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Theorem sbthlem4 7667
Description: Lemma for sbth 7674. (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 5015 . . . . 5  |-  dom  (
g  |`  ( B  \ 
( f " U. D ) ) )  =  ran  `' ( g  |`  ( B  \  ( f " U. D ) ) )
2 difss 3569 . . . . . . 7  |-  ( B 
\  ( f " U. D ) )  C_  B
3 sseq2 3463 . . . . . . 7  |-  ( dom  g  =  B  -> 
( ( B  \ 
( f " U. D ) )  C_  dom  g  <->  ( B  \ 
( f " U. D ) )  C_  B ) )
42, 3mpbiri 233 . . . . . 6  |-  ( dom  g  =  B  -> 
( B  \  (
f " U. D
) )  C_  dom  g )
5 ssdmres 5114 . . . . . 6  |-  ( ( B  \  ( f
" U. D ) )  C_  dom  g  <->  dom  ( g  |`  ( B  \  (
f " U. D
) ) )  =  ( B  \  (
f " U. D
) ) )
64, 5sylib 196 . . . . 5  |-  ( dom  g  =  B  ->  dom  ( g  |`  ( B  \  ( f " U. D ) ) )  =  ( B  \ 
( f " U. D ) ) )
71, 6syl5reqr 2458 . . . 4  |-  ( dom  g  =  B  -> 
( B  \  (
f " U. D
) )  =  ran  `' ( g  |`  ( B  \  ( f " U. D ) ) ) )
8 funcnvres 5637 . . . . . 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 7666 . . . . . . 7  |-  ( ran  g  C_  A  ->  ( g " ( B 
\  ( f " U. D ) ) )  =  ( A  \  U. D ) )
1211reseq2d 5093 . . . . . 6  |-  ( ran  g  C_  A  ->  ( `' g  |`  ( g
" ( B  \ 
( f " U. D ) ) ) )  =  ( `' g  |`  ( A  \ 
U. D ) ) )
138, 12sylan9eqr 2465 . . . . 5  |-  ( ( ran  g  C_  A  /\  Fun  `' g )  ->  `' ( g  |`  ( B  \  (
f " U. D
) ) )  =  ( `' g  |`  ( A  \  U. D
) ) )
1413rneqd 5050 . . . 4  |-  ( ( ran  g  C_  A  /\  Fun  `' g )  ->  ran  `' (
g  |`  ( B  \ 
( f " U. D ) ) )  =  ran  ( `' g  |`  ( A  \ 
U. D ) ) )
157, 14sylan9eq 2463 . . 3  |-  ( ( dom  g  =  B  /\  ( ran  g  C_  A  /\  Fun  `' g ) )  -> 
( B  \  (
f " U. D
) )  =  ran  ( `' g  |`  ( A 
\  U. D ) ) )
1615anassrs 646 . 2  |-  ( ( ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( B  \  ( f " U. D ) )  =  ran  ( `' g  |`  ( A  \  U. D ) ) )
17 df-ima 4835 . 2  |-  ( `' g " ( A 
\  U. D ) )  =  ran  ( `' g  |`  ( A  \ 
U. D ) )
1816, 17syl6reqr 2462 1  |-  ( ( ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( `' g " ( A  \  U. D ) )  =  ( B 
\  ( f " U. D ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   {cab 2387   _Vcvv 3058    \ cdif 3410    C_ wss 3413   U.cuni 4190   `'ccnv 4821   dom cdm 4822   ran crn 4823    |` cres 4824   "cima 4825   Fun wfun 5562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-fun 5570
This theorem is referenced by:  sbthlem6  7669  sbthlem8  7671
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