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Theorem ordtypelem4 7756
Description: Lemma for ordtype 7767. (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 6874 . . . . . . 7  |-  ( Fun 
F  /\  Lim  dom  F
)
32simpli 458 . . . . . 6  |-  Fun  F
4 funres 5478 . . . . . 6  |-  ( Fun 
F  ->  Fun  ( F  |`  T ) )
53, 4mp1i 12 . . . . 5  |-  ( ph  ->  Fun  ( F  |`  T ) )
6 funfn 5468 . . . . 5  |-  ( Fun  ( F  |`  T )  <-> 
( F  |`  T )  Fn  dom  ( F  |`  T ) )
75, 6sylib 196 . . . 4  |-  ( ph  ->  ( F  |`  T )  Fn  dom  ( F  |`  T ) )
8 dmres 5152 . . . . 5  |-  dom  ( F  |`  T )  =  ( T  i^i  dom  F )
98fneq2i 5527 . . . 4  |-  ( ( F  |`  T )  Fn  dom  ( F  |`  T )  <->  ( F  |`  T )  Fn  ( T  i^i  dom  F )
)
107, 9sylib 196 . . 3  |-  ( ph  ->  ( F  |`  T )  Fn  ( T  i^i  dom 
F ) )
11 inss1 3591 . . . . . . 7  |-  ( T  i^i  dom  F )  C_  T
12 simpr 461 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( T  i^i  dom  F
) )  ->  a  e.  ( T  i^i  dom  F ) )
1311, 12sseldi 3375 . . . . . 6  |-  ( (
ph  /\  a  e.  ( T  i^i  dom  F
) )  ->  a  e.  T )
14 fvres 5725 . . . . . 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 3458 . . . . . . 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 3458 . . . . . . 7  |-  { w  e.  A  |  A. j  e.  ( F " a ) j R w }  C_  A
1816, 17sstri 3386 . . . . . 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 7755 . . . . . 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 3375 . . . . 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 2820 . . 3  |-  ( ph  ->  A. a  e.  ( T  i^i  dom  F
) ( ( F  |`  T ) `  a
)  e.  A )
29 ffnfv 5890 . . 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 664 . 2  |-  ( ph  ->  ( F  |`  T ) : ( T  i^i  dom 
F ) --> A )
311, 19, 20, 21, 22, 23, 24ordtypelem1 7753 . . 3  |-  ( ph  ->  O  =  ( F  |`  T ) )
3231feq1d 5567 . 2  |-  ( ph  ->  ( O : ( T  i^i  dom  F
) --> A  <->  ( F  |`  T ) : ( T  i^i  dom  F
) --> A ) )
3330, 32mpbird 232 1  |-  ( ph  ->  O : ( T  i^i  dom  F ) --> A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2736   E.wrex 2737   {crab 2740   _Vcvv 2993    i^i cin 3348   class class class wbr 4313    e. cmpt 4371   Se wse 4698    We wwe 4699   Oncon0 4740   Lim wlim 4741   dom cdm 4861   ran crn 4862    |` cres 4863   "cima 4864   Fun wfun 5433    Fn wfn 5434   -->wf 5435   ` cfv 5439   iota_crio 6072  recscrecs 6852  OrdIsocoi 7744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-se 4701  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-recs 6853  df-oi 7745
This theorem is referenced by:  ordtypelem5  7757  ordtypelem6  7758  ordtypelem7  7759  ordtypelem8  7760  ordtypelem9  7761  ordtypelem10  7762
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