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Theorem sbthlemi8 6673
Description: Lemma for isbth 6676. (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
sbthlemi8  |-  ( ( (EXMID 
/\  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 sbthlemi8
StepHypRef Expression
1 funres11 5086 . . . 4  |-  ( Fun  `' f  ->  Fun  `' ( f  |`  U. D
) )
21ad2antlr 473 . . 3  |-  ( ( (EXMID 
/\  Fun  `' f
)  /\  ( (
( Fun  g  /\  dom  g  =  B
)  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  Fun  `' ( f  |`  U. D ) )
3 funcnvcnv 5073 . . . . . 6  |-  ( Fun  g  ->  Fun  `' `' g )
4 funres11 5086 . . . . . 6  |-  ( Fun  `' `' g  ->  Fun  `' ( `' g  |`  ( A 
\  U. D ) ) )
53, 4syl 14 . . . . 5  |-  ( Fun  g  ->  Fun  `' ( `' g  |`  ( A 
\  U. D ) ) )
65ad2antrr 472 . . . 4  |-  ( ( ( Fun  g  /\  dom  g  =  B
)  /\  ran  g  C_  A )  ->  Fun  `' ( `' g  |`  ( A  \  U. D
) ) )
76ad2antrl 474 . . 3  |-  ( ( (EXMID 
/\  Fun  `' f
)  /\  ( (
( Fun  g  /\  dom  g  =  B
)  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  Fun  `' ( `' g  |`  ( A  \  U. D ) ) )
8 simpll 496 . . . 4  |-  ( ( (EXMID 
/\  Fun  `' f
)  /\  ( (
( Fun  g  /\  dom  g  =  B
)  /\  ran  g  C_  A )  /\  Fun  `' g ) )  -> EXMID )
9 simprll 504 . . . . 5  |-  ( ( (EXMID 
/\  Fun  `' f
)  /\  ( (
( Fun  g  /\  dom  g  =  B
)  /\  ran  g  C_  A )  /\  Fun  `' g ) )  -> 
( Fun  g  /\  dom  g  =  B
) )
109simprd 112 . . . 4  |-  ( ( (EXMID 
/\  Fun  `' f
)  /\  ( (
( Fun  g  /\  dom  g  =  B
)  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  dom  g  =  B
)
11 simprlr 505 . . . 4  |-  ( ( (EXMID 
/\  Fun  `' f
)  /\  ( (
( Fun  g  /\  dom  g  =  B
)  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  ran  g  C_  A )
12 simprr 499 . . . 4  |-  ( ( (EXMID 
/\  Fun  `' f
)  /\  ( (
( Fun  g  /\  dom  g  =  B
)  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  Fun  `' g )
13 df-ima 4451 . . . . . . 7  |-  ( f
" U. D )  =  ran  ( f  |`  U. D )
14 df-rn 4449 . . . . . . 7  |-  ran  (
f  |`  U. D )  =  dom  `' ( f  |`  U. D )
1513, 14eqtr2i 2109 . . . . . 6  |-  dom  `' ( f  |`  U. D
)  =  ( f
" U. D )
16 df-ima 4451 . . . . . . . 8  |-  ( `' g " ( A 
\  U. D ) )  =  ran  ( `' g  |`  ( A  \ 
U. D ) )
17 df-rn 4449 . . . . . . . 8  |-  ran  ( `' g  |`  ( A 
\  U. D ) )  =  dom  `' ( `' g  |`  ( A 
\  U. D ) )
1816, 17eqtri 2108 . . . . . . 7  |-  ( `' g " ( A 
\  U. D ) )  =  dom  `' ( `' g  |`  ( A 
\  U. D ) )
19 sbthlem.1 . . . . . . . 8  |-  A  e. 
_V
20 sbthlem.2 . . . . . . . 8  |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f "
x ) ) ) 
C_  ( A  \  x ) ) }
2119, 20sbthlemi4 6669 . . . . . . 7  |-  ( (EXMID  /\  ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( `' g " ( A  \  U. D ) )  =  ( B 
\  ( f " U. D ) ) )
2218, 21syl5eqr 2134 . . . . . 6  |-  ( (EXMID  /\  ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  dom  `' ( `' g  |`  ( A  \  U. D
) )  =  ( B  \  ( f
" U. D ) ) )
23 ineq12 3196 . . . . . 6  |-  ( ( 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 ) ) ) )
2415, 22, 23sylancr 405 . . . . 5  |-  ( (EXMID  /\  ( 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 ) ) ) )
25 disjdif 3355 . . . . 5  |-  ( ( f " U. D
)  i^i  ( B  \  ( f " U. D ) ) )  =  (/)
2624, 25syl6eq 2136 . . . 4  |-  ( (EXMID  /\  ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( dom  `' ( f  |`  U. D )  i^i  dom  `' ( `' g  |`  ( A  \  U. D
) ) )  =  (/) )
278, 10, 11, 12, 26syl121anc 1179 . . 3  |-  ( ( (EXMID 
/\  Fun  `' f
)  /\  ( (
( Fun  g  /\  dom  g  =  B
)  /\  ran  g  C_  A )  /\  Fun  `' g ) )  -> 
( dom  `' (
f  |`  U. D )  i^i  dom  `' ( `' g  |`  ( A 
\  U. D ) ) )  =  (/) )
28 funun 5058 . . 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 ) ) ) )
292, 7, 27, 28syl21anc 1173 . 2  |-  ( ( (EXMID 
/\  Fun  `' f
)  /\  ( (
( Fun  g  /\  dom  g  =  B
)  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  Fun  ( `' ( f  |`  U. D )  u.  `' ( `' g  |`  ( A  \  U. D ) ) ) )
30 sbthlem.3 . . . . 5  |-  H  =  ( ( f  |`  U. D )  u.  ( `' g  |`  ( A 
\  U. D ) ) )
3130cnveqi 4611 . . . 4  |-  `' H  =  `' ( ( f  |`  U. D )  u.  ( `' g  |`  ( A  \  U. D
) ) )
32 cnvun 4837 . . . 4  |-  `' ( ( f  |`  U. D
)  u.  ( `' g  |`  ( A  \ 
U. D ) ) )  =  ( `' ( f  |`  U. D
)  u.  `' ( `' g  |`  ( A 
\  U. D ) ) )
3331, 32eqtri 2108 . . 3  |-  `' H  =  ( `' ( f  |`  U. D )  u.  `' ( `' g  |`  ( A  \ 
U. D ) ) )
3433funeqi 5036 . 2  |-  ( Fun  `' H  <->  Fun  ( `' ( f  |`  U. D )  u.  `' ( `' g  |`  ( A  \ 
U. D ) ) ) )
3529, 34sylibr 132 1  |-  ( ( (EXMID 
/\  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 102    /\ w3a 924    = wceq 1289    e. wcel 1438   {cab 2074   _Vcvv 2619    \ cdif 2996    u. cun 2997    i^i cin 2998    C_ wss 2999   (/)c0 3286   U.cuni 3653  EXMIDwem 4029   `'ccnv 4437   dom cdm 4438   ran crn 4439    |` cres 4440   "cima 4441   Fun wfun 5009
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-stab 776  df-dc 781  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-exmid 4030  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-fun 5017
This theorem is referenced by:  sbthlemi9  6674
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