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Theorem sbthlemi8 6910
Description: Lemma for isbth 6913. (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 5244 . . . 4  |-  ( Fun  `' f  ->  Fun  `' ( f  |`  U. D
) )
21ad2antlr 481 . . 3  |-  ( ( (EXMID 
/\  Fun  `' f
)  /\  ( (
( Fun  g  /\  dom  g  =  B
)  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  Fun  `' ( f  |`  U. D ) )
3 funcnvcnv 5231 . . . . . 6  |-  ( Fun  g  ->  Fun  `' `' g )
4 funres11 5244 . . . . . 6  |-  ( Fun  `' `' g  ->  Fun  `' ( `' g  |`  ( A 
\  U. D ) ) )
53, 4syl 14 . . . . 5  |-  ( Fun  g  ->  Fun  `' ( `' g  |`  ( A 
\  U. D ) ) )
65ad2antrr 480 . . . 4  |-  ( ( ( Fun  g  /\  dom  g  =  B
)  /\  ran  g  C_  A )  ->  Fun  `' ( `' g  |`  ( A  \  U. D
) ) )
76ad2antrl 482 . . 3  |-  ( ( (EXMID 
/\  Fun  `' f
)  /\  ( (
( Fun  g  /\  dom  g  =  B
)  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  Fun  `' ( `' g  |`  ( A  \  U. D ) ) )
8 simpll 519 . . . 4  |-  ( ( (EXMID 
/\  Fun  `' f
)  /\  ( (
( Fun  g  /\  dom  g  =  B
)  /\  ran  g  C_  A )  /\  Fun  `' g ) )  -> EXMID )
9 simprll 527 . . . . 5  |-  ( ( (EXMID 
/\  Fun  `' f
)  /\  ( (
( Fun  g  /\  dom  g  =  B
)  /\  ran  g  C_  A )  /\  Fun  `' g ) )  -> 
( Fun  g  /\  dom  g  =  B
) )
109simprd 113 . . . 4  |-  ( ( (EXMID 
/\  Fun  `' f
)  /\  ( (
( Fun  g  /\  dom  g  =  B
)  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  dom  g  =  B
)
11 simprlr 528 . . . 4  |-  ( ( (EXMID 
/\  Fun  `' f
)  /\  ( (
( Fun  g  /\  dom  g  =  B
)  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  ran  g  C_  A )
12 simprr 522 . . . 4  |-  ( ( (EXMID 
/\  Fun  `' f
)  /\  ( (
( Fun  g  /\  dom  g  =  B
)  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  Fun  `' g )
13 df-ima 4601 . . . . . . 7  |-  ( f
" U. D )  =  ran  ( f  |`  U. D )
14 df-rn 4599 . . . . . . 7  |-  ran  (
f  |`  U. D )  =  dom  `' ( f  |`  U. D )
1513, 14eqtr2i 2179 . . . . . 6  |-  dom  `' ( f  |`  U. D
)  =  ( f
" U. D )
16 df-ima 4601 . . . . . . . 8  |-  ( `' g " ( A 
\  U. D ) )  =  ran  ( `' g  |`  ( A  \ 
U. D ) )
17 df-rn 4599 . . . . . . . 8  |-  ran  ( `' g  |`  ( A 
\  U. D ) )  =  dom  `' ( `' g  |`  ( A 
\  U. D ) )
1816, 17eqtri 2178 . . . . . . 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 6906 . . . . . . 7  |-  ( (EXMID  /\  ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( `' g " ( A  \  U. D ) )  =  ( B 
\  ( f " U. D ) ) )
2218, 21eqtr3id 2204 . . . . . 6  |-  ( (EXMID  /\  ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  dom  `' ( `' g  |`  ( A  \  U. D
) )  =  ( B  \  ( f
" U. D ) ) )
23 ineq12 3304 . . . . . 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 411 . . . . 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 3467 . . . . 5  |-  ( ( f " U. D
)  i^i  ( B  \  ( f " U. D ) ) )  =  (/)
2624, 25eqtrdi 2206 . . . 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 1225 . . 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 5216 . . 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 1219 . 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 4763 . . . 4  |-  `' H  =  `' ( ( f  |`  U. D )  u.  ( `' g  |`  ( A  \  U. D
) ) )
32 cnvun 4993 . . . 4  |-  `' ( ( f  |`  U. D
)  u.  ( `' g  |`  ( A  \ 
U. D ) ) )  =  ( `' ( f  |`  U. D
)  u.  `' ( `' g  |`  ( A 
\  U. D ) ) )
3331, 32eqtri 2178 . . 3  |-  `' H  =  ( `' ( f  |`  U. D )  u.  `' ( `' g  |`  ( A  \ 
U. D ) ) )
3433funeqi 5193 . 2  |-  ( Fun  `' H  <->  Fun  ( `' ( f  |`  U. D )  u.  `' ( `' g  |`  ( A  \ 
U. D ) ) ) )
3529, 34sylibr 133 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 103    /\ w3a 963    = wceq 1335    e. wcel 2128   {cab 2143   _Vcvv 2712    \ cdif 3099    u. cun 3100    i^i cin 3101    C_ wss 3102   (/)c0 3395   U.cuni 3774  EXMIDwem 4157   `'ccnv 4587   dom cdm 4588   ran crn 4589    |` cres 4590   "cima 4591   Fun wfun 5166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-nul 4092  ax-pow 4137  ax-pr 4171
This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-br 3968  df-opab 4028  df-exmid 4158  df-id 4255  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 5174
This theorem is referenced by:  sbthlemi9  6911
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