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Theorem bdayelon 24405
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 24401 . . 3  |-  Fun  bday
2 fvelrn 5677 . . . 4  |-  ( ( Fun  bday  /\  A  e. 
dom  bday )  ->  ( bday `  A )  e. 
ran  bday )
3 bdayrn 24402 . . . 4  |-  ran  bday  =  On
42, 3syl6eleq 2386 . . 3  |-  ( ( Fun  bday  /\  A  e. 
dom  bday )  ->  ( bday `  A )  e.  On )
51, 4mpan 651 . 2  |-  ( A  e.  dom  bday  ->  (
bday `  A )  e.  On )
6 ndmfv 5568 . . 3  |-  ( -.  A  e.  dom  bday  -> 
( bday `  A )  =  (/) )
7 0elon 4461 . . 3  |-  (/)  e.  On
86, 7syl6eqel 2384 . 2  |-  ( -.  A  e.  dom  bday  -> 
( bday `  A )  e.  On )
95, 8pm2.61i 156 1  |-  ( bday `  A )  e.  On
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    e. wcel 1696   (/)c0 3468   Oncon0 4408   dom cdm 4705   ran crn 4706   Fun wfun 5265   ` cfv 5271   bdaycbday 24367
This theorem is referenced by:  fvnobday  24407  nodenselem3  24408  nodenselem4  24409  nodenselem6  24411  nodense  24414  nocvxminlem  24415  nobndlem2  24418  nobndlem4  24420  nobndlem5  24421  nobndlem6  24422  nobndlem8  24424
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6495  df-no 24368  df-bday 24370
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