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Theorem bdayfo 30635
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 6743 . . . 4  |-  ( x  e.  No  ->  dom  x  e.  _V )
21rgen 2766 . . 3  |-  A. x  e.  No  dom  x  e. 
_V
3 df-bday 30603 . . . 4  |-  bday  =  ( x  e.  No  |->  dom  x )
43mptfng 5713 . . 3  |-  ( A. x  e.  No  dom  x  e.  _V  <->  bday  Fn  No )
52, 4mpbi 213 . 2  |-  bday  Fn  No
63rnmpt 5086 . . 3  |-  ran  bday  =  { y  |  E. x  e.  No  y  =  dom  x }
7 noxp1o 30624 . . . . . 6  |-  ( y  e.  On  ->  (
y  X.  { 1o } )  e.  No )
8 1on 7207 . . . . . . . . . 10  |-  1o  e.  On
98elexi 3041 . . . . . . . . 9  |-  1o  e.  _V
109snnz 4081 . . . . . . . 8  |-  { 1o }  =/=  (/)
11 dmxp 5059 . . . . . . . 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 5040 . . . . . . . 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 3136 . . . . . 6  |-  ( ( ( y  X.  { 1o } )  e.  No  /\  y  =  dom  (
y  X.  { 1o } ) )  ->  E. x  e.  No  y  =  dom  x )
177, 13, 16sylancl 675 . . . . 5  |-  ( y  e.  On  ->  E. x  e.  No  y  =  dom  x )
18 nodmon 30608 . . . . . . 7  |-  ( x  e.  No  ->  dom  x  e.  On )
19 eleq1a 2544 . . . . . . 7  |-  ( dom  x  e.  On  ->  ( y  =  dom  x  ->  y  e.  On ) )
2018, 19syl 17 . . . . . 6  |-  ( x  e.  No  ->  (
y  =  dom  x  ->  y  e.  On ) )
2120rexlimiv 2867 . . . . 5  |-  ( E. x  e.  No  y  =  dom  x  ->  y  e.  On )
2217, 21impbii 192 . . . 4  |-  ( y  e.  On  <->  E. x  e.  No  y  =  dom  x )
2322abbi2i 2586 . . 3  |-  On  =  { y  |  E. x  e.  No  y  =  dom  x }
246, 23eqtr4i 2496 . 2  |-  ran  bday  =  On
25 df-fo 5595 . 2  |-  ( bday
: No -onto-> On  <->  ( bday  Fn  No  /\  ran  bday  =  On ) )
265, 24, 25mpbir2an 934 1  |-  bday : No -onto-> On
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1452    e. wcel 1904   {cab 2457    =/= wne 2641   A.wral 2756   E.wrex 2757   _Vcvv 3031   (/)c0 3722   {csn 3959    X. cxp 4837   dom cdm 4839   ran crn 4840   Oncon0 5430    Fn wfn 5584   -onto->wfo 5587   1oc1o 7193   Nocsur 30598   bdaycbday 30600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-ord 5433  df-on 5434  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-1o 7200  df-no 30601  df-bday 30603
This theorem is referenced by:  bdayfun  30636  bdayrn  30637  bdaydm  30638  noprc  30641
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