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Theorem oemapso 7594
Description: The relation  T is a strict order on  S (a corollary of wemapso2 7477). (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
oemapso  |-  ( ph  ->  T  Or  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 oemapso
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 cantnfs.3 . . 3  |-  ( ph  ->  B  e.  On )
2 eloni 4551 . . . . . 6  |-  ( B  e.  On  ->  Ord  B )
31, 2syl 16 . . . . 5  |-  ( ph  ->  Ord  B )
4 ordwe 4554 . . . . 5  |-  ( Ord 
B  ->  _E  We  B )
5 weso 4533 . . . . 5  |-  (  _E  We  B  ->  _E  Or  B )
63, 4, 53syl 19 . . . 4  |-  ( ph  ->  _E  Or  B )
7 cnvso 5370 . . . 4  |-  (  _E  Or  B  <->  `'  _E  Or  B )
86, 7sylib 189 . . 3  |-  ( ph  ->  `'  _E  Or  B )
9 cantnfs.2 . . . . 5  |-  ( ph  ->  A  e.  On )
10 eloni 4551 . . . . 5  |-  ( A  e.  On  ->  Ord  A )
119, 10syl 16 . . . 4  |-  ( ph  ->  Ord  A )
12 ordwe 4554 . . . 4  |-  ( Ord 
A  ->  _E  We  A )
13 weso 4533 . . . 4  |-  (  _E  We  A  ->  _E  Or  A )
1411, 12, 133syl 19 . . 3  |-  ( ph  ->  _E  Or  A )
15 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
) ) ) }
16 fvex 5701 . . . . . . . . 9  |-  ( y `
 z )  e. 
_V
1716epelc 4456 . . . . . . . 8  |-  ( ( x `  z )  _E  ( y `  z )  <->  ( x `  z )  e.  ( y `  z ) )
18 vex 2919 . . . . . . . . . . . 12  |-  w  e. 
_V
19 vex 2919 . . . . . . . . . . . 12  |-  z  e. 
_V
2018, 19brcnv 5014 . . . . . . . . . . 11  |-  ( w `'  _E  z  <->  z  _E  w )
21 epel 4457 . . . . . . . . . . 11  |-  ( z  _E  w  <->  z  e.  w )
2220, 21bitri 241 . . . . . . . . . 10  |-  ( w `'  _E  z  <->  z  e.  w )
2322imbi1i 316 . . . . . . . . 9  |-  ( ( w `'  _E  z  ->  ( x `  w
)  =  ( y `
 w ) )  <-> 
( z  e.  w  ->  ( x `  w
)  =  ( y `
 w ) ) )
2423ralbii 2690 . . . . . . . 8  |-  ( A. w  e.  B  (
w `'  _E  z  ->  ( x `  w
)  =  ( y `
 w ) )  <->  A. w  e.  B  ( z  e.  w  ->  ( x `  w
)  =  ( y `
 w ) ) )
2517, 24anbi12i 679 . . . . . . 7  |-  ( ( ( x `  z
)  _E  ( y `
 z )  /\  A. w  e.  B  ( w `'  _E  z  ->  ( x `  w
)  =  ( y `
 w ) ) )  <->  ( ( x `
 z )  e.  ( y `  z
)  /\  A. w  e.  B  ( z  e.  w  ->  ( x `
 w )  =  ( y `  w
) ) ) )
2625rexbii 2691 . . . . . 6  |-  ( E. z  e.  B  ( ( x `  z
)  _E  ( y `
 z )  /\  A. w  e.  B  ( w `'  _E  z  ->  ( x `  w
)  =  ( y `
 w ) ) )  <->  E. z  e.  B  ( ( x `  z )  e.  ( y `  z )  /\  A. w  e.  B  ( z  e.  w  ->  ( x `  w )  =  ( y `  w ) ) ) )
2726opabbii 4232 . . . . 5  |-  { <. x ,  y >.  |  E. z  e.  B  (
( x `  z
)  _E  ( y `
 z )  /\  A. w  e.  B  ( w `'  _E  z  ->  ( x `  w
)  =  ( y `
 w ) ) ) }  =  { <. x ,  y >.  |  E. z  e.  B  ( ( x `  z )  e.  ( y `  z )  /\  A. w  e.  B  ( z  e.  w  ->  ( x `  w )  =  ( y `  w ) ) ) }
2815, 27eqtr4i 2427 . . . 4  |-  T  =  { <. x ,  y
>.  |  E. z  e.  B  ( (
x `  z )  _E  ( y `  z
)  /\  A. w  e.  B  ( w `'  _E  z  ->  (
x `  w )  =  ( y `  w ) ) ) }
29 cnveq 5005 . . . . . . . 8  |-  ( g  =  x  ->  `' g  =  `' x
)
3029imaeq1d 5161 . . . . . . 7  |-  ( g  =  x  ->  ( `' g " ( _V  \  1o ) )  =  ( `' x " ( _V  \  1o ) ) )
31 df1o2 6695 . . . . . . . . 9  |-  1o  =  { (/) }
3231difeq2i 3422 . . . . . . . 8  |-  ( _V 
\  1o )  =  ( _V  \  { (/)
} )
3332imaeq2i 5160 . . . . . . 7  |-  ( `' x " ( _V 
\  1o ) )  =  ( `' x " ( _V  \  { (/)
} ) )
3430, 33syl6eq 2452 . . . . . 6  |-  ( g  =  x  ->  ( `' g " ( _V  \  1o ) )  =  ( `' x " ( _V  \  { (/)
} ) ) )
3534eleq1d 2470 . . . . 5  |-  ( g  =  x  ->  (
( `' g "
( _V  \  1o ) )  e.  Fin  <->  ( `' x " ( _V 
\  { (/) } ) )  e.  Fin )
)
3635cbvrabv 2915 . . . 4  |-  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin }  =  { x  e.  ( A  ^m  B )  |  ( `' x "
( _V  \  { (/)
} ) )  e. 
Fin }
3728, 36wemapso2 7477 . . 3  |-  ( ( B  e.  On  /\  `'  _E  Or  B  /\  _E  Or  A )  ->  T  Or  { g  e.  ( A  ^m  B
)  |  ( `' g " ( _V 
\  1o ) )  e.  Fin } )
381, 8, 14, 37syl3anc 1184 . 2  |-  ( ph  ->  T  Or  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin } )
39 cantnfs.1 . . . 4  |-  S  =  dom  ( A CNF  B
)
40 eqid 2404 . . . . 5  |-  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin }  =  { g  e.  ( A  ^m  B )  |  ( `' g
" ( _V  \  1o ) )  e.  Fin }
4140, 9, 1cantnfdm 7575 . . . 4  |-  ( ph  ->  dom  ( A CNF  B
)  =  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin } )
4239, 41syl5eq 2448 . . 3  |-  ( ph  ->  S  =  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin } )
43 soeq2 4483 . . 3  |-  ( S  =  { g  e.  ( A  ^m  B
)  |  ( `' g " ( _V 
\  1o ) )  e.  Fin }  ->  ( T  Or  S  <->  T  Or  { g  e.  ( A  ^m  B )  |  ( `' g "
( _V  \  1o ) )  e.  Fin } ) )
4442, 43syl 16 . 2  |-  ( ph  ->  ( T  Or  S  <->  T  Or  { g  e.  ( A  ^m  B
)  |  ( `' g " ( _V 
\  1o ) )  e.  Fin } ) )
4538, 44mpbird 224 1  |-  ( ph  ->  T  Or  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667   {crab 2670   _Vcvv 2916    \ cdif 3277   (/)c0 3588   {csn 3774   class class class wbr 4172   {copab 4225    _E cep 4452    Or wor 4462    We wwe 4500   Ord word 4540   Oncon0 4541   `'ccnv 4836   dom cdm 4837   "cima 4840   ` cfv 5413  (class class class)co 6040   1oc1o 6676    ^m cmap 6977   Fincfn 7068   CNF ccnf 7572
This theorem is referenced by:  cantnf  7605
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-seqom 6664  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-fin 7072  df-oi 7435  df-cnf 7573
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