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Theorem bdayrn 27137
Description: The birthday function's range is On. (Contributed by Scott Fenton, 14-Jun-2011.)
Assertion
Ref Expression
bdayrn ran bday = On

Proof of Theorem bdayrn
StepHypRef Expression
1 bdayfo 27041 . 2 bday : No onto→On
2 forn 6760 . 2 ( bday : No onto→On → ran bday = On)
31, 2ax-mp 5 1 ran bday = On
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  ran crn 5635  Oncon0 6318  ontowfo 6495   No csur 27004   bday cbday 27006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-1o 8413  df-no 27007  df-bday 27009
This theorem is referenced by:  nocvxminlem  27139  nocvxmin  27140  etasslt2  27175  scutbdaybnd2lim  27178  bday0s  27189  lrrecfr  27277
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