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Theorem bdayfo 25595
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 5122 . . . 4  |-  ( x  e.  No  ->  dom  x  e.  _V )
21rgen 2763 . . 3  |-  A. x  e.  No  dom  x  e. 
_V
3 df-bday 25565 . . . 4  |-  bday  =  ( x  e.  No  |->  dom  x )
43mptfng 5562 . . 3  |-  ( A. x  e.  No  dom  x  e.  _V  <->  bday  Fn  No )
52, 4mpbi 200 . 2  |-  bday  Fn  No
63rnmpt 5108 . . 3  |-  ran  bday  =  { y  |  E. x  e.  No  y  =  dom  x }
7 noxp1o 25586 . . . . . 6  |-  ( y  e.  On  ->  (
y  X.  { 1o } )  e.  No )
8 1on 6723 . . . . . . . . . 10  |-  1o  e.  On
98elexi 2957 . . . . . . . . 9  |-  1o  e.  _V
109snnz 3914 . . . . . . . 8  |-  { 1o }  =/=  (/)
11 dmxp 5080 . . . . . . . 8  |-  ( { 1o }  =/=  (/)  ->  dom  ( y  X.  { 1o } )  =  y )
1210, 11ax-mp 8 . . . . . . 7  |-  dom  (
y  X.  { 1o } )  =  y
1312eqcomi 2439 . . . . . 6  |-  y  =  dom  ( y  X. 
{ 1o } )
14 dmeq 5062 . . . . . . . 8  |-  ( x  =  ( y  X. 
{ 1o } )  ->  dom  x  =  dom  ( y  X.  { 1o } ) )
1514eqeq2d 2446 . . . . . . 7  |-  ( x  =  ( y  X. 
{ 1o } )  ->  ( y  =  dom  x  <->  y  =  dom  ( y  X.  { 1o } ) ) )
1615rspcev 3044 . . . . . 6  |-  ( ( ( y  X.  { 1o } )  e.  No  /\  y  =  dom  (
y  X.  { 1o } ) )  ->  E. x  e.  No  y  =  dom  x )
177, 13, 16sylancl 644 . . . . 5  |-  ( y  e.  On  ->  E. x  e.  No  y  =  dom  x )
18 nodmon 25570 . . . . . . 7  |-  ( x  e.  No  ->  dom  x  e.  On )
19 eleq1a 2504 . . . . . . 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 2816 . . . . 5  |-  ( E. x  e.  No  y  =  dom  x  ->  y  e.  On )
2217, 21impbii 181 . . . 4  |-  ( y  e.  On  <->  E. x  e.  No  y  =  dom  x )
2322abbi2i 2546 . . 3  |-  On  =  { y  |  E. x  e.  No  y  =  dom  x }
246, 23eqtr4i 2458 . 2  |-  ran  bday  =  On
25 df-fo 5452 . 2  |-  ( bday
: No -onto-> On  <->  ( bday  Fn  No  /\  ran  bday  =  On ) )
265, 24, 25mpbir2an 887 1  |-  bday : No -onto-> On
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   {cab 2421    =/= wne 2598   A.wral 2697   E.wrex 2698   _Vcvv 2948   (/)c0 3620   {csn 3806   Oncon0 4573    X. cxp 4868   dom cdm 4870   ran crn 4871    Fn wfn 5441   -onto->wfo 5444   1oc1o 6709   Nocsur 25560   bdaycbday 25562
This theorem is referenced by:  bdayfun  25596  bdayrn  25597  bdaydm  25598  noprc  25601
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-13 1727  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  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  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-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-suc 4579  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-1o 6716  df-no 25563  df-bday 25565
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