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Theorem ordtypelem4 7252
Description: Lemma for ordtype 7263. (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 6426 . . . . . . 7  |-  ( Fun 
F  /\  Lim  dom  F
)
32simpli 444 . . . . . 6  |-  Fun  F
4 funres 5309 . . . . . 6  |-  ( Fun 
F  ->  Fun  ( F  |`  T ) )
53, 4mp1i 11 . . . . 5  |-  ( ph  ->  Fun  ( F  |`  T ) )
6 funfn 5299 . . . . 5  |-  ( Fun  ( F  |`  T )  <-> 
( F  |`  T )  Fn  dom  ( F  |`  T ) )
75, 6sylib 188 . . . 4  |-  ( ph  ->  ( F  |`  T )  Fn  dom  ( F  |`  T ) )
8 dmres 4992 . . . . 5  |-  dom  ( F  |`  T )  =  ( T  i^i  dom  F )
98fneq2i 5355 . . . 4  |-  ( ( F  |`  T )  Fn  dom  ( F  |`  T )  <->  ( F  |`  T )  Fn  ( T  i^i  dom  F )
)
107, 9sylib 188 . . 3  |-  ( ph  ->  ( F  |`  T )  Fn  ( T  i^i  dom 
F ) )
11 inss1 3402 . . . . . . 7  |-  ( T  i^i  dom  F )  C_  T
12 simpr 447 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( T  i^i  dom  F
) )  ->  a  e.  ( T  i^i  dom  F ) )
1311, 12sseldi 3191 . . . . . 6  |-  ( (
ph  /\  a  e.  ( T  i^i  dom  F
) )  ->  a  e.  T )
14 fvres 5558 . . . . . 6  |-  ( a  e.  T  ->  (
( F  |`  T ) `
 a )  =  ( F `  a
) )
1513, 14syl 15 . . . . 5  |-  ( (
ph  /\  a  e.  ( T  i^i  dom  F
) )  ->  (
( F  |`  T ) `
 a )  =  ( F `  a
) )
16 ssrab2 3271 . . . . . . 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 3271 . . . . . . 7  |-  { w  e.  A  |  A. j  e.  ( F " a ) j R w }  C_  A
1816, 17sstri 3201 . . . . . 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 7251 . . . . . 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 3191 . . . . 5  |-  ( (
ph  /\  a  e.  ( T  i^i  dom  F
) )  ->  ( F `  a )  e.  A )
2715, 26eqeltrd 2370 . . . 4  |-  ( (
ph  /\  a  e.  ( T  i^i  dom  F
) )  ->  (
( F  |`  T ) `
 a )  e.  A )
2827ralrimiva 2639 . . 3  |-  ( ph  ->  A. a  e.  ( T  i^i  dom  F
) ( ( F  |`  T ) `  a
)  e.  A )
29 ffnfv 5701 . . 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 645 . 2  |-  ( ph  ->  ( F  |`  T ) : ( T  i^i  dom 
F ) --> A )
311, 19, 20, 21, 22, 23, 24ordtypelem1 7249 . . 3  |-  ( ph  ->  O  =  ( F  |`  T ) )
3231feq1d 5395 . 2  |-  ( ph  ->  ( O : ( T  i^i  dom  F
) --> A  <->  ( F  |`  T ) : ( T  i^i  dom  F
) --> A ) )
3330, 32mpbird 223 1  |-  ( ph  ->  O : ( T  i^i  dom  F ) --> A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   {crab 2560   _Vcvv 2801    i^i cin 3164   class class class wbr 4039    e. cmpt 4093   Se wse 4366    We wwe 4367   Oncon0 4408   Lim wlim 4409   dom cdm 4705   ran crn 4706    |` cres 4707   "cima 4708   Fun wfun 5265    Fn wfn 5266   -->wf 5267   ` cfv 5271   iota_crio 6313  recscrecs 6403  OrdIsocoi 7240
This theorem is referenced by:  ordtypelem5  7253  ordtypelem6  7254  ordtypelem7  7255  ordtypelem8  7256  ordtypelem9  7257  ordtypelem10  7258
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 6320  df-recs 6404  df-oi 7241
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