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Theorem sbthlemi8 6941
Description: Lemma for isbth 6944. (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 5270 . . . 4  |-  ( Fun  `' f  ->  Fun  `' ( f  |`  U. D
) )
21ad2antlr 486 . . 3  |-  ( ( (EXMID 
/\  Fun  `' f
)  /\  ( (
( Fun  g  /\  dom  g  =  B
)  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  Fun  `' ( f  |`  U. D ) )
3 funcnvcnv 5257 . . . . . 6  |-  ( Fun  g  ->  Fun  `' `' g )
4 funres11 5270 . . . . . 6  |-  ( Fun  `' `' g  ->  Fun  `' ( `' g  |`  ( A 
\  U. D ) ) )
53, 4syl 14 . . . . 5  |-  ( Fun  g  ->  Fun  `' ( `' g  |`  ( A 
\  U. D ) ) )
65ad2antrr 485 . . . 4  |-  ( ( ( Fun  g  /\  dom  g  =  B
)  /\  ran  g  C_  A )  ->  Fun  `' ( `' g  |`  ( A  \  U. D
) ) )
76ad2antrl 487 . . 3  |-  ( ( (EXMID 
/\  Fun  `' f
)  /\  ( (
( Fun  g  /\  dom  g  =  B
)  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  Fun  `' ( `' g  |`  ( A  \  U. D ) ) )
8 simpll 524 . . . 4  |-  ( ( (EXMID 
/\  Fun  `' f
)  /\  ( (
( Fun  g  /\  dom  g  =  B
)  /\  ran  g  C_  A )  /\  Fun  `' g ) )  -> EXMID )
9 simprll 532 . . . . 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 533 . . . 4  |-  ( ( (EXMID 
/\  Fun  `' f
)  /\  ( (
( Fun  g  /\  dom  g  =  B
)  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  ran  g  C_  A )
12 simprr 527 . . . 4  |-  ( ( (EXMID 
/\  Fun  `' f
)  /\  ( (
( Fun  g  /\  dom  g  =  B
)  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  Fun  `' g )
13 df-ima 4624 . . . . . . 7  |-  ( f
" U. D )  =  ran  ( f  |`  U. D )
14 df-rn 4622 . . . . . . 7  |-  ran  (
f  |`  U. D )  =  dom  `' ( f  |`  U. D )
1513, 14eqtr2i 2192 . . . . . 6  |-  dom  `' ( f  |`  U. D
)  =  ( f
" U. D )
16 df-ima 4624 . . . . . . . 8  |-  ( `' g " ( A 
\  U. D ) )  =  ran  ( `' g  |`  ( A  \ 
U. D ) )
17 df-rn 4622 . . . . . . . 8  |-  ran  ( `' g  |`  ( A 
\  U. D ) )  =  dom  `' ( `' g  |`  ( A 
\  U. D ) )
1816, 17eqtri 2191 . . . . . . 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 6937 . . . . . . 7  |-  ( (EXMID  /\  ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( `' g " ( A  \  U. D ) )  =  ( B 
\  ( f " U. D ) ) )
2218, 21eqtr3id 2217 . . . . . 6  |-  ( (EXMID  /\  ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  dom  `' ( `' g  |`  ( A  \  U. D
) )  =  ( B  \  ( f
" U. D ) ) )
23 ineq12 3323 . . . . . 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 412 . . . . 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 3487 . . . . 5  |-  ( ( f " U. D
)  i^i  ( B  \  ( f " U. D ) ) )  =  (/)
2624, 25eqtrdi 2219 . . . 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 1238 . . 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 5242 . . 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 1232 . 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 4786 . . . 4  |-  `' H  =  `' ( ( f  |`  U. D )  u.  ( `' g  |`  ( A  \  U. D
) ) )
32 cnvun 5016 . . . 4  |-  `' ( ( f  |`  U. D
)  u.  ( `' g  |`  ( A  \ 
U. D ) ) )  =  ( `' ( f  |`  U. D
)  u.  `' ( `' g  |`  ( A 
\  U. D ) ) )
3331, 32eqtri 2191 . . 3  |-  `' H  =  ( `' ( f  |`  U. D )  u.  `' ( `' g  |`  ( A  \ 
U. D ) ) )
3433funeqi 5219 . 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 973    = wceq 1348    e. wcel 2141   {cab 2156   _Vcvv 2730    \ cdif 3118    u. cun 3119    i^i cin 3120    C_ wss 3121   (/)c0 3414   U.cuni 3796  EXMIDwem 4180   `'ccnv 4610   dom cdm 4611   ran crn 4612    |` cres 4613   "cima 4614   Fun wfun 5192
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-exmid 4181  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-fun 5200
This theorem is referenced by:  sbthlemi9  6942
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