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Theorem bdayfo 29675
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 6704 . . . 4  |-  ( x  e.  No  ->  dom  x  e.  _V )
21rgen 2814 . . 3  |-  A. x  e.  No  dom  x  e. 
_V
3 df-bday 29645 . . . 4  |-  bday  =  ( x  e.  No  |->  dom  x )
43mptfng 5688 . . 3  |-  ( A. x  e.  No  dom  x  e.  _V  <->  bday  Fn  No )
52, 4mpbi 208 . 2  |-  bday  Fn  No
63rnmpt 5237 . . 3  |-  ran  bday  =  { y  |  E. x  e.  No  y  =  dom  x }
7 noxp1o 29666 . . . . . 6  |-  ( y  e.  On  ->  (
y  X.  { 1o } )  e.  No )
8 1on 7129 . . . . . . . . . 10  |-  1o  e.  On
98elexi 3116 . . . . . . . . 9  |-  1o  e.  _V
109snnz 4134 . . . . . . . 8  |-  { 1o }  =/=  (/)
11 dmxp 5210 . . . . . . . 8  |-  ( { 1o }  =/=  (/)  ->  dom  ( y  X.  { 1o } )  =  y )
1210, 11ax-mp 5 . . . . . . 7  |-  dom  (
y  X.  { 1o } )  =  y
1312eqcomi 2467 . . . . . 6  |-  y  =  dom  ( y  X. 
{ 1o } )
14 dmeq 5192 . . . . . . . 8  |-  ( x  =  ( y  X. 
{ 1o } )  ->  dom  x  =  dom  ( y  X.  { 1o } ) )
1514eqeq2d 2468 . . . . . . 7  |-  ( x  =  ( y  X. 
{ 1o } )  ->  ( y  =  dom  x  <->  y  =  dom  ( y  X.  { 1o } ) ) )
1615rspcev 3207 . . . . . 6  |-  ( ( ( y  X.  { 1o } )  e.  No  /\  y  =  dom  (
y  X.  { 1o } ) )  ->  E. x  e.  No  y  =  dom  x )
177, 13, 16sylancl 660 . . . . 5  |-  ( y  e.  On  ->  E. x  e.  No  y  =  dom  x )
18 nodmon 29650 . . . . . . 7  |-  ( x  e.  No  ->  dom  x  e.  On )
19 eleq1a 2537 . . . . . . 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 2940 . . . . 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 2587 . . 3  |-  On  =  { y  |  E. x  e.  No  y  =  dom  x }
246, 23eqtr4i 2486 . 2  |-  ran  bday  =  On
25 df-fo 5576 . 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 1398    e. wcel 1823   {cab 2439    =/= wne 2649   A.wral 2804   E.wrex 2805   _Vcvv 3106   (/)c0 3783   {csn 4016   Oncon0 4867    X. cxp 4986   dom cdm 4988   ran crn 4989    Fn wfn 5565   -onto->wfo 5568   1oc1o 7115   Nocsur 29640   bdaycbday 29642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-1o 7122  df-no 29643  df-bday 29645
This theorem is referenced by:  bdayfun  29676  bdayrn  29677  bdaydm  29678  noprc  29681
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