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Theorem oemapso 7638
Description: The relation  T is a strict order on  S (a corollary of wemapso2 7521). (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 4591 . . . . . 6  |-  ( B  e.  On  ->  Ord  B )
31, 2syl 16 . . . . 5  |-  ( ph  ->  Ord  B )
4 ordwe 4594 . . . . 5  |-  ( Ord 
B  ->  _E  We  B )
5 weso 4573 . . . . 5  |-  (  _E  We  B  ->  _E  Or  B )
63, 4, 53syl 19 . . . 4  |-  ( ph  ->  _E  Or  B )
7 cnvso 5411 . . . 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 4591 . . . . 5  |-  ( A  e.  On  ->  Ord  A )
119, 10syl 16 . . . 4  |-  ( ph  ->  Ord  A )
12 ordwe 4594 . . . 4  |-  ( Ord 
A  ->  _E  We  A )
13 weso 4573 . . . 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 5742 . . . . . . . . 9  |-  ( y `
 z )  e. 
_V
1716epelc 4496 . . . . . . . 8  |-  ( ( x `  z )  _E  ( y `  z )  <->  ( x `  z )  e.  ( y `  z ) )
18 vex 2959 . . . . . . . . . . . 12  |-  w  e. 
_V
19 vex 2959 . . . . . . . . . . . 12  |-  z  e. 
_V
2018, 19brcnv 5055 . . . . . . . . . . 11  |-  ( w `'  _E  z  <->  z  _E  w )
21 epel 4497 . . . . . . . . . . 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 2729 . . . . . . . 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 2730 . . . . . 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 4272 . . . . 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 2459 . . . 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 5046 . . . . . . . 8  |-  ( g  =  x  ->  `' g  =  `' x
)
3029imaeq1d 5202 . . . . . . 7  |-  ( g  =  x  ->  ( `' g " ( _V  \  1o ) )  =  ( `' x " ( _V  \  1o ) ) )
31 df1o2 6736 . . . . . . . . 9  |-  1o  =  { (/) }
3231difeq2i 3462 . . . . . . . 8  |-  ( _V 
\  1o )  =  ( _V  \  { (/)
} )
3332imaeq2i 5201 . . . . . . 7  |-  ( `' x " ( _V 
\  1o ) )  =  ( `' x " ( _V  \  { (/)
} ) )
3430, 33syl6eq 2484 . . . . . 6  |-  ( g  =  x  ->  ( `' g " ( _V  \  1o ) )  =  ( `' x " ( _V  \  { (/)
} ) ) )
3534eleq1d 2502 . . . . 5  |-  ( g  =  x  ->  (
( `' g "
( _V  \  1o ) )  e.  Fin  <->  ( `' x " ( _V 
\  { (/) } ) )  e.  Fin )
)
3635cbvrabv 2955 . . . 4  |-  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin }  =  { x  e.  ( A  ^m  B )  |  ( `' x "
( _V  \  { (/)
} ) )  e. 
Fin }
3728, 36wemapso2 7521 . . 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 2436 . . . . 5  |-  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin }  =  { g  e.  ( A  ^m  B )  |  ( `' g
" ( _V  \  1o ) )  e.  Fin }
4140, 9, 1cantnfdm 7619 . . . 4  |-  ( ph  ->  dom  ( A CNF  B
)  =  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin } )
4239, 41syl5eq 2480 . . 3  |-  ( ph  ->  S  =  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin } )
43 soeq2 4523 . . 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 1652    e. wcel 1725   A.wral 2705   E.wrex 2706   {crab 2709   _Vcvv 2956    \ cdif 3317   (/)c0 3628   {csn 3814   class class class wbr 4212   {copab 4265    _E cep 4492    Or wor 4502    We wwe 4540   Ord word 4580   Oncon0 4581   `'ccnv 4877   dom cdm 4878   "cima 4881   ` cfv 5454  (class class class)co 6081   1oc1o 6717    ^m cmap 7018   Fincfn 7109   CNF ccnf 7616
This theorem is referenced by:  cantnf  7649
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-seqom 6705  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-fin 7113  df-oi 7479  df-cnf 7617
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