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Theorem elno2 25609
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 25604 . . 3  |-  ( A  e.  No  ->  Fun  A )
2 nodmon 25605 . . 3  |-  ( A  e.  No  ->  dom  A  e.  On )
3 norn 25606 . . 3  |-  ( A  e.  No  ->  ran  A 
C_  { 1o ,  2o } )
41, 2, 33jca 1134 . 2  |-  ( A  e.  No  ->  ( Fun  A  /\  dom  A  e.  On  /\  ran  A  C_ 
{ 1o ,  2o } ) )
5 simp2 958 . . . 4  |-  ( ( Fun  A  /\  dom  A  e.  On  /\  ran  A 
C_  { 1o ,  2o } )  ->  dom  A  e.  On )
6 simpl 444 . . . . . . . 8  |-  ( ( Fun  A  /\  dom  A  e.  On )  ->  Fun  A )
7 eqidd 2437 . . . . . . . 8  |-  ( ( Fun  A  /\  dom  A  e.  On )  ->  dom  A  =  dom  A
)
8 df-fn 5457 . . . . . . . 8  |-  ( A  Fn  dom  A  <->  ( Fun  A  /\  dom  A  =  dom  A ) )
96, 7, 8sylanbrc 646 . . . . . . 7  |-  ( ( Fun  A  /\  dom  A  e.  On )  ->  A  Fn  dom  A )
109anim1i 552 . . . . . 6  |-  ( ( ( Fun  A  /\  dom  A  e.  On )  /\  ran  A  C_  { 1o ,  2o }
)  ->  ( A  Fn  dom  A  /\  ran  A 
C_  { 1o ,  2o } ) )
11103impa 1148 . . . . 5  |-  ( ( Fun  A  /\  dom  A  e.  On  /\  ran  A 
C_  { 1o ,  2o } )  ->  ( A  Fn  dom  A  /\  ran  A  C_  { 1o ,  2o } ) )
12 df-f 5458 . . . . 5  |-  ( A : dom  A --> { 1o ,  2o }  <->  ( A  Fn  dom  A  /\  ran  A 
C_  { 1o ,  2o } ) )
1311, 12sylibr 204 . . . 4  |-  ( ( Fun  A  /\  dom  A  e.  On  /\  ran  A 
C_  { 1o ,  2o } )  ->  A : dom  A --> { 1o ,  2o } )
14 feq2 5577 . . . . 5  |-  ( x  =  dom  A  -> 
( A : x --> { 1o ,  2o } 
<->  A : dom  A --> { 1o ,  2o }
) )
1514rspcev 3052 . . . 4  |-  ( ( dom  A  e.  On  /\  A : dom  A --> { 1o ,  2o }
)  ->  E. x  e.  On  A : x --> { 1o ,  2o } )
165, 13, 15syl2anc 643 . . 3  |-  ( ( Fun  A  /\  dom  A  e.  On  /\  ran  A 
C_  { 1o ,  2o } )  ->  E. x  e.  On  A : x --> { 1o ,  2o } )
17 elno 25601 . . 3  |-  ( A  e.  No  <->  E. x  e.  On  A : x --> { 1o ,  2o } )
1816, 17sylibr 204 . 2  |-  ( ( Fun  A  /\  dom  A  e.  On  /\  ran  A 
C_  { 1o ,  2o } )  ->  A  e.  No )
194, 18impbii 181 1  |-  ( A  e.  No  <->  ( Fun  A  /\  dom  A  e.  On  /\  ran  A  C_ 
{ 1o ,  2o } ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   E.wrex 2706    C_ wss 3320   {cpr 3815   Oncon0 4581   dom cdm 4878   ran crn 4879   Fun wfun 5448    Fn wfn 5449   -->wf 5450   1oc1o 6717   2oc2o 6718   Nocsur 25595
This theorem is referenced by:  elno3  25610
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-no 25598
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