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Theorem cantnffval2 7653
Description: An alternative definition of df-cnf 7619 which relies on cantnf 7651. (Note that although the use of  S seems self-referential, one can use cantnfdm 7621 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 7651 . . . 4  |-  ( ph  ->  ( A CNF  B ) 
Isom  T ,  _E  ( S ,  ( A  ^o  B ) ) )
6 isof1o 6047 . . . 4  |-  ( ( A CNF  B )  Isom  T ,  _E  ( S ,  ( A  ^o  B ) )  -> 
( A CNF  B ) : S -1-1-onto-> ( A  ^o  B
) )
7 f1orel 5679 . . . 4  |-  ( ( A CNF  B ) : S -1-1-onto-> ( A  ^o  B
)  ->  Rel  ( A CNF 
B ) )
85, 6, 73syl 19 . . 3  |-  ( ph  ->  Rel  ( A CNF  B
) )
9 dfrel2 5323 . . 3  |-  ( Rel  ( A CNF  B )  <->  `' `' ( A CNF  B
)  =  ( A CNF 
B ) )
108, 9sylib 190 . 2  |-  ( ph  ->  `' `' ( A CNF  B
)  =  ( A CNF 
B ) )
11 oecl 6783 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ^o  B
)  e.  On )
122, 3, 11syl2anc 644 . . . . . 6  |-  ( ph  ->  ( A  ^o  B
)  e.  On )
13 eloni 4593 . . . . . 6  |-  ( ( A  ^o  B )  e.  On  ->  Ord  ( A  ^o  B ) )
1412, 13syl 16 . . . . 5  |-  ( ph  ->  Ord  ( A  ^o  B ) )
15 isocnv 6052 . . . . . 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 7652 . . . . . . 7  |-  ( ph  ->  ( T  We  S  /\  dom OrdIso ( T ,  S )  =  ( A  ^o  B ) ) )
1817simpld 447 . . . . . 6  |-  ( ph  ->  T  We  S )
19 ovex 6108 . . . . . . . . 9  |-  ( A CNF 
B )  e.  _V
2019dmex 5134 . . . . . . . 8  |-  dom  ( A CNF  B )  e.  _V
211, 20eqeltri 2508 . . . . . . 7  |-  S  e. 
_V
22 exse 4548 . . . . . . 7  |-  ( S  e.  _V  ->  T Se  S )
2321, 22ax-mp 8 . . . . . 6  |-  T Se  S
24 eqid 2438 . . . . . . 7  |- OrdIso ( T ,  S )  = OrdIso
( T ,  S
)
2524oieu 7510 . . . . . 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 645 . . . . 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 889 . . . 4  |-  ( ph  ->  ( ( A  ^o  B )  =  dom OrdIso ( T ,  S )  /\  `' ( A CNF 
B )  = OrdIso ( T ,  S )
) )
2827simprd 451 . . 3  |-  ( ph  ->  `' ( A CNF  B
)  = OrdIso ( T ,  S ) )
2928cnveqd 5050 . 2  |-  ( ph  ->  `' `' ( A CNF  B
)  =  `'OrdIso ( T ,  S )
)
3010, 29eqtr3d 2472 1  |-  ( ph  ->  ( A CNF  B )  =  `'OrdIso ( T ,  S
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   _Vcvv 2958   {copab 4267    _E cep 4494   Se wse 4541    We wwe 4542   Ord word 4582   Oncon0 4583   `'ccnv 4879   dom cdm 4880   Rel wrel 4885   -1-1-onto->wf1o 5455   ` cfv 5456    Isom wiso 5457  (class class class)co 6083    ^o coe 6725  OrdIsocoi 7480   CNF ccnf 7618
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-seqom 6707  df-1o 6726  df-2o 6727  df-oadd 6730  df-omul 6731  df-oexp 6732  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-oi 7481  df-cnf 7619
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