MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cantnffval2OLD Structured version   Unicode version

Theorem cantnffval2OLD 8148
Description: An alternative definition of df-cnf 8091 which relies on cantnfOLD 8146. (Note that although the use of  S seems self-referential, one can use cantnfdmOLD 8095 to eliminate it.) (Contributed by Mario Carneiro, 28-May-2015.) Obsolete version of cantnffval2 8126 as of 2-Jul-2019. (New usage is discouraged.)
Hypotheses
Ref Expression
cantnfsOLD.1  |-  S  =  dom  ( A CNF  B
)
cantnfsOLD.2  |-  ( ph  ->  A  e.  On )
cantnfsOLD.3  |-  ( ph  ->  B  e.  On )
oemapvalOLD.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
cantnffval2OLD  |-  ( 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 cantnffval2OLD
StepHypRef Expression
1 cantnfsOLD.1 . . . . 5  |-  S  =  dom  ( A CNF  B
)
2 cantnfsOLD.2 . . . . 5  |-  ( ph  ->  A  e.  On )
3 cantnfsOLD.3 . . . . 5  |-  ( ph  ->  B  e.  On )
4 oemapvalOLD.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, 4cantnfOLD 8146 . . . 4  |-  ( ph  ->  ( A CNF  B ) 
Isom  T ,  _E  ( S ,  ( A  ^o  B ) ) )
6 isof1o 6220 . . . 4  |-  ( ( A CNF  B )  Isom  T ,  _E  ( S ,  ( A  ^o  B ) )  -> 
( A CNF  B ) : S -1-1-onto-> ( A  ^o  B
) )
7 f1orel 5825 . . . 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 5463 . . 3  |-  ( Rel  ( A CNF  B )  <->  `' `' ( A CNF  B
)  =  ( A CNF 
B ) )
108, 9sylib 196 . 2  |-  ( ph  ->  `' `' ( A CNF  B
)  =  ( A CNF 
B ) )
11 oecl 7199 . . . . . . 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 4894 . . . . . 6  |-  ( ( A  ^o  B )  e.  On  ->  Ord  ( A  ^o  B ) )
1412, 13syl 16 . . . . 5  |-  ( ph  ->  Ord  ( A  ^o  B ) )
15 isocnv 6225 . . . . . 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 8125 . . . . . . 7  |-  ( ph  ->  ( T  We  S  /\  dom OrdIso ( T ,  S )  =  ( A  ^o  B ) ) )
1817simpld 459 . . . . . 6  |-  ( ph  ->  T  We  S )
19 ovex 6320 . . . . . . . . 9  |-  ( A CNF 
B )  e.  _V
2019dmex 6728 . . . . . . . 8  |-  dom  ( A CNF  B )  e.  _V
211, 20eqeltri 2551 . . . . . . 7  |-  S  e. 
_V
22 exse 4849 . . . . . . 7  |-  ( S  e.  _V  ->  T Se  S )
2321, 22ax-mp 5 . . . . . 6  |-  T Se  S
24 eqid 2467 . . . . . . 7  |- OrdIso ( T ,  S )  = OrdIso
( T ,  S
)
2524oieu 7976 . . . . . 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 919 . . . 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 5184 . 2  |-  ( ph  ->  `' `' ( A CNF  B
)  =  `'OrdIso ( T ,  S )
)
3010, 29eqtr3d 2510 1  |-  ( ph  ->  ( A CNF  B )  =  `'OrdIso ( T ,  S
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   E.wrex 2818   _Vcvv 3118   {copab 4510    _E cep 4795   Se wse 4842    We wwe 4843   Ord word 4883   Oncon0 4884   `'ccnv 5004   dom cdm 5005   Rel wrel 5010   -1-1-onto->wf1o 5593   ` cfv 5594    Isom wiso 5595  (class class class)co 6295    ^o coe 7141  OrdIsocoi 7946   CNF ccnf 8090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-seqom 7125  df-1o 7142  df-2o 7143  df-oadd 7146  df-omul 7147  df-oexp 7148  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-oi 7947  df-cnf 8091
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator