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Theorem sbthlem4 7183
Description: Lemma for sbth 7190. (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 5026 . . . . 5  |-  dom  (
g  |`  ( B  \ 
( f " U. D ) ) )  =  ran  `' ( g  |`  ( B  \  ( f " U. D ) ) )
2 difss 3438 . . . . . . 7  |-  ( B 
\  ( f " U. D ) )  C_  B
3 sseq2 3334 . . . . . . 7  |-  ( dom  g  =  B  -> 
( ( B  \ 
( f " U. D ) )  C_  dom  g  <->  ( B  \ 
( f " U. D ) )  C_  B ) )
42, 3mpbiri 225 . . . . . 6  |-  ( dom  g  =  B  -> 
( B  \  (
f " U. D
) )  C_  dom  g )
5 ssdmres 5131 . . . . . 6  |-  ( ( B  \  ( f
" U. D ) )  C_  dom  g  <->  dom  ( g  |`  ( B  \  (
f " U. D
) ) )  =  ( B  \  (
f " U. D
) ) )
64, 5sylib 189 . . . . 5  |-  ( dom  g  =  B  ->  dom  ( g  |`  ( B  \  ( f " U. D ) ) )  =  ( B  \ 
( f " U. D ) ) )
71, 6syl5reqr 2455 . . . 4  |-  ( dom  g  =  B  -> 
( B  \  (
f " U. D
) )  =  ran  `' ( g  |`  ( B  \  ( f " U. D ) ) ) )
8 funcnvres 5485 . . . . . 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 7182 . . . . . . 7  |-  ( ran  g  C_  A  ->  ( g " ( B 
\  ( f " U. D ) ) )  =  ( A  \  U. D ) )
1211reseq2d 5109 . . . . . 6  |-  ( ran  g  C_  A  ->  ( `' g  |`  ( g
" ( B  \ 
( f " U. D ) ) ) )  =  ( `' g  |`  ( A  \ 
U. D ) ) )
138, 12sylan9eqr 2462 . . . . 5  |-  ( ( ran  g  C_  A  /\  Fun  `' g )  ->  `' ( g  |`  ( B  \  (
f " U. D
) ) )  =  ( `' g  |`  ( A  \  U. D
) ) )
1413rneqd 5060 . . . 4  |-  ( ( ran  g  C_  A  /\  Fun  `' g )  ->  ran  `' (
g  |`  ( B  \ 
( f " U. D ) ) )  =  ran  ( `' g  |`  ( A  \ 
U. D ) ) )
157, 14sylan9eq 2460 . . 3  |-  ( ( dom  g  =  B  /\  ( ran  g  C_  A  /\  Fun  `' g ) )  -> 
( B  \  (
f " U. D
) )  =  ran  ( `' g  |`  ( A 
\  U. D ) ) )
1615anassrs 630 . 2  |-  ( ( ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( B  \  ( f " U. D ) )  =  ran  ( `' g  |`  ( A  \  U. D ) ) )
17 df-ima 4854 . 2  |-  ( `' g " ( A 
\  U. D ) )  =  ran  ( `' g  |`  ( A  \ 
U. D ) )
1816, 17syl6reqr 2459 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 359    = wceq 1649    e. wcel 1721   {cab 2394   _Vcvv 2920    \ cdif 3281    C_ wss 3284   U.cuni 3979   `'ccnv 4840   dom cdm 4841   ran crn 4842    |` cres 4843   "cima 4844   Fun wfun 5411
This theorem is referenced by:  sbthlem6  7185  sbthlem8  7187
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 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-fun 5419
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