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Theorem cantnffval2 7413
Description: An alternative definition of df-cnf 7379 which relies on cantnf 7411. (Note that although the use of  S seems self-referential, one can use cantnfdm 7381 to eliminate it.) (Contributed by Mario Carneiro, 28-May-2015.)
Hypotheses
Ref Expression
cantnfs.1  |-  S  =  dom  ( A CNF  B
)
cantnfs.2  |-  ( ph  ->  A  e.  On )
cantnfs.3  |-  ( ph  ->  B  e.  On )
oemapval.t  |-  T  =  { <. x ,  y
>.  |  E. z  e.  B  ( (
x `  z )  e.  ( y `  z
)  /\  A. w  e.  B  ( z  e.  w  ->  ( x `
 w )  =  ( y `  w
) ) ) }
Assertion
Ref Expression
cantnffval2  |-  ( ph  ->  ( A CNF  B )  =  `'OrdIso ( T ,  S
) )
Distinct variable groups:    x, w, y, z, B    w, A, x, y, z    x, S, y, z    ph, x, y, z
Allowed substitution hints:    ph( w)    S( w)    T( x, y, z, w)

Proof of Theorem cantnffval2
StepHypRef Expression
1 cantnfs.1 . . . . 5  |-  S  =  dom  ( A CNF  B
)
2 cantnfs.2 . . . . 5  |-  ( ph  ->  A  e.  On )
3 cantnfs.3 . . . . 5  |-  ( ph  ->  B  e.  On )
4 oemapval.t . . . . 5  |-  T  =  { <. x ,  y
>.  |  E. z  e.  B  ( (
x `  z )  e.  ( y `  z
)  /\  A. w  e.  B  ( z  e.  w  ->  ( x `
 w )  =  ( y `  w
) ) ) }
51, 2, 3, 4cantnf 7411 . . . 4  |-  ( ph  ->  ( A CNF  B ) 
Isom  T ,  _E  ( S ,  ( A  ^o  B ) ) )
6 isof1o 5838 . . . 4  |-  ( ( A CNF  B )  Isom  T ,  _E  ( S ,  ( A  ^o  B ) )  -> 
( A CNF  B ) : S -1-1-onto-> ( A  ^o  B
) )
7 f1orel 5491 . . . 4  |-  ( ( A CNF  B ) : S -1-1-onto-> ( A  ^o  B
)  ->  Rel  ( A CNF 
B ) )
85, 6, 73syl 18 . . 3  |-  ( ph  ->  Rel  ( A CNF  B
) )
9 dfrel2 5140 . . 3  |-  ( Rel  ( A CNF  B )  <->  `' `' ( A CNF  B
)  =  ( A CNF 
B ) )
108, 9sylib 188 . 2  |-  ( ph  ->  `' `' ( A CNF  B
)  =  ( A CNF 
B ) )
11 oecl 6552 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ^o  B
)  e.  On )
122, 3, 11syl2anc 642 . . . . . 6  |-  ( ph  ->  ( A  ^o  B
)  e.  On )
13 eloni 4418 . . . . . 6  |-  ( ( A  ^o  B )  e.  On  ->  Ord  ( A  ^o  B ) )
1412, 13syl 15 . . . . 5  |-  ( ph  ->  Ord  ( A  ^o  B ) )
15 isocnv 5843 . . . . . 6  |-  ( ( A CNF  B )  Isom  T ,  _E  ( S ,  ( A  ^o  B ) )  ->  `' ( A CNF  B
)  Isom  _E  ,  T  ( ( A  ^o  B ) ,  S
) )
165, 15syl 15 . . . . 5  |-  ( ph  ->  `' ( A CNF  B
)  Isom  _E  ,  T  ( ( A  ^o  B ) ,  S
) )
171, 2, 3, 4oemapwe 7412 . . . . . . 7  |-  ( ph  ->  ( T  We  S  /\  dom OrdIso ( T ,  S )  =  ( A  ^o  B ) ) )
1817simpld 445 . . . . . 6  |-  ( ph  ->  T  We  S )
19 ovex 5899 . . . . . . . . 9  |-  ( A CNF 
B )  e.  _V
2019dmex 4957 . . . . . . . 8  |-  dom  ( A CNF  B )  e.  _V
211, 20eqeltri 2366 . . . . . . 7  |-  S  e. 
_V
22 exse 4373 . . . . . . 7  |-  ( S  e.  _V  ->  T Se  S )
2321, 22ax-mp 8 . . . . . 6  |-  T Se  S
24 eqid 2296 . . . . . . 7  |- OrdIso ( T ,  S )  = OrdIso
( T ,  S
)
2524oieu 7270 . . . . . 6  |-  ( ( T  We  S  /\  T Se  S )  ->  (
( Ord  ( A  ^o  B )  /\  `' ( A CNF  B )  Isom  _E  ,  T  ( ( A  ^o  B
) ,  S ) )  <->  ( ( A  ^o  B )  =  dom OrdIso ( T ,  S )  /\  `' ( A CNF  B )  = OrdIso ( T ,  S
) ) ) )
2618, 23, 25sylancl 643 . . . . 5  |-  ( ph  ->  ( ( Ord  ( A  ^o  B )  /\  `' ( A CNF  B
)  Isom  _E  ,  T  ( ( A  ^o  B ) ,  S
) )  <->  ( ( A  ^o  B )  =  dom OrdIso ( T ,  S )  /\  `' ( A CNF  B )  = OrdIso ( T ,  S
) ) ) )
2714, 16, 26mpbi2and 887 . . . 4  |-  ( ph  ->  ( ( A  ^o  B )  =  dom OrdIso ( T ,  S )  /\  `' ( A CNF 
B )  = OrdIso ( T ,  S )
) )
2827simprd 449 . . 3  |-  ( ph  ->  `' ( A CNF  B
)  = OrdIso ( T ,  S ) )
2928cnveqd 4873 . 2  |-  ( ph  ->  `' `' ( A CNF  B
)  =  `'OrdIso ( T ,  S )
)
3010, 29eqtr3d 2330 1  |-  ( ph  ->  ( A CNF  B )  =  `'OrdIso ( T ,  S
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   _Vcvv 2801   {copab 4092    _E cep 4319   Se wse 4366    We wwe 4367   Ord word 4407   Oncon0 4408   `'ccnv 4704   dom cdm 4705   Rel wrel 4710   -1-1-onto->wf1o 5270   ` cfv 5271    Isom wiso 5272  (class class class)co 5874    ^o coe 6494  OrdIsocoi 7240   CNF ccnf 7378
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
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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