Metamath Proof Explorer


Definition df-cnf

Description: Define the Cantor normal form function, which takes as input a finitely supported function from y to x and outputs the corresponding member of the ordinal exponential x ^o y . The content of the original Cantor Normal Form theorem is that for x =om this function is a bijection onto om ^o y for any ordinal y (or, since the function restricts naturally to different ordinals, the statement that the composite function is a bijection to On ). 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 of this function in terms of df-oi . (Contributed by Mario Carneiro, 25-May-2015) (Revised by AV, 28-Jun-2019)

Ref Expression
Assertion df-cnf CNF = ( 𝑥 ∈ On , 𝑦 ∈ On ↦ ( 𝑓 ∈ { 𝑔 ∈ ( 𝑥m 𝑦 ) ∣ 𝑔 finSupp ∅ } ↦ OrdIso ( E , ( 𝑓 supp ∅ ) ) / ( seqω ( ( 𝑘 ∈ V , 𝑧 ∈ V ↦ ( ( ( 𝑥o ( 𝑘 ) ) ·o ( 𝑓 ‘ ( 𝑘 ) ) ) +o 𝑧 ) ) , ∅ ) ‘ dom ) ) )

Detailed syntax breakdown

Step Hyp Ref Expression
0 ccnf CNF
1 vx 𝑥
2 con0 On
3 vy 𝑦
4 vf 𝑓
5 vg 𝑔
6 1 cv 𝑥
7 cmap m
8 3 cv 𝑦
9 6 8 7 co ( 𝑥m 𝑦 )
10 5 cv 𝑔
11 cfsupp finSupp
12 c0
13 10 12 11 wbr 𝑔 finSupp ∅
14 13 5 9 crab { 𝑔 ∈ ( 𝑥m 𝑦 ) ∣ 𝑔 finSupp ∅ }
15 cep E
16 4 cv 𝑓
17 csupp supp
18 16 12 17 co ( 𝑓 supp ∅ )
19 18 15 coi OrdIso ( E , ( 𝑓 supp ∅ ) )
20 vh
21 vk 𝑘
22 cvv V
23 vz 𝑧
24 coe o
25 20 cv
26 21 cv 𝑘
27 26 25 cfv ( 𝑘 )
28 6 27 24 co ( 𝑥o ( 𝑘 ) )
29 comu ·o
30 27 16 cfv ( 𝑓 ‘ ( 𝑘 ) )
31 28 30 29 co ( ( 𝑥o ( 𝑘 ) ) ·o ( 𝑓 ‘ ( 𝑘 ) ) )
32 coa +o
33 23 cv 𝑧
34 31 33 32 co ( ( ( 𝑥o ( 𝑘 ) ) ·o ( 𝑓 ‘ ( 𝑘 ) ) ) +o 𝑧 )
35 21 23 22 22 34 cmpo ( 𝑘 ∈ V , 𝑧 ∈ V ↦ ( ( ( 𝑥o ( 𝑘 ) ) ·o ( 𝑓 ‘ ( 𝑘 ) ) ) +o 𝑧 ) )
36 35 12 cseqom seqω ( ( 𝑘 ∈ V , 𝑧 ∈ V ↦ ( ( ( 𝑥o ( 𝑘 ) ) ·o ( 𝑓 ‘ ( 𝑘 ) ) ) +o 𝑧 ) ) , ∅ )
37 25 cdm dom
38 37 36 cfv ( seqω ( ( 𝑘 ∈ V , 𝑧 ∈ V ↦ ( ( ( 𝑥o ( 𝑘 ) ) ·o ( 𝑓 ‘ ( 𝑘 ) ) ) +o 𝑧 ) ) , ∅ ) ‘ dom )
39 20 19 38 csb OrdIso ( E , ( 𝑓 supp ∅ ) ) / ( seqω ( ( 𝑘 ∈ V , 𝑧 ∈ V ↦ ( ( ( 𝑥o ( 𝑘 ) ) ·o ( 𝑓 ‘ ( 𝑘 ) ) ) +o 𝑧 ) ) , ∅ ) ‘ dom )
40 4 14 39 cmpt ( 𝑓 ∈ { 𝑔 ∈ ( 𝑥m 𝑦 ) ∣ 𝑔 finSupp ∅ } ↦ OrdIso ( E , ( 𝑓 supp ∅ ) ) / ( seqω ( ( 𝑘 ∈ V , 𝑧 ∈ V ↦ ( ( ( 𝑥o ( 𝑘 ) ) ·o ( 𝑓 ‘ ( 𝑘 ) ) ) +o 𝑧 ) ) , ∅ ) ‘ dom ) )
41 1 3 2 2 40 cmpo ( 𝑥 ∈ On , 𝑦 ∈ On ↦ ( 𝑓 ∈ { 𝑔 ∈ ( 𝑥m 𝑦 ) ∣ 𝑔 finSupp ∅ } ↦ OrdIso ( E , ( 𝑓 supp ∅ ) ) / ( seqω ( ( 𝑘 ∈ V , 𝑧 ∈ V ↦ ( ( ( 𝑥o ( 𝑘 ) ) ·o ( 𝑓 ‘ ( 𝑘 ) ) ) +o 𝑧 ) ) , ∅ ) ‘ dom ) ) )
42 0 41 wceq CNF = ( 𝑥 ∈ On , 𝑦 ∈ On ↦ ( 𝑓 ∈ { 𝑔 ∈ ( 𝑥m 𝑦 ) ∣ 𝑔 finSupp ∅ } ↦ OrdIso ( E , ( 𝑓 supp ∅ ) ) / ( seqω ( ( 𝑘 ∈ V , 𝑧 ∈ V ↦ ( ( ( 𝑥o ( 𝑘 ) ) ·o ( 𝑓 ‘ ( 𝑘 ) ) ) +o 𝑧 ) ) , ∅ ) ‘ dom ) ) )