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Theorem sbthlem8 7631
Description: Lemma for sbth 7634. (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 5654 . . . 4  |-  ( Fun  `' f  ->  Fun  `' ( f  |`  U. D
) )
2 funcnvcnv 5644 . . . . . 6  |-  ( Fun  g  ->  Fun  `' `' g )
3 funres11 5654 . . . . . 6  |-  ( Fun  `' `' g  ->  Fun  `' ( `' g  |`  ( A 
\  U. D ) ) )
42, 3syl 16 . . . . 5  |-  ( Fun  g  ->  Fun  `' ( `' g  |`  ( A 
\  U. D ) ) )
54ad3antrrr 729 . . . 4  |-  ( ( ( ( Fun  g  /\  dom  g  =  B )  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  Fun  `' ( `' g  |`  ( A  \  U. D
) ) )
61, 5anim12i 566 . . 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 5012 . . . . . . . 8  |-  ( f
" U. D )  =  ran  ( f  |`  U. D )
8 df-rn 5010 . . . . . . . 8  |-  ran  (
f  |`  U. D )  =  dom  `' ( f  |`  U. D )
97, 8eqtr2i 2497 . . . . . . 7  |-  dom  `' ( f  |`  U. D
)  =  ( f
" U. D )
10 df-ima 5012 . . . . . . . . 9  |-  ( `' g " ( A 
\  U. D ) )  =  ran  ( `' g  |`  ( A  \ 
U. D ) )
11 df-rn 5010 . . . . . . . . 9  |-  ran  ( `' g  |`  ( A 
\  U. D ) )  =  dom  `' ( `' g  |`  ( A 
\  U. D ) )
1210, 11eqtri 2496 . . . . . . . 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 7627 . . . . . . . 8  |-  ( ( ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( `' g " ( A  \  U. D ) )  =  ( B 
\  ( f " U. D ) ) )
1612, 15syl5eqr 2522 . . . . . . 7  |-  ( ( ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  dom  `' ( `' g  |`  ( A  \  U. D
) )  =  ( B  \  ( f
" U. D ) ) )
17 ineq12 3695 . . . . . . 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 663 . . . . . 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 3899 . . . . . 6  |-  ( ( f " U. D
)  i^i  ( B  \  ( f " U. D ) ) )  =  (/)
2018, 19syl6eq 2524 . . . . 5  |-  ( ( ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( dom  `' ( f  |`  U. D )  i^i  dom  `' ( `' g  |`  ( A  \  U. D
) ) )  =  (/) )
2120adantlll 717 . . . 4  |-  ( ( ( ( Fun  g  /\  dom  g  =  B )  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( dom  `' ( f  |`  U. D )  i^i  dom  `' ( `' g  |`  ( A  \  U. D
) ) )  =  (/) )
2221adantl 466 . . 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 5628 . . 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 661 . 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 5175 . . . 4  |-  `' H  =  `' ( ( f  |`  U. D )  u.  ( `' g  |`  ( A  \  U. D
) ) )
27 cnvun 5409 . . . 4  |-  `' ( ( f  |`  U. D
)  u.  ( `' g  |`  ( A  \ 
U. D ) ) )  =  ( `' ( f  |`  U. D
)  u.  `' ( `' g  |`  ( A 
\  U. D ) ) )
2826, 27eqtri 2496 . . 3  |-  `' H  =  ( `' ( f  |`  U. D )  u.  `' ( `' g  |`  ( A  \ 
U. D ) ) )
2928funeqi 5606 . 2  |-  ( Fun  `' H  <->  Fun  ( `' ( f  |`  U. D )  u.  `' ( `' g  |`  ( A  \ 
U. D ) ) ) )
3024, 29sylibr 212 1  |-  ( ( Fun  `' f  /\  ( ( ( Fun  g  /\  dom  g  =  B )  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  Fun  `' H
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452   _Vcvv 3113    \ cdif 3473    u. cun 3474    i^i cin 3475    C_ wss 3476   (/)c0 3785   U.cuni 4245   `'ccnv 4998   dom cdm 4999   ran crn 5000    |` cres 5001   "cima 5002   Fun wfun 5580
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-fun 5588
This theorem is referenced by:  sbthlem9  7632
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