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Theorem sbthlem4 7117
Description: Lemma for sbth 7124. (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
sbthlem4  |-  ( ( ( 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 sbthlem4
StepHypRef Expression
1 dfdm4 4975 . . . . 5  |-  dom  (
g  |`  ( B  \ 
( f " U. D ) ) )  =  ran  `' ( g  |`  ( B  \  ( f " U. D ) ) )
2 difss 3390 . . . . . . 7  |-  ( B 
\  ( f " U. D ) )  C_  B
3 sseq2 3286 . . . . . . 7  |-  ( dom  g  =  B  -> 
( ( B  \ 
( f " U. D ) )  C_  dom  g  <->  ( B  \ 
( f " U. D ) )  C_  B ) )
42, 3mpbiri 224 . . . . . 6  |-  ( dom  g  =  B  -> 
( B  \  (
f " U. D
) )  C_  dom  g )
5 ssdmres 5080 . . . . . 6  |-  ( ( B  \  ( f
" U. D ) )  C_  dom  g  <->  dom  ( g  |`  ( B  \  (
f " U. D
) ) )  =  ( B  \  (
f " U. D
) ) )
64, 5sylib 188 . . . . 5  |-  ( dom  g  =  B  ->  dom  ( g  |`  ( B  \  ( f " U. D ) ) )  =  ( B  \ 
( f " U. D ) ) )
71, 6syl5reqr 2413 . . . 4  |-  ( dom  g  =  B  -> 
( B  \  (
f " U. D
) )  =  ran  `' ( g  |`  ( B  \  ( f " U. D ) ) ) )
8 funcnvres 5426 . . . . . 6  |-  ( Fun  `' g  ->  `' ( g  |`  ( B  \  ( f " U. D ) ) )  =  ( `' g  |`  ( g " ( B  \  ( f " U. D ) ) ) ) )
9 sbthlem.1 . . . . . . . 8  |-  A  e. 
_V
10 sbthlem.2 . . . . . . . 8  |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f "
x ) ) ) 
C_  ( A  \  x ) ) }
119, 10sbthlem3 7116 . . . . . . 7  |-  ( ran  g  C_  A  ->  ( g " ( B 
\  ( f " U. D ) ) )  =  ( A  \  U. D ) )
1211reseq2d 5058 . . . . . 6  |-  ( ran  g  C_  A  ->  ( `' g  |`  ( g
" ( B  \ 
( f " U. D ) ) ) )  =  ( `' g  |`  ( A  \ 
U. D ) ) )
138, 12sylan9eqr 2420 . . . . 5  |-  ( ( ran  g  C_  A  /\  Fun  `' g )  ->  `' ( g  |`  ( B  \  (
f " U. D
) ) )  =  ( `' g  |`  ( A  \  U. D
) ) )
1413rneqd 5009 . . . 4  |-  ( ( ran  g  C_  A  /\  Fun  `' g )  ->  ran  `' (
g  |`  ( B  \ 
( f " U. D ) ) )  =  ran  ( `' g  |`  ( A  \ 
U. D ) ) )
157, 14sylan9eq 2418 . . 3  |-  ( ( dom  g  =  B  /\  ( ran  g  C_  A  /\  Fun  `' g ) )  -> 
( B  \  (
f " U. D
) )  =  ran  ( `' g  |`  ( A 
\  U. D ) ) )
1615anassrs 629 . 2  |-  ( ( ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( B  \  ( f " U. D ) )  =  ran  ( `' g  |`  ( A  \  U. D ) ) )
17 df-ima 4805 . 2  |-  ( `' g " ( A 
\  U. D ) )  =  ran  ( `' g  |`  ( A  \ 
U. D ) )
1816, 17syl6reqr 2417 1  |-  ( ( ( 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 358    = wceq 1647    e. wcel 1715   {cab 2352   _Vcvv 2873    \ cdif 3235    C_ wss 3238   U.cuni 3929   `'ccnv 4791   dom cdm 4792   ran crn 4793    |` cres 4794   "cima 4795   Fun wfun 5352
This theorem is referenced by:  sbthlem6  7119  sbthlem8  7121
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-fun 5360
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