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Theorem ordtypelem2 7234
Description: Lemma for ordtype 7247. (Contributed by Mario Carneiro, 24-Jun-2015.)
Hypotheses
Ref Expression
ordtypelem.1  |-  F  = recs ( G )
ordtypelem.2  |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }
ordtypelem.3  |-  G  =  ( h  e.  _V  |->  ( iota_ v  e.  C A. u  e.  C  -.  u R v ) )
ordtypelem.5  |-  T  =  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t }
ordtypelem.6  |-  O  = OrdIso
( R ,  A
)
ordtypelem.7  |-  ( ph  ->  R  We  A )
ordtypelem.8  |-  ( ph  ->  R Se  A )
Assertion
Ref Expression
ordtypelem2  |-  ( ph  ->  Ord  T )
Distinct variable groups:    v, u, C    h, j, t, u, v, w, x, z, R    A, h, j, t, u, v, w, x, z    t, O, u, v, x    ph, t, x    h, F, j, t, u, v, w, x, z
Allowed substitution hints:    ph( z, w, v, u, h, j)    C( x, z, w, t, h, j)    T( x, z, w, v, u, t, h, j)    G( x, z, w, v, u, t, h, j)    O( z, w, h, j)

Proof of Theorem ordtypelem2
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 ordtypelem.5 . . . . . . . . . 10  |-  T  =  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t }
2 ssrab2 3258 . . . . . . . . . 10  |-  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x
) z R t }  C_  On
31, 2eqsstri 3208 . . . . . . . . 9  |-  T  C_  On
43a1i 10 . . . . . . . 8  |-  ( ph  ->  T  C_  On )
54sselda 3180 . . . . . . 7  |-  ( (
ph  /\  a  e.  T )  ->  a  e.  On )
6 onss 4582 . . . . . . 7  |-  ( a  e.  On  ->  a  C_  On )
75, 6syl 15 . . . . . 6  |-  ( (
ph  /\  a  e.  T )  ->  a  C_  On )
8 eloni 4402 . . . . . . . 8  |-  ( a  e.  On  ->  Ord  a )
95, 8syl 15 . . . . . . 7  |-  ( (
ph  /\  a  e.  T )  ->  Ord  a )
10 imaeq2 5008 . . . . . . . . . . . 12  |-  ( x  =  a  ->  ( F " x )  =  ( F " a
) )
1110raleqdv 2742 . . . . . . . . . . 11  |-  ( x  =  a  ->  ( A. z  e.  ( F " x ) z R t  <->  A. z  e.  ( F " a
) z R t ) )
1211rexbidv 2564 . . . . . . . . . 10  |-  ( x  =  a  ->  ( E. t  e.  A  A. z  e.  ( F " x ) z R t  <->  E. t  e.  A  A. z  e.  ( F " a
) z R t ) )
1312, 1elrab2 2925 . . . . . . . . 9  |-  ( a  e.  T  <->  ( a  e.  On  /\  E. t  e.  A  A. z  e.  ( F " a
) z R t ) )
1413simprbi 450 . . . . . . . 8  |-  ( a  e.  T  ->  E. t  e.  A  A. z  e.  ( F " a
) z R t )
1514adantl 452 . . . . . . 7  |-  ( (
ph  /\  a  e.  T )  ->  E. t  e.  A  A. z  e.  ( F " a
) z R t )
16 ordelss 4408 . . . . . . . . 9  |-  ( ( Ord  a  /\  x  e.  a )  ->  x  C_  a )
17 imass2 5049 . . . . . . . . 9  |-  ( x 
C_  a  ->  ( F " x )  C_  ( F " a ) )
18 ssralv 3237 . . . . . . . . . 10  |-  ( ( F " x ) 
C_  ( F "
a )  ->  ( A. z  e.  ( F " a ) z R t  ->  A. z  e.  ( F " x
) z R t ) )
1918reximdv 2654 . . . . . . . . 9  |-  ( ( F " x ) 
C_  ( F "
a )  ->  ( E. t  e.  A  A. z  e.  ( F " a ) z R t  ->  E. t  e.  A  A. z  e.  ( F " x
) z R t ) )
2016, 17, 193syl 18 . . . . . . . 8  |-  ( ( Ord  a  /\  x  e.  a )  ->  ( E. t  e.  A  A. z  e.  ( F " a ) z R t  ->  E. t  e.  A  A. z  e.  ( F " x
) z R t ) )
2120ralrimdva 2633 . . . . . . 7  |-  ( Ord  a  ->  ( E. t  e.  A  A. z  e.  ( F " a ) z R t  ->  A. x  e.  a  E. t  e.  A  A. z  e.  ( F " x
) z R t ) )
229, 15, 21sylc 56 . . . . . 6  |-  ( (
ph  /\  a  e.  T )  ->  A. x  e.  a  E. t  e.  A  A. z  e.  ( F " x
) z R t )
23 ssrab 3251 . . . . . 6  |-  ( a 
C_  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x
) z R t }  <->  ( a  C_  On  /\  A. x  e.  a  E. t  e.  A  A. z  e.  ( F " x
) z R t ) )
247, 22, 23sylanbrc 645 . . . . 5  |-  ( (
ph  /\  a  e.  T )  ->  a  C_ 
{ x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t } )
2524, 1syl6sseqr 3225 . . . 4  |-  ( (
ph  /\  a  e.  T )  ->  a  C_  T )
2625ralrimiva 2626 . . 3  |-  ( ph  ->  A. a  e.  T  a  C_  T )
27 dftr3 4117 . . 3  |-  ( Tr  T  <->  A. a  e.  T  a  C_  T )
2826, 27sylibr 203 . 2  |-  ( ph  ->  Tr  T )
29 ordon 4574 . . 3  |-  Ord  On
30 trssord 4409 . . 3  |-  ( ( Tr  T  /\  T  C_  On  /\  Ord  On )  ->  Ord  T )
313, 29, 30mp3an23 1269 . 2  |-  ( Tr  T  ->  Ord  T )
3228, 31syl 15 1  |-  ( ph  ->  Ord  T )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788    C_ wss 3152   class class class wbr 4023    e. cmpt 4077   Tr wtr 4113   Se wse 4350    We wwe 4351   Ord word 4391   Oncon0 4392   ran crn 4690   "cima 4692   iota_crio 6297  recscrecs 6387  OrdIsocoi 7224
This theorem is referenced by:  ordtypelem5  7237  ordtypelem6  7238  ordtypelem7  7239  ordtypelem8  7240  ordtypelem9  7241
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-sep 4141  ax-nul 4149  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-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702
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