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Theorem elno2 28991
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 28986 . . 3  |-  ( A  e.  No  ->  Fun  A )
2 nodmon 28987 . . 3  |-  ( A  e.  No  ->  dom  A  e.  On )
3 norn 28988 . . 3  |-  ( A  e.  No  ->  ran  A 
C_  { 1o ,  2o } )
41, 2, 33jca 1176 . 2  |-  ( A  e.  No  ->  ( Fun  A  /\  dom  A  e.  On  /\  ran  A  C_ 
{ 1o ,  2o } ) )
5 simp2 997 . . . 4  |-  ( ( Fun  A  /\  dom  A  e.  On  /\  ran  A 
C_  { 1o ,  2o } )  ->  dom  A  e.  On )
6 simpl 457 . . . . . . . 8  |-  ( ( Fun  A  /\  dom  A  e.  On )  ->  Fun  A )
7 eqidd 2468 . . . . . . . 8  |-  ( ( Fun  A  /\  dom  A  e.  On )  ->  dom  A  =  dom  A
)
8 df-fn 5589 . . . . . . . 8  |-  ( A  Fn  dom  A  <->  ( Fun  A  /\  dom  A  =  dom  A ) )
96, 7, 8sylanbrc 664 . . . . . . 7  |-  ( ( Fun  A  /\  dom  A  e.  On )  ->  A  Fn  dom  A )
109anim1i 568 . . . . . 6  |-  ( ( ( Fun  A  /\  dom  A  e.  On )  /\  ran  A  C_  { 1o ,  2o }
)  ->  ( A  Fn  dom  A  /\  ran  A 
C_  { 1o ,  2o } ) )
11103impa 1191 . . . . 5  |-  ( ( Fun  A  /\  dom  A  e.  On  /\  ran  A 
C_  { 1o ,  2o } )  ->  ( A  Fn  dom  A  /\  ran  A  C_  { 1o ,  2o } ) )
12 df-f 5590 . . . . 5  |-  ( A : dom  A --> { 1o ,  2o }  <->  ( A  Fn  dom  A  /\  ran  A 
C_  { 1o ,  2o } ) )
1311, 12sylibr 212 . . . 4  |-  ( ( Fun  A  /\  dom  A  e.  On  /\  ran  A 
C_  { 1o ,  2o } )  ->  A : dom  A --> { 1o ,  2o } )
14 feq2 5712 . . . . 5  |-  ( x  =  dom  A  -> 
( A : x --> { 1o ,  2o } 
<->  A : dom  A --> { 1o ,  2o }
) )
1514rspcev 3214 . . . 4  |-  ( ( dom  A  e.  On  /\  A : dom  A --> { 1o ,  2o }
)  ->  E. x  e.  On  A : x --> { 1o ,  2o } )
165, 13, 15syl2anc 661 . . 3  |-  ( ( Fun  A  /\  dom  A  e.  On  /\  ran  A 
C_  { 1o ,  2o } )  ->  E. x  e.  On  A : x --> { 1o ,  2o } )
17 elno 28983 . . 3  |-  ( A  e.  No  <->  E. x  e.  On  A : x --> { 1o ,  2o } )
1816, 17sylibr 212 . 2  |-  ( ( Fun  A  /\  dom  A  e.  On  /\  ran  A 
C_  { 1o ,  2o } )  ->  A  e.  No )
194, 18impbii 188 1  |-  ( A  e.  No  <->  ( Fun  A  /\  dom  A  e.  On  /\  ran  A  C_ 
{ 1o ,  2o } ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   E.wrex 2815    C_ wss 3476   {cpr 4029   Oncon0 4878   dom cdm 4999   ran crn 5000   Fun wfun 5580    Fn wfn 5581   -->wf 5582   1oc1o 7120   2oc2o 7121   Nocsur 28977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-no 28980
This theorem is referenced by:  elno3  28992
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