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Theorem bdayelon 29657
Description: The value of the birthday function is always an ordinal. (Contributed by Scott Fenton, 14-Jun-2011.)
Assertion
Ref Expression
bdayelon  |-  ( bday `  A )  e.  On

Proof of Theorem bdayelon
StepHypRef Expression
1 bdayfun 29653 . . 3  |-  Fun  bday
2 fvelrn 6025 . . . 4  |-  ( ( Fun  bday  /\  A  e. 
dom  bday )  ->  ( bday `  A )  e. 
ran  bday )
3 bdayrn 29654 . . . 4  |-  ran  bday  =  On
42, 3syl6eleq 2555 . . 3  |-  ( ( Fun  bday  /\  A  e. 
dom  bday )  ->  ( bday `  A )  e.  On )
51, 4mpan 670 . 2  |-  ( A  e.  dom  bday  ->  (
bday `  A )  e.  On )
6 ndmfv 5896 . . 3  |-  ( -.  A  e.  dom  bday  -> 
( bday `  A )  =  (/) )
7 0elon 4940 . . 3  |-  (/)  e.  On
86, 7syl6eqel 2553 . 2  |-  ( -.  A  e.  dom  bday  -> 
( bday `  A )  e.  On )
95, 8pm2.61i 164 1  |-  ( bday `  A )  e.  On
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    e. wcel 1819   (/)c0 3793   Oncon0 4887   dom cdm 5008   ran crn 5009   Fun wfun 5588   ` cfv 5594   bdaycbday 29619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-1o 7148  df-no 29620  df-bday 29622
This theorem is referenced by:  fvnobday  29659  nodenselem3  29660  nodenselem4  29661  nodenselem6  29663  nodense  29666  nocvxminlem  29667  nobndlem2  29670  nobndlem4  29672  nobndlem5  29673  nobndlem6  29674  nobndlem8  29676
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