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Theorem sbthlem8 7216
Description: Lemma for sbth 7219. (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 5513 . . . 4  |-  ( Fun  `' f  ->  Fun  `' ( f  |`  U. D
) )
2 funcnvcnv 5501 . . . . . 6  |-  ( Fun  g  ->  Fun  `' `' g )
3 funres11 5513 . . . . . 6  |-  ( Fun  `' `' g  ->  Fun  `' ( `' g  |`  ( A 
\  U. D ) ) )
42, 3syl 16 . . . . 5  |-  ( Fun  g  ->  Fun  `' ( `' g  |`  ( A 
\  U. D ) ) )
54ad3antrrr 711 . . . 4  |-  ( ( ( ( Fun  g  /\  dom  g  =  B )  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  Fun  `' ( `' g  |`  ( A  \  U. D
) ) )
61, 5anim12i 550 . . 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 4883 . . . . . . . 8  |-  ( f
" U. D )  =  ran  ( f  |`  U. D )
8 df-rn 4881 . . . . . . . 8  |-  ran  (
f  |`  U. D )  =  dom  `' ( f  |`  U. D )
97, 8eqtr2i 2456 . . . . . . 7  |-  dom  `' ( f  |`  U. D
)  =  ( f
" U. D )
10 df-ima 4883 . . . . . . . . 9  |-  ( `' g " ( A 
\  U. D ) )  =  ran  ( `' g  |`  ( A  \ 
U. D ) )
11 df-rn 4881 . . . . . . . . 9  |-  ran  ( `' g  |`  ( A 
\  U. D ) )  =  dom  `' ( `' g  |`  ( A 
\  U. D ) )
1210, 11eqtri 2455 . . . . . . . 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 7212 . . . . . . . 8  |-  ( ( ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( `' g " ( A  \  U. D ) )  =  ( B 
\  ( f " U. D ) ) )
1612, 15syl5eqr 2481 . . . . . . 7  |-  ( ( ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  dom  `' ( `' g  |`  ( A  \  U. D
) )  =  ( B  \  ( f
" U. D ) ) )
17 ineq12 3529 . . . . . . 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 645 . . . . . 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 3692 . . . . . 6  |-  ( ( f " U. D
)  i^i  ( B  \  ( f " U. D ) ) )  =  (/)
2018, 19syl6eq 2483 . . . . 5  |-  ( ( ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( dom  `' ( f  |`  U. D )  i^i  dom  `' ( `' g  |`  ( A  \  U. D
) ) )  =  (/) )
2120adantlll 699 . . . 4  |-  ( ( ( ( Fun  g  /\  dom  g  =  B )  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( dom  `' ( f  |`  U. D )  i^i  dom  `' ( `' g  |`  ( A  \  U. D
) ) )  =  (/) )
2221adantl 453 . . 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 5487 . . 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 643 . 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 5039 . . . 4  |-  `' H  =  `' ( ( f  |`  U. D )  u.  ( `' g  |`  ( A  \  U. D
) ) )
27 cnvun 5269 . . . 4  |-  `' ( ( f  |`  U. D
)  u.  ( `' g  |`  ( A  \ 
U. D ) ) )  =  ( `' ( f  |`  U. D
)  u.  `' ( `' g  |`  ( A 
\  U. D ) ) )
2826, 27eqtri 2455 . . 3  |-  `' H  =  ( `' ( f  |`  U. D )  u.  `' ( `' g  |`  ( A  \ 
U. D ) ) )
2928funeqi 5466 . 2  |-  ( Fun  `' H  <->  Fun  ( `' ( f  |`  U. D )  u.  `' ( `' g  |`  ( A  \ 
U. D ) ) ) )
3024, 29sylibr 204 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 359    = wceq 1652    e. wcel 1725   {cab 2421   _Vcvv 2948    \ cdif 3309    u. cun 3310    i^i cin 3311    C_ wss 3312   (/)c0 3620   U.cuni 4007   `'ccnv 4869   dom cdm 4870   ran crn 4871    |` cres 4872   "cima 4873   Fun wfun 5440
This theorem is referenced by:  sbthlem9  7217
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-fun 5448
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