Theorem bdayrn 28164

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

Step	Hyp	Ref	Expression
1		bdayfo 28061	. 2 ⊢ bday:No⟶ontoOn
2		forn 6818	. 2 ⊢ (bday:No⟶ontoOn → ran bday = On)
3	1, 2	ax-mp 5	1 ⊢ ran bday = On

This proof depends on syntax axioms:   = wceq 1570  ran crn 5656  Oncon0 6377  ⟶ontowfo 6553  Nocsur 28024  bdaycbday 28026

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-suc 6383  df-fun 6557  df-fn 6558  df-f 6559  df-fo 6561  df-1o 8501  df-no 28027  df-bday 28029

This theorem is used by:  nobdaymin  28166  nocvxminlem  28167  noeta2  28174  etaslts2  28207  cutbdaybnd2lim  28210  bday0  28224  lrrecfr  28356  oniso  28684  bdayons  28689