Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elno2 Structured version   Unicode version

Theorem elno2 30101
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 30096 . . 3  |-  ( A  e.  No  ->  Fun  A )
2 nodmon 30097 . . 3  |-  ( A  e.  No  ->  dom  A  e.  On )
3 norn 30098 . . 3  |-  ( A  e.  No  ->  ran  A 
C_  { 1o ,  2o } )
41, 2, 33jca 1177 . 2  |-  ( A  e.  No  ->  ( Fun  A  /\  dom  A  e.  On  /\  ran  A  C_ 
{ 1o ,  2o } ) )
5 simp2 998 . . . 4  |-  ( ( Fun  A  /\  dom  A  e.  On  /\  ran  A 
C_  { 1o ,  2o } )  ->  dom  A  e.  On )
6 simpl 455 . . . . . . . 8  |-  ( ( Fun  A  /\  dom  A  e.  On )  ->  Fun  A )
7 eqidd 2403 . . . . . . . 8  |-  ( ( Fun  A  /\  dom  A  e.  On )  ->  dom  A  =  dom  A
)
8 df-fn 5571 . . . . . . . 8  |-  ( A  Fn  dom  A  <->  ( Fun  A  /\  dom  A  =  dom  A ) )
96, 7, 8sylanbrc 662 . . . . . . 7  |-  ( ( Fun  A  /\  dom  A  e.  On )  ->  A  Fn  dom  A )
109anim1i 566 . . . . . 6  |-  ( ( ( Fun  A  /\  dom  A  e.  On )  /\  ran  A  C_  { 1o ,  2o }
)  ->  ( A  Fn  dom  A  /\  ran  A 
C_  { 1o ,  2o } ) )
11103impa 1192 . . . . 5  |-  ( ( Fun  A  /\  dom  A  e.  On  /\  ran  A 
C_  { 1o ,  2o } )  ->  ( A  Fn  dom  A  /\  ran  A  C_  { 1o ,  2o } ) )
12 df-f 5572 . . . . 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 5696 . . . . 5  |-  ( x  =  dom  A  -> 
( A : x --> { 1o ,  2o } 
<->  A : dom  A --> { 1o ,  2o }
) )
1514rspcev 3159 . . . 4  |-  ( ( dom  A  e.  On  /\  A : dom  A --> { 1o ,  2o }
)  ->  E. x  e.  On  A : x --> { 1o ,  2o } )
165, 13, 15syl2anc 659 . . 3  |-  ( ( Fun  A  /\  dom  A  e.  On  /\  ran  A 
C_  { 1o ,  2o } )  ->  E. x  e.  On  A : x --> { 1o ,  2o } )
17 elno 30093 . . 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842   E.wrex 2754    C_ wss 3413   {cpr 3973   dom cdm 4822   ran crn 4823   Oncon0 5409   Fun wfun 5562    Fn wfn 5563   -->wf 5564   1oc1o 7159   2oc2o 7160   Nocsur 30087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-no 30090
This theorem is referenced by:  elno3  30102
  Copyright terms: Public domain W3C validator