Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-cnf Unicode version

Definition df-cnf 8100
 Description: Define the Cantor normal form function, which takes as input a finitely supported function from to and outputs the corresponding member of the ordinal exponential . The content of the original Cantor Normal Form theorem is that for this function is a bijection onto for any ordinal (or, since the function restricts naturally to different ordinals, the statement that the composite function is a bijection to ). More can be said about the function, however, and in particular it is an order isomorphism for a certain easily defined well-ordering of the finitely supported functions, which gives an alternate definition cantnffval2 8135 of this function in terms of df-oi 7956. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
Assertion
Ref Expression
df-cnf
Distinct variable group:   ,,,,,,

Detailed syntax breakdown of Definition df-cnf
StepHypRef Expression
1 ccnf 8099 . 2
2 vx . . 3
3 vy . . 3
4 con0 4883 . . 3
5 vf . . . 4
6 vg . . . . . . 7
76cv 1394 . . . . . 6
8 c0 3784 . . . . . 6
9 cfsupp 7849 . . . . . 6
107, 8, 9wbr 4452 . . . . 5
112cv 1394 . . . . . 6
123cv 1394 . . . . . 6
13 cmap 7439 . . . . . 6
1411, 12, 13co 6296 . . . . 5
1510, 6, 14crab 2811 . . . 4
16 vh . . . . 5
175cv 1394 . . . . . . 7
18 csupp 6918 . . . . . . 7
1917, 8, 18co 6296 . . . . . 6
20 cep 4794 . . . . . 6
2119, 20coi 7955 . . . . 5
2216cv 1394 . . . . . . 7
2322cdm 5004 . . . . . 6
24 vk . . . . . . . 8
25 vz . . . . . . . 8
26 cvv 3109 . . . . . . . 8
2724cv 1394 . . . . . . . . . . . 12
2827, 22cfv 5593 . . . . . . . . . . 11
29 coe 7148 . . . . . . . . . . 11
3011, 28, 29co 6296 . . . . . . . . . 10
3128, 17cfv 5593 . . . . . . . . . 10
32 comu 7147 . . . . . . . . . 10
3330, 31, 32co 6296 . . . . . . . . 9
3425cv 1394 . . . . . . . . 9
35 coa 7146 . . . . . . . . 9
3633, 34, 35co 6296 . . . . . . . 8
3724, 25, 26, 26, 36cmpt2 6298 . . . . . . 7
3837, 8cseqom 7131 . . . . . 6
3923, 38cfv 5593 . . . . 5
4016, 21, 39csb 3434 . . . 4
415, 15, 40cmpt 4510 . . 3
422, 3, 4, 4, 41cmpt2 6298 . 2
431, 42wceq 1395 1
 Colors of variables: wff setvar class This definition is referenced by:  cantnffval  8101
 Copyright terms: Public domain W3C validator