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Theorem cnfcom3cOLD 8149
Description: Wrap the construction of cnfcom3OLD 8147 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.) Obsolete version of cnfcom3c 8141 as of 4-Jul-2019. (New usage is discouraged.)
Assertion
Ref Expression
cnfcom3cOLD  |-  ( 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 cnfcom3cOLD
Dummy variables  f 
k  u  v  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2462 . 2  |-  dom  ( om CNF  A )  =  dom  ( om CNF  A )
2 eqid 2462 . 2  |-  ( `' ( om CNF  A ) `  b )  =  ( `' ( om CNF  A
) `  b )
3 eqid 2462 . 2  |- OrdIso (  _E  ,  ( `' ( `' ( om CNF  A
) `  b ) " ( _V  \  1o ) ) )  = OrdIso
(  _E  ,  ( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) )
4 eqid 2462 . 2  |- seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( om  ^o  (OrdIso (  _E  ,  ( `' ( `' ( om CNF  A
) `  b ) " ( _V  \  1o ) ) ) `  k ) )  .o  ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  ( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) )  = seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( om  ^o  (OrdIso (  _E  ,  ( `' ( `' ( om CNF  A
) `  b ) " ( _V  \  1o ) ) ) `  k ) )  .o  ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  ( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) )
5 eqid 2462 . 2  |- seq𝜔 ( ( k  e. 
_V ,  f  e. 
_V  |->  ( ( x  e.  ( ( om 
^o  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  k ) )  .o  ( ( `' ( om CNF  A
) `  b ) `  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  k ) ) )  |->  ( dom  f  +o  x ) )  u.  `' ( x  e.  dom  f  |->  ( ( ( om 
^o  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  k ) )  .o  ( ( `' ( om CNF  A
) `  b ) `  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  k ) ) )  +o  x
) ) ) ) ,  (/) )  = seq𝜔 ( ( k  e.  _V , 
f  e.  _V  |->  ( ( x  e.  ( ( om  ^o  (OrdIso (  _E  ,  ( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  k ) )  .o  ( ( `' ( om CNF  A
) `  b ) `  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  k ) ) )  |->  ( dom  f  +o  x ) )  u.  `' ( x  e.  dom  f  |->  ( ( ( om 
^o  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  k ) )  .o  ( ( `' ( om CNF  A
) `  b ) `  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  k ) ) )  +o  x
) ) ) ) ,  (/) )
6 eqid 2462 . 2  |-  ( ( om  ^o  (OrdIso (  _E  ,  ( `' ( `' ( om CNF  A
) `  b ) " ( _V  \  1o ) ) ) `  k ) )  .o  ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  ( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  k ) ) )  =  ( ( om  ^o  (OrdIso (  _E  ,  ( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  k ) )  .o  ( ( `' ( om CNF  A
) `  b ) `  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  k ) ) )
7 eqid 2462 . 2  |-  ( ( x  e.  ( ( om  ^o  (OrdIso (  _E  ,  ( `' ( `' ( om CNF  A
) `  b ) " ( _V  \  1o ) ) ) `  k ) )  .o  ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  ( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  k ) ) )  |->  ( dom  f  +o  x ) )  u.  `' ( x  e.  dom  f  |->  ( ( ( om 
^o  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  k ) )  .o  ( ( `' ( om CNF  A
) `  b ) `  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  k ) ) )  +o  x
) ) )  =  ( ( x  e.  ( ( om  ^o  (OrdIso (  _E  ,  ( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  k ) )  .o  ( ( `' ( om CNF  A
) `  b ) `  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  k ) ) )  |->  ( dom  f  +o  x ) )  u.  `' ( x  e.  dom  f  |->  ( ( ( om 
^o  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  k ) )  .o  ( ( `' ( om CNF  A
) `  b ) `  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  k ) ) )  +o  x
) ) )
8 eqid 2462 . 2  |-  (OrdIso (  _E  ,  ( `' ( `' ( om CNF  A
) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) )  =  (OrdIso (  _E  ,  ( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  ,  ( `' ( `' ( om CNF 
A ) `  b
) " ( _V 
\  1o ) ) ) )
9 eqid 2462 . 2  |-  ( u  e.  ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  ( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  ,  ( `' ( `' ( om CNF 
A ) `  b
) " ( _V 
\  1o ) ) ) ) ) ,  v  e.  ( om 
^o  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  ,  ( `' ( `' ( om CNF 
A ) `  b
) " ( _V 
\  1o ) ) ) ) )  |->  ( ( ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  ( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  ,  ( `' ( `' ( om CNF 
A ) `  b
) " ( _V 
\  1o ) ) ) ) )  .o  v )  +o  u
) )  =  ( u  e.  ( ( `' ( om CNF  A
) `  b ) `  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  ,  ( `' ( `' ( om CNF 
A ) `  b
) " ( _V 
\  1o ) ) ) ) ) ,  v  e.  ( om 
^o  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  ,  ( `' ( `' ( om CNF 
A ) `  b
) " ( _V 
\  1o ) ) ) ) )  |->  ( ( ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  ( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  ,  ( `' ( `' ( om CNF 
A ) `  b
) " ( _V 
\  1o ) ) ) ) )  .o  v )  +o  u
) )
10 eqid 2462 . 2  |-  ( u  e.  ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  ( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  ,  ( `' ( `' ( om CNF 
A ) `  b
) " ( _V 
\  1o ) ) ) ) ) ,  v  e.  ( om 
^o  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  ,  ( `' ( `' ( om CNF 
A ) `  b
) " ( _V 
\  1o ) ) ) ) )  |->  ( ( ( om  ^o  (OrdIso (  _E  ,  ( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  ,  ( `' ( `' ( om CNF 
A ) `  b
) " ( _V 
\  1o ) ) ) ) )  .o  u )  +o  v
) )  =  ( u  e.  ( ( `' ( om CNF  A
) `  b ) `  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  ,  ( `' ( `' ( om CNF 
A ) `  b
) " ( _V 
\  1o ) ) ) ) ) ,  v  e.  ( om 
^o  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  ,  ( `' ( `' ( om CNF 
A ) `  b
) " ( _V 
\  1o ) ) ) ) )  |->  ( ( ( om  ^o  (OrdIso (  _E  ,  ( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  ,  ( `' ( `' ( om CNF 
A ) `  b
) " ( _V 
\  1o ) ) ) ) )  .o  u )  +o  v
) )
11 eqid 2462 . 2  |-  ( ( ( u  e.  ( ( `' ( om CNF 
A ) `  b
) `  (OrdIso (  _E  ,  ( `' ( `' ( om CNF  A
) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) ) ) ,  v  e.  ( om 
^o  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  ,  ( `' ( `' ( om CNF 
A ) `  b
) " ( _V 
\  1o ) ) ) ) )  |->  ( ( ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  ( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  ,  ( `' ( `' ( om CNF 
A ) `  b
) " ( _V 
\  1o ) ) ) ) )  .o  v )  +o  u
) )  o.  `' ( u  e.  (
( `' ( om CNF 
A ) `  b
) `  (OrdIso (  _E  ,  ( `' ( `' ( om CNF  A
) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) ) ) ,  v  e.  ( om 
^o  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  ,  ( `' ( `' ( om CNF 
A ) `  b
) " ( _V 
\  1o ) ) ) ) )  |->  ( ( ( om  ^o  (OrdIso (  _E  ,  ( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  ,  ( `' ( `' ( om CNF 
A ) `  b
) " ( _V 
\  1o ) ) ) ) )  .o  u )  +o  v
) ) )  o.  (seq𝜔 ( ( k  e. 
_V ,  f  e. 
_V  |->  ( ( x  e.  ( ( om 
^o  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  k ) )  .o  ( ( `' ( om CNF  A
) `  b ) `  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  k ) ) )  |->  ( dom  f  +o  x ) )  u.  `' ( x  e.  dom  f  |->  ( ( ( om 
^o  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  k ) )  .o  ( ( `' ( om CNF  A
) `  b ) `  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  k ) ) )  +o  x
) ) ) ) ,  (/) ) `  dom OrdIso (  _E  ,  ( `' ( `' ( om CNF 
A ) `  b
) " ( _V 
\  1o ) ) ) ) )  =  ( ( ( u  e.  ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  ( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  ,  ( `' ( `' ( om CNF 
A ) `  b
) " ( _V 
\  1o ) ) ) ) ) ,  v  e.  ( om 
^o  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  ,  ( `' ( `' ( om CNF 
A ) `  b
) " ( _V 
\  1o ) ) ) ) )  |->  ( ( ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  ( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  ,  ( `' ( `' ( om CNF 
A ) `  b
) " ( _V 
\  1o ) ) ) ) )  .o  v )  +o  u
) )  o.  `' ( u  e.  (
( `' ( om CNF 
A ) `  b
) `  (OrdIso (  _E  ,  ( `' ( `' ( om CNF  A
) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) ) ) ,  v  e.  ( om 
^o  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  ,  ( `' ( `' ( om CNF 
A ) `  b
) " ( _V 
\  1o ) ) ) ) )  |->  ( ( ( om  ^o  (OrdIso (  _E  ,  ( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  ,  ( `' ( `' ( om CNF 
A ) `  b
) " ( _V 
\  1o ) ) ) ) )  .o  u )  +o  v
) ) )  o.  (seq𝜔 ( ( k  e. 
_V ,  f  e. 
_V  |->  ( ( x  e.  ( ( om 
^o  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  k ) )  .o  ( ( `' ( om CNF  A
) `  b ) `  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  k ) ) )  |->  ( dom  f  +o  x ) )  u.  `' ( x  e.  dom  f  |->  ( ( ( om 
^o  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  k ) )  .o  ( ( `' ( om CNF  A
) `  b ) `  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  k ) ) )  +o  x
) ) ) ) ,  (/) ) `  dom OrdIso (  _E  ,  ( `' ( `' ( om CNF 
A ) `  b
) " ( _V 
\  1o ) ) ) ) )
12 eqid 2462 . 2  |-  ( b  e.  ( om  ^o  A )  |->  ( ( ( u  e.  ( ( `' ( om CNF 
A ) `  b
) `  (OrdIso (  _E  ,  ( `' ( `' ( om CNF  A
) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) ) ) ,  v  e.  ( om 
^o  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  ,  ( `' ( `' ( om CNF 
A ) `  b
) " ( _V 
\  1o ) ) ) ) )  |->  ( ( ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  ( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  ,  ( `' ( `' ( om CNF 
A ) `  b
) " ( _V 
\  1o ) ) ) ) )  .o  v )  +o  u
) )  o.  `' ( u  e.  (
( `' ( om CNF 
A ) `  b
) `  (OrdIso (  _E  ,  ( `' ( `' ( om CNF  A
) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) ) ) ,  v  e.  ( om 
^o  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  ,  ( `' ( `' ( om CNF 
A ) `  b
) " ( _V 
\  1o ) ) ) ) )  |->  ( ( ( om  ^o  (OrdIso (  _E  ,  ( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  ,  ( `' ( `' ( om CNF 
A ) `  b
) " ( _V 
\  1o ) ) ) ) )  .o  u )  +o  v
) ) )  o.  (seq𝜔 ( ( k  e. 
_V ,  f  e. 
_V  |->  ( ( x  e.  ( ( om 
^o  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  k ) )  .o  ( ( `' ( om CNF  A
) `  b ) `  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  k ) ) )  |->  ( dom  f  +o  x ) )  u.  `' ( x  e.  dom  f  |->  ( ( ( om 
^o  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  k ) )  .o  ( ( `' ( om CNF  A
) `  b ) `  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  k ) ) )  +o  x
) ) ) ) ,  (/) ) `  dom OrdIso (  _E  ,  ( `' ( `' ( om CNF 
A ) `  b
) " ( _V 
\  1o ) ) ) ) ) )  =  ( b  e.  ( om  ^o  A
)  |->  ( ( ( u  e.  ( ( `' ( om CNF  A
) `  b ) `  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  ,  ( `' ( `' ( om CNF 
A ) `  b
) " ( _V 
\  1o ) ) ) ) ) ,  v  e.  ( om 
^o  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  ,  ( `' ( `' ( om CNF 
A ) `  b
) " ( _V 
\  1o ) ) ) ) )  |->  ( ( ( ( `' ( om CNF  A ) `  b ) `  (OrdIso (  _E  ,  ( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  ,  ( `' ( `' ( om CNF 
A ) `  b
) " ( _V 
\  1o ) ) ) ) )  .o  v )  +o  u
) )  o.  `' ( u  e.  (
( `' ( om CNF 
A ) `  b
) `  (OrdIso (  _E  ,  ( `' ( `' ( om CNF  A
) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) ) ) ,  v  e.  ( om 
^o  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  ,  ( `' ( `' ( om CNF 
A ) `  b
) " ( _V 
\  1o ) ) ) ) )  |->  ( ( ( om  ^o  (OrdIso (  _E  ,  ( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  U. dom OrdIso (  _E  ,  ( `' ( `' ( om CNF 
A ) `  b
) " ( _V 
\  1o ) ) ) ) )  .o  u )  +o  v
) ) )  o.  (seq𝜔 ( ( k  e. 
_V ,  f  e. 
_V  |->  ( ( x  e.  ( ( om 
^o  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  k ) )  .o  ( ( `' ( om CNF  A
) `  b ) `  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  k ) ) )  |->  ( dom  f  +o  x ) )  u.  `' ( x  e.  dom  f  |->  ( ( ( om 
^o  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  k ) )  .o  ( ( `' ( om CNF  A
) `  b ) `  (OrdIso (  _E  , 
( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) ) `  k ) ) )  +o  x
) ) ) ) ,  (/) ) `  dom OrdIso (  _E  ,  ( `' ( `' ( om CNF 
A ) `  b
) " ( _V 
\  1o ) ) ) ) ) )
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12cnfcom3clemOLD 8148 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 1591    e. wcel 1762   A.wral 2809   E.wrex 2810   _Vcvv 3108    \ cdif 3468    u. cun 3469    C_ wss 3471   (/)c0 3780   U.cuni 4240    |-> cmpt 4500    _E cep 4784   Oncon0 4873   `'ccnv 4993   dom cdm 4994   "cima 4997    o. ccom 4998   -1-1-onto->wf1o 5580   ` cfv 5581  (class class class)co 6277    |-> cmpt2 6279   omcom 6673  seq𝜔cseqom 7104   1oc1o 7115    +o coa 7119    .o comu 7120    ^o coe 7121  OrdIsocoi 7925   CNF ccnf 8069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-se 4834  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-supp 6894  df-recs 7034  df-rdg 7068  df-seqom 7105  df-1o 7122  df-2o 7123  df-oadd 7126  df-omul 7127  df-oexp 7128  df-er 7303  df-map 7414  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-fsupp 7821  df-oi 7926  df-cnf 8070
This theorem is referenced by:  infxpenc2OLD  8394
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