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Theorem cnfcom3c 8210
Description: Wrap the construction of cnfcom3 8208 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 2429 . 2  |-  dom  ( om CNF  A )  =  dom  ( om CNF  A )
2 eqid 2429 . 2  |-  ( `' ( om CNF  A ) `  b )  =  ( `' ( om CNF  A
) `  b )
3 eqid 2429 . 2  |- OrdIso (  _E  ,  ( ( `' ( om CNF  A ) `  b ) supp  (/) ) )  = OrdIso (  _E  , 
( ( `' ( om CNF  A ) `  b ) supp  (/) ) )
4 eqid 2429 . 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 2429 . 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 2429 . 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 2429 . 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 2429 . 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 2429 . 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 2429 . 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 2429 . 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 2429 . 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 8209 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 1659    e. wcel 1870   A.wral 2782   E.wrex 2783   _Vcvv 3087    \ cdif 3439    u. cun 3440    C_ wss 3442   (/)c0 3767   U.cuni 4222    |-> cmpt 4484    _E cep 4763   `'ccnv 4853   dom cdm 4854    o. ccom 4858   Oncon0 5442   -1-1-onto->wf1o 5600   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307   omcom 6706   supp csupp 6925  seq𝜔cseqom 7172   1oc1o 7183    +o coa 7187    .o comu 7188    ^o coe 7189  OrdIsocoi 8024   CNF ccnf 8165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-seqom 7173  df-1o 7190  df-2o 7191  df-oadd 7194  df-omul 7195  df-oexp 7196  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-oi 8025  df-cnf 8166
This theorem is referenced by:  infxpenc2  8451
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