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Mirrors > Home > MPE Home > Th. List > bdayrn | Structured version Visualization version GIF version |
Description: The birthday function's range is On. (Contributed by Scott Fenton, 14-Jun-2011.) |
Ref | Expression |
---|---|
bdayrn | ⊢ ran bday = On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdayfo 27737 | . 2 ⊢ bday : No –onto→On | |
2 | forn 6824 | . 2 ⊢ ( bday : No –onto→On → ran bday = On) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ran bday = On |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ran crn 5690 Oncon0 6386 –onto→wfo 6561 No csur 27699 bday cbday 27701 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-suc 6392 df-fun 6565 df-fn 6566 df-f 6567 df-fo 6569 df-1o 8505 df-no 27702 df-bday 27704 |
This theorem is referenced by: nocvxminlem 27837 nocvxmin 27838 etasslt2 27874 scutbdaybnd2lim 27877 bday0s 27888 lrrecfr 27991 zs12bday 28439 |
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