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Theorem elno3 24309
Description: Another condition for membership in  No. (Contributed by Scott Fenton, 14-Apr-2012.)
Assertion
Ref Expression
elno3  |-  ( A  e.  No  <->  ( A : dom  A --> { 1o ,  2o }  /\  dom  A  e.  On ) )

Proof of Theorem elno3
StepHypRef Expression
1 3anan32 946 . 2  |-  ( ( Fun  A  /\  dom  A  e.  On  /\  ran  A 
C_  { 1o ,  2o } )  <->  ( ( Fun  A  /\  ran  A  C_ 
{ 1o ,  2o } )  /\  dom  A  e.  On ) )
2 elno2 24308 . 2  |-  ( A  e.  No  <->  ( Fun  A  /\  dom  A  e.  On  /\  ran  A  C_ 
{ 1o ,  2o } ) )
3 df-f 5259 . . . 4  |-  ( A : dom  A --> { 1o ,  2o }  <->  ( A  Fn  dom  A  /\  ran  A 
C_  { 1o ,  2o } ) )
4 funfn 5283 . . . . 5  |-  ( Fun 
A  <->  A  Fn  dom  A )
54anbi1i 676 . . . 4  |-  ( ( Fun  A  /\  ran  A 
C_  { 1o ,  2o } )  <->  ( A  Fn  dom  A  /\  ran  A 
C_  { 1o ,  2o } ) )
63, 5bitr4i 243 . . 3  |-  ( A : dom  A --> { 1o ,  2o }  <->  ( Fun  A  /\  ran  A  C_  { 1o ,  2o }
) )
76anbi1i 676 . 2  |-  ( ( A : dom  A --> { 1o ,  2o }  /\  dom  A  e.  On ) 
<->  ( ( Fun  A  /\  ran  A  C_  { 1o ,  2o } )  /\  dom  A  e.  On ) )
81, 2, 73bitr4i 268 1  |-  ( A  e.  No  <->  ( A : dom  A --> { 1o ,  2o }  /\  dom  A  e.  On ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1684    C_ wss 3152   {cpr 3641   Oncon0 4392   dom cdm 4689   ran crn 4690   Fun wfun 5249    Fn wfn 5250   -->wf 5251   1oc1o 6472   2oc2o 6473   Nocsur 24294
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-no 24297
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