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Theorem ordtypelem4 7482
Description: Lemma for ordtype 7493. (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 6647 . . . . . . 7  |-  ( Fun 
F  /\  Lim  dom  F
)
32simpli 445 . . . . . 6  |-  Fun  F
4 funres 5484 . . . . . 6  |-  ( Fun 
F  ->  Fun  ( F  |`  T ) )
53, 4mp1i 12 . . . . 5  |-  ( ph  ->  Fun  ( F  |`  T ) )
6 funfn 5474 . . . . 5  |-  ( Fun  ( F  |`  T )  <-> 
( F  |`  T )  Fn  dom  ( F  |`  T ) )
75, 6sylib 189 . . . 4  |-  ( ph  ->  ( F  |`  T )  Fn  dom  ( F  |`  T ) )
8 dmres 5159 . . . . 5  |-  dom  ( F  |`  T )  =  ( T  i^i  dom  F )
98fneq2i 5532 . . . 4  |-  ( ( F  |`  T )  Fn  dom  ( F  |`  T )  <->  ( F  |`  T )  Fn  ( T  i^i  dom  F )
)
107, 9sylib 189 . . 3  |-  ( ph  ->  ( F  |`  T )  Fn  ( T  i^i  dom 
F ) )
11 inss1 3553 . . . . . . 7  |-  ( T  i^i  dom  F )  C_  T
12 simpr 448 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( T  i^i  dom  F
) )  ->  a  e.  ( T  i^i  dom  F ) )
1311, 12sseldi 3338 . . . . . 6  |-  ( (
ph  /\  a  e.  ( T  i^i  dom  F
) )  ->  a  e.  T )
14 fvres 5737 . . . . . 6  |-  ( a  e.  T  ->  (
( F  |`  T ) `
 a )  =  ( F `  a
) )
1513, 14syl 16 . . . . 5  |-  ( (
ph  /\  a  e.  ( T  i^i  dom  F
) )  ->  (
( F  |`  T ) `
 a )  =  ( F `  a
) )
16 ssrab2 3420 . . . . . . 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 3420 . . . . . . 7  |-  { w  e.  A  |  A. j  e.  ( F " a ) j R w }  C_  A
1816, 17sstri 3349 . . . . . 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 7481 . . . . . 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 3338 . . . . 5  |-  ( (
ph  /\  a  e.  ( T  i^i  dom  F
) )  ->  ( F `  a )  e.  A )
2715, 26eqeltrd 2509 . . . 4  |-  ( (
ph  /\  a  e.  ( T  i^i  dom  F
) )  ->  (
( F  |`  T ) `
 a )  e.  A )
2827ralrimiva 2781 . . 3  |-  ( ph  ->  A. a  e.  ( T  i^i  dom  F
) ( ( F  |`  T ) `  a
)  e.  A )
29 ffnfv 5886 . . 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 646 . 2  |-  ( ph  ->  ( F  |`  T ) : ( T  i^i  dom 
F ) --> A )
311, 19, 20, 21, 22, 23, 24ordtypelem1 7479 . . 3  |-  ( ph  ->  O  =  ( F  |`  T ) )
3231feq1d 5572 . 2  |-  ( ph  ->  ( O : ( T  i^i  dom  F
) --> A  <->  ( F  |`  T ) : ( T  i^i  dom  F
) --> A ) )
3330, 32mpbird 224 1  |-  ( ph  ->  O : ( T  i^i  dom  F ) --> A )
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    i^i cin 3311   class class class wbr 4204    e. cmpt 4258   Se wse 4531    We wwe 4532   Oncon0 4573   Lim wlim 4574   dom cdm 4870   ran crn 4871    |` cres 4872   "cima 4873   Fun wfun 5440    Fn wfn 5441   -->wf 5442   ` cfv 5446   iota_crio 6534  recscrecs 6624  OrdIsocoi 7470
This theorem is referenced by:  ordtypelem5  7483  ordtypelem6  7484  ordtypelem7  7485  ordtypelem8  7486  ordtypelem9  7487  ordtypelem10  7488
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-pow 4369  ax-pr 4395  ax-un 4693
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-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-riota 6541  df-recs 6625  df-oi 7471
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