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Theorem cantnffval2 8006
Description: An alternative definition of df-cnf 7971 which relies on cantnf 8004. (Note that although the use of  S seems self-referential, one can use cantnfdm 7973 to eliminate it.) (Contributed by Mario Carneiro, 28-May-2015.)
Hypotheses
Ref Expression
cantnfs.s  |-  S  =  dom  ( A CNF  B
)
cantnfs.a  |-  ( ph  ->  A  e.  On )
cantnfs.b  |-  ( 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.s . . . . 5  |-  S  =  dom  ( A CNF  B
)
2 cantnfs.a . . . . 5  |-  ( ph  ->  A  e.  On )
3 cantnfs.b . . . . 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 8004 . . . 4  |-  ( ph  ->  ( A CNF  B ) 
Isom  T ,  _E  ( S ,  ( A  ^o  B ) ) )
6 isof1o 6117 . . . 4  |-  ( ( A CNF  B )  Isom  T ,  _E  ( S ,  ( A  ^o  B ) )  -> 
( A CNF  B ) : S -1-1-onto-> ( A  ^o  B
) )
7 f1orel 5744 . . . 4  |-  ( ( A CNF  B ) : S -1-1-onto-> ( A  ^o  B
)  ->  Rel  ( A CNF 
B ) )
85, 6, 73syl 20 . . 3  |-  ( ph  ->  Rel  ( A CNF  B
) )
9 dfrel2 5388 . . 3  |-  ( Rel  ( A CNF  B )  <->  `' `' ( A CNF  B
)  =  ( A CNF 
B ) )
108, 9sylib 196 . 2  |-  ( ph  ->  `' `' ( A CNF  B
)  =  ( A CNF 
B ) )
11 oecl 7079 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ^o  B
)  e.  On )
122, 3, 11syl2anc 661 . . . . . 6  |-  ( ph  ->  ( A  ^o  B
)  e.  On )
13 eloni 4829 . . . . . 6  |-  ( ( A  ^o  B )  e.  On  ->  Ord  ( A  ^o  B ) )
1412, 13syl 16 . . . . 5  |-  ( ph  ->  Ord  ( A  ^o  B ) )
15 isocnv 6122 . . . . . 6  |-  ( ( A CNF  B )  Isom  T ,  _E  ( S ,  ( A  ^o  B ) )  ->  `' ( A CNF  B
)  Isom  _E  ,  T  ( ( A  ^o  B ) ,  S
) )
165, 15syl 16 . . . . 5  |-  ( ph  ->  `' ( A CNF  B
)  Isom  _E  ,  T  ( ( A  ^o  B ) ,  S
) )
171, 2, 3, 4oemapwe 8005 . . . . . . 7  |-  ( ph  ->  ( T  We  S  /\  dom OrdIso ( T ,  S )  =  ( A  ^o  B ) ) )
1817simpld 459 . . . . . 6  |-  ( ph  ->  T  We  S )
19 ovex 6217 . . . . . . . . 9  |-  ( A CNF 
B )  e.  _V
2019dmex 6613 . . . . . . . 8  |-  dom  ( A CNF  B )  e.  _V
211, 20eqeltri 2535 . . . . . . 7  |-  S  e. 
_V
22 exse 4784 . . . . . . 7  |-  ( S  e.  _V  ->  T Se  S )
2321, 22ax-mp 5 . . . . . 6  |-  T Se  S
24 eqid 2451 . . . . . . 7  |- OrdIso ( T ,  S )  = OrdIso
( T ,  S
)
2524oieu 7856 . . . . . 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 662 . . . . 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 912 . . . 4  |-  ( ph  ->  ( ( A  ^o  B )  =  dom OrdIso ( T ,  S )  /\  `' ( A CNF 
B )  = OrdIso ( T ,  S )
) )
2827simprd 463 . . 3  |-  ( ph  ->  `' ( A CNF  B
)  = OrdIso ( T ,  S ) )
2928cnveqd 5115 . 2  |-  ( ph  ->  `' `' ( A CNF  B
)  =  `'OrdIso ( T ,  S )
)
3010, 29eqtr3d 2494 1  |-  ( ph  ->  ( A CNF  B )  =  `'OrdIso ( T ,  S
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795   E.wrex 2796   _Vcvv 3070   {copab 4449    _E cep 4730   Se wse 4777    We wwe 4778   Ord word 4818   Oncon0 4819   `'ccnv 4939   dom cdm 4940   Rel wrel 4945   -1-1-onto->wf1o 5517   ` cfv 5518    Isom wiso 5519  (class class class)co 6192    ^o coe 7021  OrdIsocoi 7826   CNF ccnf 7970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-isom 5527  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-supp 6793  df-recs 6934  df-rdg 6968  df-seqom 7005  df-1o 7022  df-2o 7023  df-oadd 7026  df-omul 7027  df-oexp 7028  df-er 7203  df-map 7318  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-fsupp 7724  df-oi 7827  df-cnf 7971
This theorem is referenced by: (None)
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