MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbthlem8 Unicode version

Theorem sbthlem8 6978
Description: Lemma for sbth 6981. (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
sbthlem8  |-  ( ( Fun  `' f  /\  ( ( ( Fun  g  /\  dom  g  =  B )  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  Fun  `' H
)
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 sbthlem8
StepHypRef Expression
1 funres11 5320 . . . 4  |-  ( Fun  `' f  ->  Fun  `' ( f  |`  U. D
) )
2 funcnvcnv 5308 . . . . . 6  |-  ( Fun  g  ->  Fun  `' `' g )
3 funres11 5320 . . . . . 6  |-  ( Fun  `' `' g  ->  Fun  `' ( `' g  |`  ( A 
\  U. D ) ) )
42, 3syl 15 . . . . 5  |-  ( Fun  g  ->  Fun  `' ( `' g  |`  ( A 
\  U. D ) ) )
54ad3antrrr 710 . . . 4  |-  ( ( ( ( Fun  g  /\  dom  g  =  B )  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  Fun  `' ( `' g  |`  ( A  \  U. D
) ) )
61, 5anim12i 549 . . 3  |-  ( ( Fun  `' f  /\  ( ( ( Fun  g  /\  dom  g  =  B )  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  ( Fun  `' ( f  |`  U. D
)  /\  Fun  `' ( `' g  |`  ( A 
\  U. D ) ) ) )
7 df-ima 4702 . . . . . . . 8  |-  ( f
" U. D )  =  ran  ( f  |`  U. D )
8 df-rn 4700 . . . . . . . 8  |-  ran  (
f  |`  U. D )  =  dom  `' ( f  |`  U. D )
97, 8eqtr2i 2304 . . . . . . 7  |-  dom  `' ( f  |`  U. D
)  =  ( f
" U. D )
10 df-ima 4702 . . . . . . . . 9  |-  ( `' g " ( A 
\  U. D ) )  =  ran  ( `' g  |`  ( A  \ 
U. D ) )
11 df-rn 4700 . . . . . . . . 9  |-  ran  ( `' g  |`  ( A 
\  U. D ) )  =  dom  `' ( `' g  |`  ( A 
\  U. D ) )
1210, 11eqtri 2303 . . . . . . . 8  |-  ( `' g " ( A 
\  U. D ) )  =  dom  `' ( `' g  |`  ( A 
\  U. D ) )
13 sbthlem.1 . . . . . . . . 9  |-  A  e. 
_V
14 sbthlem.2 . . . . . . . . 9  |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f "
x ) ) ) 
C_  ( A  \  x ) ) }
1513, 14sbthlem4 6974 . . . . . . . 8  |-  ( ( ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( `' g " ( A  \  U. D ) )  =  ( B 
\  ( f " U. D ) ) )
1612, 15syl5eqr 2329 . . . . . . 7  |-  ( ( ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  dom  `' ( `' g  |`  ( A  \  U. D
) )  =  ( B  \  ( f
" U. D ) ) )
17 ineq12 3365 . . . . . . 7  |-  ( ( dom  `' ( f  |`  U. D )  =  ( f " U. D )  /\  dom  `' ( `' g  |`  ( A  \  U. D
) )  =  ( B  \  ( f
" U. D ) ) )  ->  ( dom  `' ( f  |`  U. D )  i^i  dom  `' ( `' g  |`  ( A  \  U. D
) ) )  =  ( ( f " U. D )  i^i  ( B  \  ( f " U. D ) ) ) )
189, 16, 17sylancr 644 . . . . . 6  |-  ( ( ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( dom  `' ( f  |`  U. D )  i^i  dom  `' ( `' g  |`  ( A  \  U. D
) ) )  =  ( ( f " U. D )  i^i  ( B  \  ( f " U. D ) ) ) )
19 disjdif 3526 . . . . . 6  |-  ( ( f " U. D
)  i^i  ( B  \  ( f " U. D ) ) )  =  (/)
2018, 19syl6eq 2331 . . . . 5  |-  ( ( ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( dom  `' ( f  |`  U. D )  i^i  dom  `' ( `' g  |`  ( A  \  U. D
) ) )  =  (/) )
2120adantlll 698 . . . 4  |-  ( ( ( ( Fun  g  /\  dom  g  =  B )  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( dom  `' ( f  |`  U. D )  i^i  dom  `' ( `' g  |`  ( A  \  U. D
) ) )  =  (/) )
2221adantl 452 . . 3  |-  ( ( Fun  `' f  /\  ( ( ( Fun  g  /\  dom  g  =  B )  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  ( dom  `' ( f  |`  U. D
)  i^i  dom  `' ( `' g  |`  ( A 
\  U. D ) ) )  =  (/) )
23 funun 5296 . . 3  |-  ( ( ( Fun  `' ( f  |`  U. D )  /\  Fun  `' ( `' g  |`  ( A 
\  U. D ) ) )  /\  ( dom  `' ( f  |`  U. D )  i^i  dom  `' ( `' g  |`  ( A  \  U. D
) ) )  =  (/) )  ->  Fun  ( `' ( f  |`  U. D )  u.  `' ( `' g  |`  ( A 
\  U. D ) ) ) )
246, 22, 23syl2anc 642 . 2  |-  ( ( Fun  `' f  /\  ( ( ( Fun  g  /\  dom  g  =  B )  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  Fun  ( `' ( f  |`  U. D
)  u.  `' ( `' g  |`  ( A 
\  U. D ) ) ) )
25 sbthlem.3 . . . . 5  |-  H  =  ( ( f  |`  U. D )  u.  ( `' g  |`  ( A 
\  U. D ) ) )
2625cnveqi 4856 . . . 4  |-  `' H  =  `' ( ( f  |`  U. D )  u.  ( `' g  |`  ( A  \  U. D
) ) )
27 cnvun 5086 . . . 4  |-  `' ( ( f  |`  U. D
)  u.  ( `' g  |`  ( A  \ 
U. D ) ) )  =  ( `' ( f  |`  U. D
)  u.  `' ( `' g  |`  ( A 
\  U. D ) ) )
2826, 27eqtri 2303 . . 3  |-  `' H  =  ( `' ( f  |`  U. D )  u.  `' ( `' g  |`  ( A  \ 
U. D ) ) )
2928funeqi 5275 . 2  |-  ( Fun  `' H  <->  Fun  ( `' ( f  |`  U. D )  u.  `' ( `' g  |`  ( A  \ 
U. D ) ) ) )
3024, 29sylibr 203 1  |-  ( ( Fun  `' f  /\  ( ( ( Fun  g  /\  dom  g  =  B )  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  Fun  `' H
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   _Vcvv 2788    \ cdif 3149    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   U.cuni 3827   `'ccnv 4688   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692   Fun wfun 5249
This theorem is referenced by:  sbthlem9  6979
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5257
  Copyright terms: Public domain W3C validator