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Theorem elno3 25326
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 948 . 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 25325 . 2  |-  ( A  e.  No  <->  ( Fun  A  /\  dom  A  e.  On  /\  ran  A  C_ 
{ 1o ,  2o } ) )
3 df-f 5391 . . . 4  |-  ( A : dom  A --> { 1o ,  2o }  <->  ( A  Fn  dom  A  /\  ran  A 
C_  { 1o ,  2o } ) )
4 funfn 5415 . . . . 5  |-  ( Fun 
A  <->  A  Fn  dom  A )
54anbi1i 677 . . . 4  |-  ( ( Fun  A  /\  ran  A 
C_  { 1o ,  2o } )  <->  ( A  Fn  dom  A  /\  ran  A 
C_  { 1o ,  2o } ) )
63, 5bitr4i 244 . . 3  |-  ( A : dom  A --> { 1o ,  2o }  <->  ( Fun  A  /\  ran  A  C_  { 1o ,  2o }
) )
76anbi1i 677 . 2  |-  ( ( A : dom  A --> { 1o ,  2o }  /\  dom  A  e.  On ) 
<->  ( ( Fun  A  /\  ran  A  C_  { 1o ,  2o } )  /\  dom  A  e.  On ) )
81, 2, 73bitr4i 269 1  |-  ( A  e.  No  <->  ( A : dom  A --> { 1o ,  2o }  /\  dom  A  e.  On ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936    e. wcel 1717    C_ wss 3256   {cpr 3751   Oncon0 4515   dom cdm 4811   ran crn 4812   Fun wfun 5381    Fn wfn 5382   -->wf 5383   1oc1o 6646   2oc2o 6647   Nocsur 25311
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-no 25314
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