ( B  \  (
f " U. D
) ) ) )
10 funcnvres 5132 . . . . . . 7  |-  ( Fun  `' g  ->  `' ( g  |`  ( B  \  ( f " U. D ) ) )  =  ( `' g  |`  ( g " ( B  \  ( f " U. D ) ) ) ) )
11103ad2ant3 972 . . . . . 6  |-  ( (EXMID  /\ 
ran  g  C_  A  /\  Fun  `' g )  ->  `' ( g  |`  ( B  \  (
f " U. D
) ) )  =  ( `' g  |`  ( g " ( B  \  ( f " U. D ) ) ) ) )
12 sbthlem.1 . . . . . . . . 9  |-  A  e. 
_V
13 sbthlem.2 . . . . . . . . 9  |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f "
x ) ) ) 
C_  ( A  \  x ) ) }
1412, 13sbthlemi3 6775 . . . . . . . 8  |-  ( (EXMID  /\ 
ran  g  C_  A
)  ->  ( g " ( B  \ 
( f " U. D ) ) )  =  ( A  \  U. D ) )
1514reseq2d 4755 . . . . . . 7  |-  ( (EXMID  /\ 
ran  g  C_  A
)  ->  ( `' g  |`  ( g "
( B  \  (
f " U. D
) ) ) )  =  ( `' g  |`  ( A  \  U. D ) ) )
16153adant3 969 . . . . . 6  |-  ( (EXMID  /\ 
ran  g  C_  A  /\  Fun  `' g )  ->  ( `' g  |`  ( g " ( B  \  ( f " U. D ) ) ) )  =  ( `' g  |`  ( A  \ 
U. D ) ) )
1711, 16eqtrd 2132 . . . . 5  |-  ( (EXMID  /\ 
ran  g  C_  A  /\  Fun  `' g )  ->  `' ( g  |`  ( B  \  (
f " U. D
) ) )  =  ( `' g  |`  ( A  \  U. D
) ) )
1817rneqd 4706 . . . 4  |-  ( (EXMID  /\ 
ran  g  C_  A  /\  Fun  `' g )  ->  ran  `' (
g  |`  ( B  \ 
( f " U. D ) ) )  =  ran  ( `' g  |`  ( A  \ 
U. D ) ) )
19183adant2l 1178 . . 3  |-  ( (EXMID  /\  ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ran  `' ( g  |`  ( B  \  ( f " U. D ) ) )  =  ran  ( `' g  |`  ( A  \ 
U. D ) ) )
209, 19eqtrd 2132 . 2  |-  ( (EXMID  /\  ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( B  \  ( f " U. D ) )  =  ran  ( `' g  |`  ( A  \  U. D ) ) )
21 df-ima 4490 . 2  |-  ( `' g " ( A 
\  U. D ) )  =  ran  ( `' g  |`  ( A  \ 
U. D ) )
2220, 21syl6reqr 2151 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 103    /\ w3a 930    = wceq 1299    e. wcel 1448   {cab 2086   _Vcvv 2641    \ cdif 3018    C_ wss 3021   U.cuni 3683  EXMIDwem 4058   `'ccnv 4476   dom cdm 4477   ran crn 4478    |` cres 4479   "cima 4480   Fun wfun 5053
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 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069
This theorem depends on definitions:  df-bi 116  df-stab 782  df-dc 787  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-exmid 4059  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-fun 5061
This theorem is referenced by:  sbthlemi6  6778  sbthlemi8  6780
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