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Theorem elno3 25602
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 25601 . 2  |-  ( A  e.  No  <->  ( Fun  A  /\  dom  A  e.  On  /\  ran  A  C_ 
{ 1o ,  2o } ) )
3 df-f 5450 . . . 4  |-  ( A : dom  A --> { 1o ,  2o }  <->  ( A  Fn  dom  A  /\  ran  A 
C_  { 1o ,  2o } ) )
4 funfn 5474 . . . . 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 1725    C_ wss 3312   {cpr 3807   Oncon0 4573   dom cdm 4870   ran crn 4871   Fun wfun 5440    Fn wfn 5441   -->wf 5442   1oc1o 6709   2oc2o 6710   Nocsur 25587
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pr 4395
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-no 25590
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