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Theorem ordtypelem2 7477
Description: Lemma for ordtype 7490. (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 3420 . . . . . . . . . 10  |-  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x
) z R t }  C_  On
31, 2eqsstri 3370 . . . . . . . . 9  |-  T  C_  On
43a1i 11 . . . . . . . 8  |-  ( ph  ->  T  C_  On )
54sselda 3340 . . . . . . 7  |-  ( (
ph  /\  a  e.  T )  ->  a  e.  On )
6 onss 4762 . . . . . . 7  |-  ( a  e.  On  ->  a  C_  On )
75, 6syl 16 . . . . . 6  |-  ( (
ph  /\  a  e.  T )  ->  a  C_  On )
8 eloni 4583 . . . . . . . 8  |-  ( a  e.  On  ->  Ord  a )
95, 8syl 16 . . . . . . 7  |-  ( (
ph  /\  a  e.  T )  ->  Ord  a )
10 imaeq2 5190 . . . . . . . . . . . 12  |-  ( x  =  a  ->  ( F " x )  =  ( F " a
) )
1110raleqdv 2902 . . . . . . . . . . 11  |-  ( x  =  a  ->  ( A. z  e.  ( F " x ) z R t  <->  A. z  e.  ( F " a
) z R t ) )
1211rexbidv 2718 . . . . . . . . . 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 3086 . . . . . . . . 9  |-  ( a  e.  T  <->  ( a  e.  On  /\  E. t  e.  A  A. z  e.  ( F " a
) z R t ) )
1413simprbi 451 . . . . . . . 8  |-  ( a  e.  T  ->  E. t  e.  A  A. z  e.  ( F " a
) z R t )
1514adantl 453 . . . . . . 7  |-  ( (
ph  /\  a  e.  T )  ->  E. t  e.  A  A. z  e.  ( F " a
) z R t )
16 ordelss 4589 . . . . . . . . 9  |-  ( ( Ord  a  /\  x  e.  a )  ->  x  C_  a )
17 imass2 5231 . . . . . . . . 9  |-  ( x 
C_  a  ->  ( F " x )  C_  ( F " a ) )
18 ssralv 3399 . . . . . . . . . 10  |-  ( ( F " x ) 
C_  ( F "
a )  ->  ( A. z  e.  ( F " a ) z R t  ->  A. z  e.  ( F " x
) z R t ) )
1918reximdv 2809 . . . . . . . . 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 19 . . . . . . . 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 2788 . . . . . . 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 58 . . . . . 6  |-  ( (
ph  /\  a  e.  T )  ->  A. x  e.  a  E. t  e.  A  A. z  e.  ( F " x
) z R t )
23 ssrab 3413 . . . . . 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 646 . . . . 5  |-  ( (
ph  /\  a  e.  T )  ->  a  C_ 
{ x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t } )
2524, 1syl6sseqr 3387 . . . 4  |-  ( (
ph  /\  a  e.  T )  ->  a  C_  T )
2625ralrimiva 2781 . . 3  |-  ( ph  ->  A. a  e.  T  a  C_  T )
27 dftr3 4298 . . 3  |-  ( Tr  T  <->  A. a  e.  T  a  C_  T )
2826, 27sylibr 204 . 2  |-  ( ph  ->  Tr  T )
29 ordon 4754 . . 3  |-  Ord  On
30 trssord 4590 . . 3  |-  ( ( Tr  T  /\  T  C_  On  /\  Ord  On )  ->  Ord  T )
313, 29, 30mp3an23 1271 . 2  |-  ( Tr  T  ->  Ord  T )
3228, 31syl 16 1  |-  ( ph  ->  Ord  T )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   {crab 2701   _Vcvv 2948    C_ wss 3312   class class class wbr 4204    e. cmpt 4258   Tr wtr 4294   Se wse 4531    We wwe 4532   Ord word 4572   Oncon0 4573   ran crn 4870   "cima 4872   iota_crio 6533  recscrecs 6623  OrdIsocoi 7467
This theorem is referenced by:  ordtypelem5  7480  ordtypelem6  7481  ordtypelem7  7482  ordtypelem8  7483  ordtypelem9  7484
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 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4692
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-xp 4875  df-cnv 4877  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882
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