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Theorem ordtypelem4 8036
Description: Lemma for ordtype 8047. (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
ordtypelem4  |-  ( ph  ->  O : ( T  i^i  dom  F ) --> A )
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 ordtypelem4
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 ordtypelem.1 . . . . . . . 8  |-  F  = recs ( G )
21tfr1a 7120 . . . . . . 7  |-  ( Fun 
F  /\  Lim  dom  F
)
32simpli 459 . . . . . 6  |-  Fun  F
4 funres 5640 . . . . . 6  |-  ( Fun 
F  ->  Fun  ( F  |`  T ) )
53, 4mp1i 13 . . . . 5  |-  ( ph  ->  Fun  ( F  |`  T ) )
6 funfn 5630 . . . . 5  |-  ( Fun  ( F  |`  T )  <-> 
( F  |`  T )  Fn  dom  ( F  |`  T ) )
75, 6sylib 199 . . . 4  |-  ( ph  ->  ( F  |`  T )  Fn  dom  ( F  |`  T ) )
8 dmres 5145 . . . . 5  |-  dom  ( F  |`  T )  =  ( T  i^i  dom  F )
98fneq2i 5689 . . . 4  |-  ( ( F  |`  T )  Fn  dom  ( F  |`  T )  <->  ( F  |`  T )  Fn  ( T  i^i  dom  F )
)
107, 9sylib 199 . . 3  |-  ( ph  ->  ( F  |`  T )  Fn  ( T  i^i  dom 
F ) )
11 inss1 3688 . . . . . . 7  |-  ( T  i^i  dom  F )  C_  T
12 simpr 462 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( T  i^i  dom  F
) )  ->  a  e.  ( T  i^i  dom  F ) )
1311, 12sseldi 3468 . . . . . 6  |-  ( (
ph  /\  a  e.  ( T  i^i  dom  F
) )  ->  a  e.  T )
14 fvres 5895 . . . . . 6  |-  ( a  e.  T  ->  (
( F  |`  T ) `
 a )  =  ( F `  a
) )
1513, 14syl 17 . . . . 5  |-  ( (
ph  /\  a  e.  ( T  i^i  dom  F
) )  ->  (
( F  |`  T ) `
 a )  =  ( F `  a
) )
16 ssrab2 3552 . . . . . . 7  |-  { v  e.  { w  e.  A  |  A. j  e.  ( F " a
) j R w }  |  A. u  e.  { w  e.  A  |  A. j  e.  ( F " a ) j R w }  -.  u R v } 
C_  { w  e.  A  |  A. j  e.  ( F " a
) j R w }
17 ssrab2 3552 . . . . . . 7  |-  { w  e.  A  |  A. j  e.  ( F " a ) j R w }  C_  A
1816, 17sstri 3479 . . . . . 6  |-  { v  e.  { w  e.  A  |  A. j  e.  ( F " a
) j R w }  |  A. u  e.  { w  e.  A  |  A. j  e.  ( F " a ) j R w }  -.  u R v } 
C_  A
19 ordtypelem.2 . . . . . . 7  |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }
20 ordtypelem.3 . . . . . . 7  |-  G  =  ( h  e.  _V  |->  ( iota_ v  e.  C  A. u  e.  C  -.  u R v ) )
21 ordtypelem.5 . . . . . . 7  |-  T  =  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t }
22 ordtypelem.6 . . . . . . 7  |-  O  = OrdIso
( R ,  A
)
23 ordtypelem.7 . . . . . . 7  |-  ( ph  ->  R  We  A )
24 ordtypelem.8 . . . . . . 7  |-  ( ph  ->  R Se  A )
251, 19, 20, 21, 22, 23, 24ordtypelem3 8035 . . . . . 6  |-  ( (
ph  /\  a  e.  ( T  i^i  dom  F
) )  ->  ( F `  a )  e.  { v  e.  {
w  e.  A  |  A. j  e.  ( F " a ) j R w }  |  A. u  e.  { w  e.  A  |  A. j  e.  ( F " a ) j R w }  -.  u R v } )
2618, 25sseldi 3468 . . . . 5  |-  ( (
ph  /\  a  e.  ( T  i^i  dom  F
) )  ->  ( F `  a )  e.  A )
2715, 26eqeltrd 2517 . . . 4  |-  ( (
ph  /\  a  e.  ( T  i^i  dom  F
) )  ->  (
( F  |`  T ) `
 a )  e.  A )
2827ralrimiva 2846 . . 3  |-  ( ph  ->  A. a  e.  ( T  i^i  dom  F
) ( ( F  |`  T ) `  a
)  e.  A )
29 ffnfv 6064 . . 3  |-  ( ( F  |`  T ) : ( T  i^i  dom 
F ) --> A  <->  ( ( F  |`  T )  Fn  ( T  i^i  dom  F )  /\  A. a  e.  ( T  i^i  dom  F ) ( ( F  |`  T ) `  a
)  e.  A ) )
3010, 28, 29sylanbrc 668 . 2  |-  ( ph  ->  ( F  |`  T ) : ( T  i^i  dom 
F ) --> A )
311, 19, 20, 21, 22, 23, 24ordtypelem1 8033 . . 3  |-  ( ph  ->  O  =  ( F  |`  T ) )
3231feq1d 5732 . 2  |-  ( ph  ->  ( O : ( T  i^i  dom  F
) --> A  <->  ( F  |`  T ) : ( T  i^i  dom  F
) --> A ) )
3330, 32mpbird 235 1  |-  ( ph  ->  O : ( T  i^i  dom  F ) --> A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782   E.wrex 2783   {crab 2786   _Vcvv 3087    i^i cin 3441   class class class wbr 4426    |-> cmpt 4484   Se wse 4811    We wwe 4812   dom cdm 4854   ran crn 4855    |` cres 4856   "cima 4857   Oncon0 5442   Lim wlim 5443   Fun wfun 5595    Fn wfn 5596   -->wf 5597   ` cfv 5601   iota_crio 6266  recscrecs 7097  OrdIsocoi 8024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-wrecs 7036  df-recs 7098  df-oi 8025
This theorem is referenced by:  ordtypelem5  8037  ordtypelem6  8038  ordtypelem7  8039  ordtypelem8  8040  ordtypelem9  8041  ordtypelem10  8042
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