MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbthlem6 Structured version   Visualization version   Unicode version

Theorem sbthlem6 7705
Description: Lemma for sbth 7710. (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 4852 . . . . 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 7703 . . . . 5  |-  ( ( ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( `' g " ( A  \  U. D ) )  =  ( B 
\  ( f " U. D ) ) )
51, 4syl5reqr 2520 . . . 4  |-  ( ( ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( B  \  ( f " U. D ) )  =  ran  ( `' g  |`  ( A  \  U. D ) ) )
65uneq2d 3579 . . 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 5250 . . . 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 5067 . . . 4  |-  ran  H  =  ran  ( ( f  |`  U. D )  u.  ( `' g  |`  ( A  \  U. D
) ) )
10 df-ima 4852 . . . . 5  |-  ( f
" U. D )  =  ran  ( f  |`  U. D )
1110uneq1i 3575 . . . 4  |-  ( ( f " U. D
)  u.  ran  ( `' g  |`  ( A 
\  U. D ) ) )  =  ( ran  ( f  |`  U. D
)  u.  ran  ( `' g  |`  ( A 
\  U. D ) ) )
127, 9, 113eqtr4i 2503 . . 3  |-  ran  H  =  ( ( f
" U. D )  u.  ran  ( `' g  |`  ( A  \ 
U. D ) ) )
136, 12syl6reqr 2524 . 2  |-  ( ( ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ran  H  =  ( ( f
" U. D )  u.  ( B  \ 
( f " U. D ) ) ) )
14 imassrn 5185 . . . 4  |-  ( f
" U. D ) 
C_  ran  f
15 sstr2 3425 . . . 4  |-  ( ( f " U. D
)  C_  ran  f  -> 
( ran  f  C_  B  ->  ( f " U. D )  C_  B
) )
1614, 15ax-mp 5 . . 3  |-  ( ran  f  C_  B  ->  ( f " U. D
)  C_  B )
17 undif 3839 . . 3  |-  ( ( f " U. D
)  C_  B  <->  ( (
f " U. D
)  u.  ( B 
\  ( f " U. D ) ) )  =  B )
1816, 17sylib 201 . 2  |-  ( ran  f  C_  B  ->  ( ( f " U. D )  u.  ( B  \  ( f " U. D ) ) )  =  B )
1913, 18sylan9eqr 2527 1  |-  ( ( ran  f  C_  B  /\  ( ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  ran  H  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   {cab 2457   _Vcvv 3031    \ cdif 3387    u. cun 3388    C_ wss 3390   U.cuni 4190   `'ccnv 4838   dom cdm 4839   ran crn 4840    |` cres 4841   "cima 4842   Fun wfun 5583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-fun 5591
This theorem is referenced by:  sbthlem9  7708
  Copyright terms: Public domain W3C validator