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Mirrors > Home > MPE Home > Th. List > bdaydm | Structured version Visualization version GIF version |
Description: The birthday function's domain is No . (Contributed by Scott Fenton, 14-Jun-2011.) |
Ref | Expression |
---|---|
bdaydm | ⊢ dom bday = No |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdayfo 27701 | . . 3 ⊢ bday : No –onto→On | |
2 | fof 6804 | . . 3 ⊢ ( bday : No –onto→On → bday : No ⟶On) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ bday : No ⟶On |
4 | 3 | fdmi 6728 | 1 ⊢ dom bday = No |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 dom cdm 5672 Oncon0 6365 ⟶wf 6539 –onto→wfo 6541 No csur 27663 bday cbday 27665 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-suc 6371 df-fun 6545 df-fn 6546 df-f 6547 df-fo 6549 df-1o 8485 df-no 27666 df-bday 27668 |
This theorem is referenced by: nocvxminlem 27801 nocvxmin 27802 bday0s 27852 leftval 27881 rightval 27882 madebdayim 27905 lrold 27914 negsbdaylem 28059 |
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