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Theorem bdayfo 29288
Description: The birthday function maps the surreals onto the ordinals. Alling's axiom (B). (Shortened proof on 2012-Apr-14, SF). (Contributed by Scott Fenton, 11-Jun-2011.)
Assertion
Ref Expression
bdayfo  |-  bday : No -onto-> On

Proof of Theorem bdayfo
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmexg 6716 . . . 4  |-  ( x  e.  No  ->  dom  x  e.  _V )
21rgen 2824 . . 3  |-  A. x  e.  No  dom  x  e. 
_V
3 df-bday 29258 . . . 4  |-  bday  =  ( x  e.  No  |->  dom  x )
43mptfng 5706 . . 3  |-  ( A. x  e.  No  dom  x  e.  _V  <->  bday  Fn  No )
52, 4mpbi 208 . 2  |-  bday  Fn  No
63rnmpt 5248 . . 3  |-  ran  bday  =  { y  |  E. x  e.  No  y  =  dom  x }
7 noxp1o 29279 . . . . . 6  |-  ( y  e.  On  ->  (
y  X.  { 1o } )  e.  No )
8 1on 7138 . . . . . . . . . 10  |-  1o  e.  On
98elexi 3123 . . . . . . . . 9  |-  1o  e.  _V
109snnz 4145 . . . . . . . 8  |-  { 1o }  =/=  (/)
11 dmxp 5221 . . . . . . . 8  |-  ( { 1o }  =/=  (/)  ->  dom  ( y  X.  { 1o } )  =  y )
1210, 11ax-mp 5 . . . . . . 7  |-  dom  (
y  X.  { 1o } )  =  y
1312eqcomi 2480 . . . . . 6  |-  y  =  dom  ( y  X. 
{ 1o } )
14 dmeq 5203 . . . . . . . 8  |-  ( x  =  ( y  X. 
{ 1o } )  ->  dom  x  =  dom  ( y  X.  { 1o } ) )
1514eqeq2d 2481 . . . . . . 7  |-  ( x  =  ( y  X. 
{ 1o } )  ->  ( y  =  dom  x  <->  y  =  dom  ( y  X.  { 1o } ) ) )
1615rspcev 3214 . . . . . 6  |-  ( ( ( y  X.  { 1o } )  e.  No  /\  y  =  dom  (
y  X.  { 1o } ) )  ->  E. x  e.  No  y  =  dom  x )
177, 13, 16sylancl 662 . . . . 5  |-  ( y  e.  On  ->  E. x  e.  No  y  =  dom  x )
18 nodmon 29263 . . . . . . 7  |-  ( x  e.  No  ->  dom  x  e.  On )
19 eleq1a 2550 . . . . . . 7  |-  ( dom  x  e.  On  ->  ( y  =  dom  x  ->  y  e.  On ) )
2018, 19syl 16 . . . . . 6  |-  ( x  e.  No  ->  (
y  =  dom  x  ->  y  e.  On ) )
2120rexlimiv 2949 . . . . 5  |-  ( E. x  e.  No  y  =  dom  x  ->  y  e.  On )
2217, 21impbii 188 . . . 4  |-  ( y  e.  On  <->  E. x  e.  No  y  =  dom  x )
2322abbi2i 2600 . . 3  |-  On  =  { y  |  E. x  e.  No  y  =  dom  x }
246, 23eqtr4i 2499 . 2  |-  ran  bday  =  On
25 df-fo 5594 . 2  |-  ( bday
: No -onto-> On  <->  ( bday  Fn  No  /\  ran  bday  =  On ) )
265, 24, 25mpbir2an 918 1  |-  bday : No -onto-> On
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   {cab 2452    =/= wne 2662   A.wral 2814   E.wrex 2815   _Vcvv 3113   (/)c0 3785   {csn 4027   Oncon0 4878    X. cxp 4997   dom cdm 4999   ran crn 5000    Fn wfn 5583   -onto->wfo 5586   1oc1o 7124   Nocsur 29253   bdaycbday 29255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-1o 7131  df-no 29256  df-bday 29258
This theorem is referenced by:  bdayfun  29289  bdayrn  29290  bdaydm  29291  noprc  29294
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