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Theorem oemapso 7429
Description: The relation  T is a strict order on  S (a corollary of wemapso2 7312). (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 4439 . . . . . 6  |-  ( B  e.  On  ->  Ord  B )
31, 2syl 15 . . . . 5  |-  ( ph  ->  Ord  B )
4 ordwe 4442 . . . . 5  |-  ( Ord 
B  ->  _E  We  B )
5 weso 4421 . . . . 5  |-  (  _E  We  B  ->  _E  Or  B )
63, 4, 53syl 18 . . . 4  |-  ( ph  ->  _E  Or  B )
7 cnvso 5251 . . . 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 4439 . . . . 5  |-  ( A  e.  On  ->  Ord  A )
119, 10syl 15 . . . 4  |-  ( ph  ->  Ord  A )
12 ordwe 4442 . . . 4  |-  ( Ord 
A  ->  _E  We  A )
13 weso 4421 . . . 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 5577 . . . . . . . . 9  |-  ( y `
 z )  e. 
_V
1716epelc 4344 . . . . . . . 8  |-  ( ( x `  z )  _E  ( y `  z )  <->  ( x `  z )  e.  ( y `  z ) )
18 vex 2825 . . . . . . . . . . . 12  |-  w  e. 
_V
19 vex 2825 . . . . . . . . . . . 12  |-  z  e. 
_V
2018, 19brcnv 4901 . . . . . . . . . . 11  |-  ( w `'  _E  z  <->  z  _E  w )
21 epel 4345 . . . . . . . . . . 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 2601 . . . . . . . 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 2602 . . . . . 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 4120 . . . . 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 2339 . . . 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 4892 . . . . . . . 8  |-  ( g  =  x  ->  `' g  =  `' x
)
3029imaeq1d 5048 . . . . . . 7  |-  ( g  =  x  ->  ( `' g " ( _V  \  1o ) )  =  ( `' x " ( _V  \  1o ) ) )
31 df1o2 6533 . . . . . . . . 9  |-  1o  =  { (/) }
3231difeq2i 3325 . . . . . . . 8  |-  ( _V 
\  1o )  =  ( _V  \  { (/)
} )
3332imaeq2i 5047 . . . . . . 7  |-  ( `' x " ( _V 
\  1o ) )  =  ( `' x " ( _V  \  { (/)
} ) )
3430, 33syl6eq 2364 . . . . . 6  |-  ( g  =  x  ->  ( `' g " ( _V  \  1o ) )  =  ( `' x " ( _V  \  { (/)
} ) ) )
3534eleq1d 2382 . . . . 5  |-  ( g  =  x  ->  (
( `' g "
( _V  \  1o ) )  e.  Fin  <->  ( `' x " ( _V 
\  { (/) } ) )  e.  Fin )
)
3635cbvrabv 2821 . . . 4  |-  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin }  =  { x  e.  ( A  ^m  B )  |  ( `' x "
( _V  \  { (/)
} ) )  e. 
Fin }
3728, 36wemapso2 7312 . . 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 2316 . . . . 5  |-  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin }  =  { g  e.  ( A  ^m  B )  |  ( `' g
" ( _V  \  1o ) )  e.  Fin }
4140, 9, 1cantnfdm 7410 . . . 4  |-  ( ph  ->  dom  ( A CNF  B
)  =  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin } )
4239, 41syl5eq 2360 . . 3  |-  ( ph  ->  S  =  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin } )
43 soeq2 4371 . . 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 1633    e. wcel 1701   A.wral 2577   E.wrex 2578   {crab 2581   _Vcvv 2822    \ cdif 3183   (/)c0 3489   {csn 3674   class class class wbr 4060   {copab 4113    _E cep 4340    Or wor 4350    We wwe 4388   Ord word 4428   Oncon0 4429   `'ccnv 4725   dom cdm 4726   "cima 4729   ` cfv 5292  (class class class)co 5900   1oc1o 6514    ^m cmap 6815   Fincfn 6906   CNF ccnf 7407
This theorem is referenced by:  cantnf  7440
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-se 4390  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-isom 5301  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-seqom 6502  df-1o 6521  df-oadd 6525  df-er 6702  df-map 6817  df-en 6907  df-fin 6910  df-oi 7270  df-cnf 7408
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