ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbthlemi9 Unicode version

Theorem sbthlemi9 6821
Description: Lemma for isbth 6823. (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 967 . . . . . . . . . 10  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  f : A -1-1-> B )
2 df-f1 5098 . . . . . . . . . 10  |-  ( f : A -1-1-> B  <->  ( f : A --> B  /\  Fun  `' f ) )
31, 2sylib 121 . . . . . . . . 9  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  ( f : A --> B  /\  Fun  `' f ) )
43simpld 111 . . . . . . . 8  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  f : A
--> B )
5 df-f 5097 . . . . . . . 8  |-  ( f : A --> B  <->  ( f  Fn  A  /\  ran  f  C_  B ) )
64, 5sylib 121 . . . . . . 7  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  ( f  Fn  A  /\  ran  f  C_  B ) )
76simpld 111 . . . . . 6  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  f  Fn  A )
8 df-fn 5096 . . . . . 6  |-  ( f  Fn  A  <->  ( Fun  f  /\  dom  f  =  A ) )
97, 8sylib 121 . . . . 5  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  ( Fun  f  /\  dom  f  =  A ) )
109simpld 111 . . . 4  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  Fun  f )
11 simp3 968 . . . . . 6  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  g : B -1-1-> A )
12 df-f1 5098 . . . . . 6  |-  ( g : B -1-1-> A  <->  ( g : B --> A  /\  Fun  `' g ) )
1311, 12sylib 121 . . . . 5  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  ( g : B --> A  /\  Fun  `' g ) )
1413simprd 113 . . . 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 6819 . . . 4  |-  ( ( Fun  f  /\  Fun  `' g )  ->  Fun  H )
1910, 14, 18syl2anc 408 . . 3  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  Fun  H )
20 simp1 966 . . . 4  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  -> EXMID )
219simprd 113 . . . 4  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  dom  f  =  A )
2213simpld 111 . . . . . 6  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  g : B
--> A )
23 df-f 5097 . . . . . 6  |-  ( g : B --> A  <->  ( g  Fn  B  /\  ran  g  C_  A ) )
2422, 23sylib 121 . . . . 5  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  ( g  Fn  B  /\  ran  g  C_  A ) )
2524simprd 113 . . . 4  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  ran  g  C_  A )
2615, 16, 17sbthlemi5 6817 . . . 4  |-  ( (EXMID  /\  ( dom  f  =  A  /\  ran  g  C_  A ) )  ->  dom  H  =  A )
2720, 21, 25, 26syl12anc 1199 . . 3  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  dom  H  =  A )
28 df-fn 5096 . . 3  |-  ( H  Fn  A  <->  ( Fun  H  /\  dom  H  =  A ) )
2919, 27, 28sylanbrc 413 . 2  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  H  Fn  A )
303simprd 113 . . . 4  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  Fun  `' f )
3124simpld 111 . . . . . 6  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  g  Fn  B )
32 df-fn 5096 . . . . . 6  |-  ( g  Fn  B  <->  ( Fun  g  /\  dom  g  =  B ) )
3331, 32sylib 121 . . . . 5  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  ( Fun  g  /\  dom  g  =  B ) )
3433, 25jca 304 . . . 4  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  ( ( Fun  g  /\  dom  g  =  B )  /\  ran  g  C_  A ) )
3515, 16, 17sbthlemi8 6820 . . . 4  |-  ( ( (EXMID 
/\  Fun  `' f
)  /\  ( (
( Fun  g  /\  dom  g  =  B
)  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  Fun  `' H )
3620, 30, 34, 14, 35syl22anc 1202 . . 3  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  Fun  `' H
)
376simprd 113 . . . 4  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  ran  f  C_  B )
3833simprd 113 . . . . 5  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  dom  g  =  B )
3938, 25jca 304 . . . 4  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  ( dom  g  =  B  /\  ran  g  C_  A ) )
40 df-rn 4520 . . . . 5  |-  ran  H  =  dom  `' H
4115, 16, 17sbthlemi6 6818 . . . . 5  |-  ( ( (EXMID 
/\  ran  f  C_  B )  /\  (
( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  ran  H  =  B )
4240, 41syl5eqr 2164 . . . 4  |-  ( ( (EXMID 
/\  ran  f  C_  B )  /\  (
( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g ) )  ->  dom  `' H  =  B
)
4320, 37, 39, 14, 42syl22anc 1202 . . 3  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  dom  `' H  =  B )
44 df-fn 5096 . . 3  |-  ( `' H  Fn  B  <->  ( Fun  `' H  /\  dom  `' H  =  B )
)
4536, 43, 44sylanbrc 413 . 2  |-  ( (EXMID  /\  f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  `' H  Fn  B )
46 dff1o4 5343 . 2  |-  ( H : A -1-1-onto-> B  <->  ( H  Fn  A  /\  `' H  Fn  B ) )
4729, 45, 46sylanbrc 413 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 103    /\ w3a 947    = wceq 1316    e. wcel 1465   {cab 2103   _Vcvv 2660    \ cdif 3038    u. cun 3039    C_ wss 3041   U.cuni 3706  EXMIDwem 4088   `'ccnv 4508   dom cdm 4509   ran crn 4510    |` cres 4511   "cima 4512   Fun wfun 5087    Fn wfn 5088   -->wf 5089   -1-1->wf1 5090   -1-1-onto->wf1o 5092
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-stab 801  df-dc 805  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-exmid 4089  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100
This theorem is referenced by:  sbthlemi10  6822
  Copyright terms: Public domain W3C validator