Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elno2 Unicode version

Theorem elno2 24379
Description: An alternative condition for membership in  No. (Contributed by Scott Fenton, 21-Mar-2012.)
Assertion
Ref Expression
elno2  |-  ( A  e.  No  <->  ( Fun  A  /\  dom  A  e.  On  /\  ran  A  C_ 
{ 1o ,  2o } ) )

Proof of Theorem elno2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 nofun 24374 . . 3  |-  ( A  e.  No  ->  Fun  A )
2 nodmon 24375 . . 3  |-  ( A  e.  No  ->  dom  A  e.  On )
3 norn 24376 . . 3  |-  ( A  e.  No  ->  ran  A 
C_  { 1o ,  2o } )
41, 2, 33jca 1132 . 2  |-  ( A  e.  No  ->  ( Fun  A  /\  dom  A  e.  On  /\  ran  A  C_ 
{ 1o ,  2o } ) )
5 simp2 956 . . . 4  |-  ( ( Fun  A  /\  dom  A  e.  On  /\  ran  A 
C_  { 1o ,  2o } )  ->  dom  A  e.  On )
6 simpl 443 . . . . . . . 8  |-  ( ( Fun  A  /\  dom  A  e.  On )  ->  Fun  A )
7 eqidd 2297 . . . . . . . 8  |-  ( ( Fun  A  /\  dom  A  e.  On )  ->  dom  A  =  dom  A
)
8 df-fn 5274 . . . . . . . 8  |-  ( A  Fn  dom  A  <->  ( Fun  A  /\  dom  A  =  dom  A ) )
96, 7, 8sylanbrc 645 . . . . . . 7  |-  ( ( Fun  A  /\  dom  A  e.  On )  ->  A  Fn  dom  A )
109anim1i 551 . . . . . 6  |-  ( ( ( Fun  A  /\  dom  A  e.  On )  /\  ran  A  C_  { 1o ,  2o }
)  ->  ( A  Fn  dom  A  /\  ran  A 
C_  { 1o ,  2o } ) )
11103impa 1146 . . . . 5  |-  ( ( Fun  A  /\  dom  A  e.  On  /\  ran  A 
C_  { 1o ,  2o } )  ->  ( A  Fn  dom  A  /\  ran  A  C_  { 1o ,  2o } ) )
12 df-f 5275 . . . . 5  |-  ( A : dom  A --> { 1o ,  2o }  <->  ( A  Fn  dom  A  /\  ran  A 
C_  { 1o ,  2o } ) )
1311, 12sylibr 203 . . . 4  |-  ( ( Fun  A  /\  dom  A  e.  On  /\  ran  A 
C_  { 1o ,  2o } )  ->  A : dom  A --> { 1o ,  2o } )
14 feq2 5392 . . . . 5  |-  ( x  =  dom  A  -> 
( A : x --> { 1o ,  2o } 
<->  A : dom  A --> { 1o ,  2o }
) )
1514rspcev 2897 . . . 4  |-  ( ( dom  A  e.  On  /\  A : dom  A --> { 1o ,  2o }
)  ->  E. x  e.  On  A : x --> { 1o ,  2o } )
165, 13, 15syl2anc 642 . . 3  |-  ( ( Fun  A  /\  dom  A  e.  On  /\  ran  A 
C_  { 1o ,  2o } )  ->  E. x  e.  On  A : x --> { 1o ,  2o } )
17 elno 24371 . . 3  |-  ( A  e.  No  <->  E. x  e.  On  A : x --> { 1o ,  2o } )
1816, 17sylibr 203 . 2  |-  ( ( Fun  A  /\  dom  A  e.  On  /\  ran  A 
C_  { 1o ,  2o } )  ->  A  e.  No )
194, 18impbii 180 1  |-  ( A  e.  No  <->  ( Fun  A  /\  dom  A  e.  On  /\  ran  A  C_ 
{ 1o ,  2o } ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   E.wrex 2557    C_ wss 3165   {cpr 3654   Oncon0 4408   dom cdm 4705   ran crn 4706   Fun wfun 5265    Fn wfn 5266   -->wf 5267   1oc1o 6488   2oc2o 6489   Nocsur 24365
This theorem is referenced by:  elno3  24380
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-no 24368
  Copyright terms: Public domain W3C validator