MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnfcom3c Structured version   Unicode version

Theorem cnfcom3c 8150
Description: Wrap the construction of cnfcom3 8148 into an existence quantifier. For any  om  C_  b, there is a bijection from  b to some power of  om. Furthermore, this bijection is canonical , which means that we can find a single function 
g which will give such bijections for every  b less than some arbitrarily large bound  A. (Contributed by Mario Carneiro, 30-May-2015.)
Assertion
Ref Expression
cnfcom3c  |-  ( A  e.  On  ->  E. g A. b  e.  A  ( om  C_  b  ->  E. w  e.  ( On 
\  1o ) ( g `  b ) : b -1-1-onto-> ( om  ^o  w
) ) )
Distinct variable group:    g, b, w, A

Proof of Theorem cnfcom3c
Dummy variables  f 
k  u  v  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2467 . 2  |-  dom  ( om CNF  A )  =  dom  ( om CNF  A )
2 eqid 2467 . 2  |-  ( `' ( om CNF  A ) `  b )  =  ( `' ( om CNF  A
) `  b )
3 eqid 2467 . 2  |- OrdIso (  _E  ,  ( ( `' ( om CNF  A ) `  b ) supp  (/) ) )  = OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) )
4 eqid 2467 . 2  |- seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( om  ^o  (OrdIso (  _E  ,  ( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) `
 k ) )  .o  ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  k ) ) )  +o  z ) ) ,  (/) )  = seq𝜔 ( ( k  e.  _V , 
z  e.  _V  |->  ( ( ( om  ^o  (OrdIso (  _E  ,  ( ( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  k ) )  .o  ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  k ) ) )  +o  z ) ) ,  (/) )
5 eqid 2467 . 2  |- seq𝜔 ( ( k  e. 
_V ,  f  e. 
_V  |->  ( ( x  e.  ( ( om 
^o  (OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) `
 k ) )  .o  ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  k ) ) ) 
|->  ( dom  f  +o  x ) )  u.  `' ( x  e. 
dom  f  |->  ( ( ( om  ^o  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  k ) )  .o  ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  k ) ) )  +o  x ) ) ) ) ,  (/) )  = seq𝜔 ( ( k  e. 
_V ,  f  e. 
_V  |->  ( ( x  e.  ( ( om 
^o  (OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) `
 k ) )  .o  ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  k ) ) ) 
|->  ( dom  f  +o  x ) )  u.  `' ( x  e. 
dom  f  |->  ( ( ( om  ^o  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  k ) )  .o  ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  k ) ) )  +o  x ) ) ) ) ,  (/) )
6 eqid 2467 . 2  |-  ( ( om  ^o  (OrdIso (  _E  ,  ( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) `
 k ) )  .o  ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  k ) ) )  =  ( ( om 
^o  (OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) `
 k ) )  .o  ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  k ) ) )
7 eqid 2467 . 2  |-  ( ( x  e.  ( ( om  ^o  (OrdIso (  _E  ,  ( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) `
 k ) )  .o  ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  k ) ) ) 
|->  ( dom  f  +o  x ) )  u.  `' ( x  e. 
dom  f  |->  ( ( ( om  ^o  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  k ) )  .o  ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  k ) ) )  +o  x ) ) )  =  ( ( x  e.  ( ( om  ^o  (OrdIso (  _E  ,  ( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) `
 k ) )  .o  ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  k ) ) ) 
|->  ( dom  f  +o  x ) )  u.  `' ( x  e. 
dom  f  |->  ( ( ( om  ^o  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  k ) )  .o  ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  k ) ) )  +o  x ) ) )
8 eqid 2467 . 2  |-  (OrdIso (  _E  ,  ( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) `
 U. dom OrdIso (  _E  ,  ( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) )  =  (OrdIso (  _E  ,  ( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) `
 U. dom OrdIso (  _E  ,  ( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) )
9 eqid 2467 . 2  |-  ( u  e.  ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  U. dom OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) ) ,  v  e.  ( om  ^o  (OrdIso (  _E  ,  ( ( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  U. dom OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) )  |->  ( ( ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  U. dom OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) )  .o  v
)  +o  u ) )  =  ( u  e.  ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  U. dom OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) ) ,  v  e.  ( om  ^o  (OrdIso (  _E  ,  ( ( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  U. dom OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) )  |->  ( ( ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  U. dom OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) )  .o  v
)  +o  u ) )
10 eqid 2467 . 2  |-  ( u  e.  ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  U. dom OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) ) ,  v  e.  ( om  ^o  (OrdIso (  _E  ,  ( ( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  U. dom OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) )  |->  ( ( ( om  ^o  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  U. dom OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) )  .o  u
)  +o  v ) )  =  ( u  e.  ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  U. dom OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) ) ,  v  e.  ( om  ^o  (OrdIso (  _E  ,  ( ( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  U. dom OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) )  |->  ( ( ( om  ^o  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  U. dom OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) )  .o  u
)  +o  v ) )
11 eqid 2467 . 2  |-  ( ( ( u  e.  ( ( `' ( om CNF 
A ) `  b
) `  (OrdIso (  _E  ,  ( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) `
 U. dom OrdIso (  _E  ,  ( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) ) ,  v  e.  ( om  ^o  (OrdIso (  _E  ,  ( ( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  U. dom OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) )  |->  ( ( ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  U. dom OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) )  .o  v
)  +o  u ) )  o.  `' ( u  e.  ( ( `' ( om CNF  A
) `  b ) `  (OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) `
 U. dom OrdIso (  _E  ,  ( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) ) ,  v  e.  ( om  ^o  (OrdIso (  _E  ,  ( ( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  U. dom OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) )  |->  ( ( ( om  ^o  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  U. dom OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) )  .o  u
)  +o  v ) ) )  o.  (seq𝜔 (
( k  e.  _V ,  f  e.  _V  |->  ( ( x  e.  ( ( om  ^o  (OrdIso (  _E  ,  ( ( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  k ) )  .o  ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  k ) ) ) 
|->  ( dom  f  +o  x ) )  u.  `' ( x  e. 
dom  f  |->  ( ( ( om  ^o  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  k ) )  .o  ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  k ) ) )  +o  x ) ) ) ) ,  (/) ) `  dom OrdIso (  _E  ,  ( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) )  =  ( ( ( u  e.  ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  U. dom OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) ) ,  v  e.  ( om  ^o  (OrdIso (  _E  ,  ( ( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  U. dom OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) )  |->  ( ( ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  U. dom OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) )  .o  v
)  +o  u ) )  o.  `' ( u  e.  ( ( `' ( om CNF  A
) `  b ) `  (OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) `
 U. dom OrdIso (  _E  ,  ( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) ) ,  v  e.  ( om  ^o  (OrdIso (  _E  ,  ( ( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  U. dom OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) )  |->  ( ( ( om  ^o  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  U. dom OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) )  .o  u
)  +o  v ) ) )  o.  (seq𝜔 (
( k  e.  _V ,  f  e.  _V  |->  ( ( x  e.  ( ( om  ^o  (OrdIso (  _E  ,  ( ( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  k ) )  .o  ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  k ) ) ) 
|->  ( dom  f  +o  x ) )  u.  `' ( x  e. 
dom  f  |->  ( ( ( om  ^o  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  k ) )  .o  ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  k ) ) )  +o  x ) ) ) ) ,  (/) ) `  dom OrdIso (  _E  ,  ( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) )
12 eqid 2467 . 2  |-  ( b  e.  ( om  ^o  A )  |->  ( ( ( u  e.  ( ( `' ( om CNF 
A ) `  b
) `  (OrdIso (  _E  ,  ( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) `
 U. dom OrdIso (  _E  ,  ( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) ) ,  v  e.  ( om  ^o  (OrdIso (  _E  ,  ( ( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  U. dom OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) )  |->  ( ( ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  U. dom OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) )  .o  v
)  +o  u ) )  o.  `' ( u  e.  ( ( `' ( om CNF  A
) `  b ) `  (OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) `
 U. dom OrdIso (  _E  ,  ( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) ) ,  v  e.  ( om  ^o  (OrdIso (  _E  ,  ( ( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  U. dom OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) )  |->  ( ( ( om  ^o  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  U. dom OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) )  .o  u
)  +o  v ) ) )  o.  (seq𝜔 (
( k  e.  _V ,  f  e.  _V  |->  ( ( x  e.  ( ( om  ^o  (OrdIso (  _E  ,  ( ( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  k ) )  .o  ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  k ) ) ) 
|->  ( dom  f  +o  x ) )  u.  `' ( x  e. 
dom  f  |->  ( ( ( om  ^o  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  k ) )  .o  ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  k ) ) )  +o  x ) ) ) ) ,  (/) ) `  dom OrdIso (  _E  ,  ( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) ) )  =  ( b  e.  ( om  ^o  A ) 
|->  ( ( ( u  e.  ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  U. dom OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) ) ,  v  e.  ( om  ^o  (OrdIso (  _E  ,  ( ( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  U. dom OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) )  |->  ( ( ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  U. dom OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) )  .o  v
)  +o  u ) )  o.  `' ( u  e.  ( ( `' ( om CNF  A
) `  b ) `  (OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) `
 U. dom OrdIso (  _E  ,  ( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) ) ,  v  e.  ( om  ^o  (OrdIso (  _E  ,  ( ( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  U. dom OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) )  |->  ( ( ( om  ^o  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  U. dom OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) )  .o  u
)  +o  v ) ) )  o.  (seq𝜔 (
( k  e.  _V ,  f  e.  _V  |->  ( ( x  e.  ( ( om  ^o  (OrdIso (  _E  ,  ( ( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  k ) )  .o  ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  k ) ) ) 
|->  ( dom  f  +o  x ) )  u.  `' ( x  e. 
dom  f  |->  ( ( ( om  ^o  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  k ) )  .o  ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  (
( `' ( om CNF 
A ) `  b
) supp  (/) ) ) `  k ) ) )  +o  x ) ) ) ) ,  (/) ) `  dom OrdIso (  _E  ,  ( ( `' ( om CNF  A ) `  b ) supp  (/) ) ) ) ) )
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12cnfcom3clem 8149 1  |-  ( A  e.  On  ->  E. g A. b  e.  A  ( om  C_  b  ->  E. w  e.  ( On 
\  1o ) ( g `  b ) : b -1-1-onto-> ( om  ^o  w
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   E.wex 1596    e. wcel 1767   A.wral 2814   E.wrex 2815   _Vcvv 3113    \ cdif 3473    u. cun 3474    C_ wss 3476   (/)c0 3785   U.cuni 4245    |-> cmpt 4505    _E cep 4789   Oncon0 4878   `'ccnv 4998   dom cdm 4999    o. ccom 5003   -1-1-onto->wf1o 5587   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286   omcom 6684   supp csupp 6901  seq𝜔cseqom 7112   1oc1o 7123    +o coa 7127    .o comu 7128    ^o coe 7129  OrdIsocoi 7934   CNF ccnf 8078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-seqom 7113  df-1o 7130  df-2o 7131  df-oadd 7134  df-omul 7135  df-oexp 7136  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-oi 7935  df-cnf 8079
This theorem is referenced by:  infxpenc2  8399
  Copyright terms: Public domain W3C validator