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Theorem sbthlem6 7181
Description: Lemma for sbth 7186. (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
sbthlem6  |-  ( ( 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 sbthlem6
StepHypRef Expression
1 df-ima 4850 . . . . 5  |-  ( `' g " ( A 
\  U. D ) )  =  ran  ( `' g  |`  ( A  \ 
U. D ) )
2 sbthlem.1 . . . . . 6  |-  A  e. 
_V
3 sbthlem.2 . . . . . 6  |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f "
x ) ) ) 
C_  ( A  \  x ) ) }
42, 3sbthlem4 7179 . . . . 5  |-  ( ( ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( `' g " ( A  \  U. D ) )  =  ( B 
\  ( f " U. D ) ) )
51, 4syl5reqr 2451 . . . 4  |-  ( ( ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( B  \  ( f " U. D ) )  =  ran  ( `' g  |`  ( A  \  U. D ) ) )
65uneq2d 3461 . . 3  |-  ( ( ( 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 ) ) ) )
7 rnun 5239 . . . 4  |-  ran  (
( f  |`  U. D
)  u.  ( `' g  |`  ( A  \ 
U. D ) ) )  =  ( ran  ( f  |`  U. D
)  u.  ran  ( `' g  |`  ( A 
\  U. D ) ) )
8 sbthlem.3 . . . . 5  |-  H  =  ( ( f  |`  U. D )  u.  ( `' g  |`  ( A 
\  U. D ) ) )
98rneqi 5055 . . . 4  |-  ran  H  =  ran  ( ( f  |`  U. D )  u.  ( `' g  |`  ( A  \  U. D
) ) )
10 df-ima 4850 . . . . 5  |-  ( f
" U. D )  =  ran  ( f  |`  U. D )
1110uneq1i 3457 . . . 4  |-  ( ( f " U. D
)  u.  ran  ( `' g  |`  ( A 
\  U. D ) ) )  =  ( ran  ( f  |`  U. D
)  u.  ran  ( `' g  |`  ( A 
\  U. D ) ) )
127, 9, 113eqtr4i 2434 . . 3  |-  ran  H  =  ( ( f
" U. D )  u.  ran  ( `' g  |`  ( A  \ 
U. D ) ) )
136, 12syl6reqr 2455 . 2  |-  ( ( ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ran  H  =  ( ( f
" U. D )  u.  ( B  \ 
( f " U. D ) ) ) )
14 imassrn 5175 . . . 4  |-  ( f
" U. D ) 
C_  ran  f
15 sstr2 3315 . . . 4  |-  ( ( f " U. D
)  C_  ran  f  -> 
( ran  f  C_  B  ->  ( f " U. D )  C_  B
) )
1614, 15ax-mp 8 . . 3  |-  ( ran  f  C_  B  ->  ( f " U. D
)  C_  B )
17 undif 3668 . . 3  |-  ( ( f " U. D
)  C_  B  <->  ( (
f " U. D
)  u.  ( B 
\  ( f " U. D ) ) )  =  B )
1816, 17sylib 189 . 2  |-  ( ran  f  C_  B  ->  ( ( f " U. D )  u.  ( B  \  ( f " U. D ) ) )  =  B )
1913, 18sylan9eqr 2458 1  |-  ( ( 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 359    = wceq 1649    e. wcel 1721   {cab 2390   _Vcvv 2916    \ cdif 3277    u. cun 3278    C_ wss 3280   U.cuni 3975   `'ccnv 4836   dom cdm 4837   ran crn 4838    |` cres 4839   "cima 4840   Fun wfun 5407
This theorem is referenced by:  sbthlem9  7184
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-fun 5415
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