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Theorem cnfcom3cOLD 8189
Description: Wrap the construction of cnfcom3OLD 8187 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 8181 as of 4-Jul-2019. (New usage is discouraged.) (Proof modification 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 2402 . 2  |-  dom  ( om CNF  A )  =  dom  ( om CNF  A )
2 eqid 2402 . 2  |-  ( `' ( om CNF  A ) `  b )  =  ( `' ( om CNF  A
) `  b )
3 eqid 2402 . 2  |- OrdIso (  _E  ,  ( `' ( `' ( om CNF  A
) `  b ) " ( _V  \  1o ) ) )  = OrdIso
(  _E  ,  ( `' ( `' ( om CNF  A ) `  b ) " ( _V  \  1o ) ) )
4 eqid 2402 . 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 2402 . 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 2402 . 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 2402 . 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 2402 . 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 2402 . 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 2402 . 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 2402 . 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 2402 . 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 8188 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 1633    e. wcel 1842   A.wral 2753   E.wrex 2754   _Vcvv 3058    \ cdif 3410    u. cun 3411    C_ wss 3413   (/)c0 3737   U.cuni 4190    |-> cmpt 4452    _E cep 4731   `'ccnv 4821   dom cdm 4822   "cima 4825    o. ccom 4826   Oncon0 5409   -1-1-onto->wf1o 5567   ` cfv 5568  (class class class)co 6277    |-> cmpt2 6279   omcom 6682  seq𝜔cseqom 7148   1oc1o 7159    +o coa 7163    .o comu 7164    ^o coe 7165  OrdIsocoi 7967   CNF ccnf 8109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-inf2 8090
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-isom 5577  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-supp 6902  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-seqom 7149  df-1o 7166  df-2o 7167  df-oadd 7170  df-omul 7171  df-oexp 7172  df-er 7347  df-map 7458  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-fsupp 7863  df-oi 7968  df-cnf 8110
This theorem is referenced by:  infxpenc2OLD  8434
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