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Theorem cnfcom3c 7425
Description: Wrap the construction of cnfcom3 7423 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 2296 . 2  |-  dom  ( om CNF  A )  =  dom  ( om CNF  A )
2 eqid 2296 . 2  |-  ( `' ( om CNF  A ) `  b )  =  ( `' ( om CNF  A
) `  b )
3 eqid 2296 . 2  |- OrdIso (  _E  ,  ( `' ( `' ( om CNF  A
) `  b ) " ( _V  \  1o ) ) )  = OrdIso
(  _E  ,  ( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) )
4 eqid 2296 . 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 2296 . 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 2296 . 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 2296 . 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 2296 . 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 2296 . 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 2296 . 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 2296 . 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 2296 . 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, 12cnfcom3clem 7424 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 set class
Syntax hints:    -> wi 4   E.wex 1531    e. wcel 1696   A.wral 2556   E.wrex 2557   _Vcvv 2801    \ cdif 3162    u. cun 3163    C_ wss 3165   (/)c0 3468   U.cuni 3843    e. cmpt 4093    _E cep 4319   Oncon0 4408   omcom 4672   `'ccnv 4704   dom cdm 4705   "cima 4708    o. ccom 4709   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876  seq𝜔cseqom 6475   1oc1o 6488    +o coa 6492    .o comu 6493    ^o coe 6494  OrdIsocoi 7240   CNF ccnf 7378
This theorem is referenced by:  infxpenc2  7665
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-seqom 6476  df-1o 6495  df-2o 6496  df-oadd 6499  df-omul 6500  df-oexp 6501  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-cnf 7379
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