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Theorem oemapso 7384
Description: The relation  T is a strict order on  S (a corollary of wemapso2 7267). (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 4402 . . . . . 6  |-  ( B  e.  On  ->  Ord  B )
31, 2syl 15 . . . . 5  |-  ( ph  ->  Ord  B )
4 ordwe 4405 . . . . 5  |-  ( Ord 
B  ->  _E  We  B )
5 weso 4384 . . . . 5  |-  (  _E  We  B  ->  _E  Or  B )
63, 4, 53syl 18 . . . 4  |-  ( ph  ->  _E  Or  B )
7 cnvso 5214 . . . 4  |-  (  _E  Or  B  <->  `'  _E  Or  B )
86, 7sylib 188 . . 3  |-  ( ph  ->  `'  _E  Or  B )
9 cantnfs.2 . . . . 5  |-  ( ph  ->  A  e.  On )
10 eloni 4402 . . . . 5  |-  ( A  e.  On  ->  Ord  A )
119, 10syl 15 . . . 4  |-  ( ph  ->  Ord  A )
12 ordwe 4405 . . . 4  |-  ( Ord 
A  ->  _E  We  A )
13 weso 4384 . . . 4  |-  (  _E  We  A  ->  _E  Or  A )
1411, 12, 133syl 18 . . 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 5539 . . . . . . . . 9  |-  ( y `
 z )  e. 
_V
1716epelc 4307 . . . . . . . 8  |-  ( ( x `  z )  _E  ( y `  z )  <->  ( x `  z )  e.  ( y `  z ) )
18 vex 2791 . . . . . . . . . . . 12  |-  w  e. 
_V
19 vex 2791 . . . . . . . . . . . 12  |-  z  e. 
_V
2018, 19brcnv 4864 . . . . . . . . . . 11  |-  ( w `'  _E  z  <->  z  _E  w )
21 epel 4308 . . . . . . . . . . 11  |-  ( z  _E  w  <->  z  e.  w )
2220, 21bitri 240 . . . . . . . . . 10  |-  ( w `'  _E  z  <->  z  e.  w )
2322imbi1i 315 . . . . . . . . 9  |-  ( ( w `'  _E  z  ->  ( x `  w
)  =  ( y `
 w ) )  <-> 
( z  e.  w  ->  ( x `  w
)  =  ( y `
 w ) ) )
2423ralbii 2567 . . . . . . . 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 678 . . . . . . 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 2568 . . . . . 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 4083 . . . . 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 2306 . . . 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 4855 . . . . . . . 8  |-  ( g  =  x  ->  `' g  =  `' x
)
3029imaeq1d 5011 . . . . . . 7  |-  ( g  =  x  ->  ( `' g " ( _V  \  1o ) )  =  ( `' x " ( _V  \  1o ) ) )
31 df1o2 6491 . . . . . . . . 9  |-  1o  =  { (/) }
3231difeq2i 3291 . . . . . . . 8  |-  ( _V 
\  1o )  =  ( _V  \  { (/)
} )
3332imaeq2i 5010 . . . . . . 7  |-  ( `' x " ( _V 
\  1o ) )  =  ( `' x " ( _V  \  { (/)
} ) )
3430, 33syl6eq 2331 . . . . . 6  |-  ( g  =  x  ->  ( `' g " ( _V  \  1o ) )  =  ( `' x " ( _V  \  { (/)
} ) ) )
3534eleq1d 2349 . . . . 5  |-  ( g  =  x  ->  (
( `' g "
( _V  \  1o ) )  e.  Fin  <->  ( `' x " ( _V 
\  { (/) } ) )  e.  Fin )
)
3635cbvrabv 2787 . . . 4  |-  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin }  =  { x  e.  ( A  ^m  B )  |  ( `' x "
( _V  \  { (/)
} ) )  e. 
Fin }
3728, 36wemapso2 7267 . . 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 1182 . 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 2283 . . . . 5  |-  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin }  =  { g  e.  ( A  ^m  B )  |  ( `' g
" ( _V  \  1o ) )  e.  Fin }
4140, 9, 1cantnfdm 7365 . . . 4  |-  ( ph  ->  dom  ( A CNF  B
)  =  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin } )
4239, 41syl5eq 2327 . . 3  |-  ( ph  ->  S  =  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin } )
43 soeq2 4334 . . 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 15 . 2  |-  ( ph  ->  ( T  Or  S  <->  T  Or  { g  e.  ( A  ^m  B
)  |  ( `' g " ( _V 
\  1o ) )  e.  Fin } ) )
4538, 44mpbird 223 1  |-  ( ph  ->  T  Or  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788    \ cdif 3149   (/)c0 3455   {csn 3640   class class class wbr 4023   {copab 4076    _E cep 4303    Or wor 4313    We wwe 4351   Ord word 4391   Oncon0 4392   `'ccnv 4688   dom cdm 4689   "cima 4692   ` cfv 5255  (class class class)co 5858   1oc1o 6472    ^m cmap 6772   Fincfn 6863   CNF ccnf 7362
This theorem is referenced by:  cantnf  7395
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-seqom 6460  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-fin 6867  df-oi 7225  df-cnf 7363
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