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Theorem sbthlemi9 6964
Description: Lemma for isbth 6966. (Contributed by NM, 28-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
sbthlemi9  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  H : A
-1-1-onto-> B )
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 sbthlemi9
StepHypRef Expression
1 simp2 998 . . . . . . . . . 10  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  f : A -1-1-> B )
2 df-f1 5222 . . . . . . . . . 10  |-  ( f : A -1-1-> B  <->  ( f : A --> B  /\  Fun  `' f ) )
31, 2sylib 122 . . . . . . . . 9  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  ( f : A --> B  /\  Fun  `' f ) )
43simpld 112 . . . . . . . 8  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  f : A
--> B )
5 df-f 5221 . . . . . . . 8  |-  ( f : A --> B  <->  ( f  Fn  A  /\  ran  f  C_  B ) )
64, 5sylib 122 . . . . . . 7  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  ( f  Fn  A  /\  ran  f  C_  B ) )
76simpld 112 . . . . . 6  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  f  Fn  A )
8 df-fn 5220 . . . . . 6  |-  ( f  Fn  A  <->  ( Fun  f  /\  dom  f  =  A ) )
97, 8sylib 122 . . . . 5  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  ( Fun  f  /\  dom  f  =  A ) )
109simpld 112 . . . 4  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  Fun  f )
11 simp3 999 . . . . . 6  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  g : B -1-1-> A )
12 df-f1 5222 . . . . . 6  |-  ( g : B -1-1-> A  <->  ( g : B --> A  /\  Fun  `' g ) )
1311, 12sylib 122 . . . . 5  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  ( g : B --> A  /\  Fun  `' g ) )
1413simprd 114 . . . 4  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  Fun  `' g )
15 sbthlem.1 . . . . 5  |-  A  e. 
_V
16 sbthlem.2 . . . . 5  |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f "
x ) ) ) 
C_  ( A  \  x ) ) }
17 sbthlem.3 . . . . 5  |-  H  =  ( ( f  |`  U. D )  u.  ( `' g  |`  ( A 
\  U. D ) ) )
1815, 16, 17sbthlem7 6962 . . . 4  |-  ( ( Fun  f  /\  Fun  `' g )  ->  Fun  H )
1910, 14, 18syl2anc 411 . . 3  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  Fun  H )
20 simp1 997 . . . 4  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  -> EXMID )
219simprd 114 . . . 4  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  dom  f  =  A )
2213simpld 112 . . . . . 6  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  g : B
--> A )
23 df-f 5221 . . . . . 6  |-  ( g : B --> A  <->  ( g  Fn  B  /\  ran  g  C_  A ) )
2422, 23sylib 122 . . . . 5  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  ( g  Fn  B  /\  ran  g  C_  A ) )
2524simprd 114 . . . 4  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  ran  g  C_  A )
2615, 16, 17sbthlemi5 6960 . . . 4  |-  ( (EXMID  /\  ( dom  f  =  A  /\  ran  g  C_  A ) )  ->  dom  H  =  A )
2720, 21, 25, 26syl12anc 1236 . . 3  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  dom  H  =  A )
28 df-fn 5220 . . 3  |-  ( H  Fn  A  <->  ( Fun  H  /\  dom  H  =  A ) )
2919, 27, 28sylanbrc 417 . 2  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  H  Fn  A )
303simprd 114 . . . 4  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  Fun  `' f )
3124simpld 112 . . . . . 6  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  g  Fn  B )
32 df-fn 5220 . . . . . 6  |-  ( g  Fn  B  <->  ( Fun  g  /\  dom  g  =  B ) )
3331, 32sylib 122 . . . . 5  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  ( Fun  g  /\  dom  g  =  B ) )
3433, 25jca 306 . . . 4  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  ( ( Fun  g  /\  dom  g  =  B )  /\  ran  g  C_  A ) )
3515, 16, 17sbthlemi8 6963 . . . 4  |-  ( ( (EXMID 
/\  Fun  `' f
)  /\  ( (
( Fun  g  /\  dom  g  =  B
)  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  Fun  `' H )
3620, 30, 34, 14, 35syl22anc 1239 . . 3  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  Fun  `' H
)
376simprd 114 . . . 4  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  ran  f  C_  B )
3833simprd 114 . . . . 5  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  dom  g  =  B )
3938, 25jca 306 . . . 4  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  ( dom  g  =  B  /\  ran  g  C_  A ) )
40 df-rn 4638 . . . . 5  |-  ran  H  =  dom  `' H
4115, 16, 17sbthlemi6 6961 . . . . 5  |-  ( ( (EXMID 
/\  ran  f  C_  B )  /\  (
( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  ran  H  =  B )
4240, 41eqtr3id 2224 . . . 4  |-  ( ( (EXMID 
/\  ran  f  C_  B )  /\  (
( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  dom  `' H  =  B
)
4320, 37, 39, 14, 42syl22anc 1239 . . 3  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  dom  `' H  =  B )
44 df-fn 5220 . . 3  |-  ( `' H  Fn  B  <->  ( Fun  `' H  /\  dom  `' H  =  B )
)
4536, 43, 44sylanbrc 417 . 2  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  `' H  Fn  B )
46 dff1o4 5470 . 2  |-  ( H : A -1-1-onto-> B  <->  ( H  Fn  A  /\  `' H  Fn  B ) )
4729, 45, 46sylanbrc 417 1  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  H : A
-1-1-onto-> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 978    = wceq 1353    e. wcel 2148   {cab 2163   _Vcvv 2738    \ cdif 3127    u. cun 3128    C_ wss 3130   U.cuni 3810  EXMIDwem 4195   `'ccnv 4626   dom cdm 4627   ran crn 4628    |` cres 4629   "cima 4630   Fun wfun 5211    Fn wfn 5212   -->wf 5213   -1-1->wf1 5214   -1-1-onto->wf1o 5216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-exmid 4196  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224
This theorem is referenced by:  sbthlemi10  6965
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