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Theorem ordtypelem2 7859
Description: Lemma for ordtype 7872. (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 3499 . . . . . . . . . 10  |-  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x
) z R t }  C_  On
31, 2eqsstri 3447 . . . . . . . . 9  |-  T  C_  On
43a1i 11 . . . . . . . 8  |-  ( ph  ->  T  C_  On )
54sselda 3417 . . . . . . 7  |-  ( (
ph  /\  a  e.  T )  ->  a  e.  On )
6 onss 6525 . . . . . . 7  |-  ( a  e.  On  ->  a  C_  On )
75, 6syl 16 . . . . . 6  |-  ( (
ph  /\  a  e.  T )  ->  a  C_  On )
8 eloni 4802 . . . . . . . 8  |-  ( a  e.  On  ->  Ord  a )
95, 8syl 16 . . . . . . 7  |-  ( (
ph  /\  a  e.  T )  ->  Ord  a )
10 imaeq2 5245 . . . . . . . . . . . 12  |-  ( x  =  a  ->  ( F " x )  =  ( F " a
) )
1110raleqdv 2985 . . . . . . . . . . 11  |-  ( x  =  a  ->  ( A. z  e.  ( F " x ) z R t  <->  A. z  e.  ( F " a
) z R t ) )
1211rexbidv 2893 . . . . . . . . . 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 3184 . . . . . . . . 9  |-  ( a  e.  T  <->  ( a  e.  On  /\  E. t  e.  A  A. z  e.  ( F " a
) z R t ) )
1413simprbi 462 . . . . . . . 8  |-  ( a  e.  T  ->  E. t  e.  A  A. z  e.  ( F " a
) z R t )
1514adantl 464 . . . . . . 7  |-  ( (
ph  /\  a  e.  T )  ->  E. t  e.  A  A. z  e.  ( F " a
) z R t )
16 ordelss 4808 . . . . . . . . 9  |-  ( ( Ord  a  /\  x  e.  a )  ->  x  C_  a )
17 imass2 5284 . . . . . . . . 9  |-  ( x 
C_  a  ->  ( F " x )  C_  ( F " a ) )
18 ssralv 3478 . . . . . . . . . 10  |-  ( ( F " x ) 
C_  ( F "
a )  ->  ( A. z  e.  ( F " a ) z R t  ->  A. z  e.  ( F " x
) z R t ) )
1918reximdv 2856 . . . . . . . . 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 20 . . . . . . . 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 2800 . . . . . . 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 60 . . . . . 6  |-  ( (
ph  /\  a  e.  T )  ->  A. x  e.  a  E. t  e.  A  A. z  e.  ( F " x
) z R t )
23 ssrab 3492 . . . . . 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 662 . . . . 5  |-  ( (
ph  /\  a  e.  T )  ->  a  C_ 
{ x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t } )
2524, 1syl6sseqr 3464 . . . 4  |-  ( (
ph  /\  a  e.  T )  ->  a  C_  T )
2625ralrimiva 2796 . . 3  |-  ( ph  ->  A. a  e.  T  a  C_  T )
27 dftr3 4464 . . 3  |-  ( Tr  T  <->  A. a  e.  T  a  C_  T )
2826, 27sylibr 212 . 2  |-  ( ph  ->  Tr  T )
29 ordon 6517 . . 3  |-  Ord  On
30 trssord 4809 . . 3  |-  ( ( Tr  T  /\  T  C_  On  /\  Ord  On )  ->  Ord  T )
313, 29, 30mp3an23 1314 . 2  |-  ( Tr  T  ->  Ord  T )
3228, 31syl 16 1  |-  ( ph  ->  Ord  T )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1826   A.wral 2732   E.wrex 2733   {crab 2736   _Vcvv 3034    C_ wss 3389   class class class wbr 4367    |-> cmpt 4425   Tr wtr 4460   Se wse 4750    We wwe 4751   Ord word 4791   Oncon0 4792   ran crn 4914   "cima 4916   iota_crio 6157  recscrecs 6959  OrdIsocoi 7849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-tr 4461  df-eprel 4705  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-xp 4919  df-cnv 4921  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926
This theorem is referenced by:  ordtypelem5  7862  ordtypelem6  7863  ordtypelem7  7864  ordtypelem8  7865  ordtypelem9  7866
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