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Theorem cnfcom3c 7409
Description: Wrap the construction of cnfcom3 7407 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 2283 . 2  |-  dom  ( om CNF  A )  =  dom  ( om CNF  A )
2 eqid 2283 . 2  |-  ( `' ( om CNF  A ) `  b )  =  ( `' ( om CNF  A
) `  b )
3 eqid 2283 . 2  |- OrdIso (  _E  ,  ( `' ( `' ( om CNF  A
) `  b ) " ( _V  \  1o ) ) )  = OrdIso
(  _E  ,  ( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) )
4 eqid 2283 . 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 2283 . 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 2283 . 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 2283 . 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 2283 . 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 2283 . 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 2283 . 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 2283 . 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 2283 . 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 7408 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 1528    e. wcel 1684   A.wral 2543   E.wrex 2544   _Vcvv 2788    \ cdif 3149    u. cun 3150    C_ wss 3152   (/)c0 3455   U.cuni 3827    e. cmpt 4077    _E cep 4303   Oncon0 4392   omcom 4656   `'ccnv 4688   dom cdm 4689   "cima 4692    o. ccom 4693   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860  seq𝜔cseqom 6459   1oc1o 6472    +o coa 6476    .o comu 6477    ^o coe 6478  OrdIsocoi 7224   CNF ccnf 7362
This theorem is referenced by:  infxpenc2  7649
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-seqom 6460  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-oexp 6485  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-cnf 7363
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