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Theorem ordtypelem4 7238
Description: Lemma for ordtype 7249. (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 6412 . . . . . . 7  |-  ( Fun 
F  /\  Lim  dom  F
)
32simpli 444 . . . . . 6  |-  Fun  F
4 funres 5295 . . . . . 6  |-  ( Fun 
F  ->  Fun  ( F  |`  T ) )
53, 4mp1i 11 . . . . 5  |-  ( ph  ->  Fun  ( F  |`  T ) )
6 funfn 5285 . . . . 5  |-  ( Fun  ( F  |`  T )  <-> 
( F  |`  T )  Fn  dom  ( F  |`  T ) )
75, 6sylib 188 . . . 4  |-  ( ph  ->  ( F  |`  T )  Fn  dom  ( F  |`  T ) )
8 dmres 4978 . . . . 5  |-  dom  ( F  |`  T )  =  ( T  i^i  dom  F )
98fneq2i 5341 . . . 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 3391 . . . . . . 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 3180 . . . . . 6  |-  ( (
ph  /\  a  e.  ( T  i^i  dom  F
) )  ->  a  e.  T )
14 fvres 5544 . . . . . 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 3260 . . . . . . 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 3260 . . . . . . 7  |-  { w  e.  A  |  A. j  e.  ( F " a ) j R w }  C_  A
1816, 17sstri 3190 . . . . . 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 7237 . . . . . 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 3180 . . . . 5  |-  ( (
ph  /\  a  e.  ( T  i^i  dom  F
) )  ->  ( F `  a )  e.  A )
2715, 26eqeltrd 2359 . . . 4  |-  ( (
ph  /\  a  e.  ( T  i^i  dom  F
) )  ->  (
( F  |`  T ) `
 a )  e.  A )
2827ralrimiva 2628 . . 3  |-  ( ph  ->  A. a  e.  ( T  i^i  dom  F
) ( ( F  |`  T ) `  a
)  e.  A )
29 ffnfv 5687 . . 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 7235 . . 3  |-  ( ph  ->  O  =  ( F  |`  T ) )
3231feq1d 5381 . 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 1625    e. wcel 1686   A.wral 2545   E.wrex 2546   {crab 2549   _Vcvv 2790    i^i cin 3153   class class class wbr 4025    e. cmpt 4079   Se wse 4352    We wwe 4353   Oncon0 4394   Lim wlim 4395   dom cdm 4691   ran crn 4692    |` cres 4693   "cima 4694   Fun wfun 5251    Fn wfn 5252   -->wf 5253   ` cfv 5257   iota_crio 6299  recscrecs 6389  OrdIsocoi 7226
This theorem is referenced by:  ordtypelem5  7239  ordtypelem6  7240  ordtypelem7  7241  ordtypelem8  7242  ordtypelem9  7243  ordtypelem10  7244
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-riota 6306  df-recs 6390  df-oi 7227
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