MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordtypelem8 Structured version   Unicode version

Theorem ordtypelem8 7760
Description: Lemma for ordtype 7767. (Contributed by Mario Carneiro, 25-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
ordtypelem8  |-  ( ph  ->  O  Isom  _E  ,  R  ( dom  O ,  ran  O ) )
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 ordtypelem8
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordtypelem.1 . . . . . 6  |-  F  = recs ( G )
2 ordtypelem.2 . . . . . 6  |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }
3 ordtypelem.3 . . . . . 6  |-  G  =  ( h  e.  _V  |->  ( iota_ v  e.  C  A. u  e.  C  -.  u R v ) )
4 ordtypelem.5 . . . . . 6  |-  T  =  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t }
5 ordtypelem.6 . . . . . 6  |-  O  = OrdIso
( R ,  A
)
6 ordtypelem.7 . . . . . 6  |-  ( ph  ->  R  We  A )
7 ordtypelem.8 . . . . . 6  |-  ( ph  ->  R Se  A )
81, 2, 3, 4, 5, 6, 7ordtypelem4 7756 . . . . 5  |-  ( ph  ->  O : ( T  i^i  dom  F ) --> A )
9 fdm 5584 . . . . 5  |-  ( O : ( T  i^i  dom 
F ) --> A  ->  dom  O  =  ( T  i^i  dom  F )
)
108, 9syl 16 . . . 4  |-  ( ph  ->  dom  O  =  ( T  i^i  dom  F
) )
11 inss1 3591 . . . . 5  |-  ( T  i^i  dom  F )  C_  T
121, 2, 3, 4, 5, 6, 7ordtypelem2 7754 . . . . . 6  |-  ( ph  ->  Ord  T )
13 ordsson 6422 . . . . . 6  |-  ( Ord 
T  ->  T  C_  On )
1412, 13syl 16 . . . . 5  |-  ( ph  ->  T  C_  On )
1511, 14syl5ss 3388 . . . 4  |-  ( ph  ->  ( T  i^i  dom  F )  C_  On )
1610, 15eqsstrd 3411 . . 3  |-  ( ph  ->  dom  O  C_  On )
17 epweon 6416 . . . 4  |-  _E  We  On
18 weso 4732 . . . 4  |-  (  _E  We  On  ->  _E  Or  On )
1917, 18ax-mp 5 . . 3  |-  _E  Or  On
20 soss 4680 . . 3  |-  ( dom 
O  C_  On  ->  (  _E  Or  On  ->  _E  Or  dom  O ) )
2116, 19, 20mpisyl 18 . 2  |-  ( ph  ->  _E  Or  dom  O
)
22 frn 5586 . . . . 5  |-  ( O : ( T  i^i  dom 
F ) --> A  ->  ran  O  C_  A )
238, 22syl 16 . . . 4  |-  ( ph  ->  ran  O  C_  A
)
24 wess 4728 . . . 4  |-  ( ran 
O  C_  A  ->  ( R  We  A  ->  R  We  ran  O ) )
2523, 6, 24sylc 60 . . 3  |-  ( ph  ->  R  We  ran  O
)
26 weso 4732 . . 3  |-  ( R  We  ran  O  ->  R  Or  ran  O )
27 sopo 4679 . . 3  |-  ( R  Or  ran  O  ->  R  Po  ran  O )
2825, 26, 273syl 20 . 2  |-  ( ph  ->  R  Po  ran  O
)
29 ffun 5582 . . . 4  |-  ( O : ( T  i^i  dom 
F ) --> A  ->  Fun  O )
308, 29syl 16 . . 3  |-  ( ph  ->  Fun  O )
31 funforn 5648 . . 3  |-  ( Fun 
O  <->  O : dom  O -onto-> ran  O )
3230, 31sylib 196 . 2  |-  ( ph  ->  O : dom  O -onto-> ran  O )
33 epel 4656 . . . . 5  |-  ( a  _E  b  <->  a  e.  b )
341, 2, 3, 4, 5, 6, 7ordtypelem6 7758 . . . . 5  |-  ( (
ph  /\  b  e.  dom  O )  ->  (
a  e.  b  -> 
( O `  a
) R ( O `
 b ) ) )
3533, 34syl5bi 217 . . . 4  |-  ( (
ph  /\  b  e.  dom  O )  ->  (
a  _E  b  -> 
( O `  a
) R ( O `
 b ) ) )
3635ralrimiva 2820 . . 3  |-  ( ph  ->  A. b  e.  dom  O ( a  _E  b  ->  ( O `  a
) R ( O `
 b ) ) )
3736ralrimivw 2821 . 2  |-  ( ph  ->  A. a  e.  dom  O A. b  e.  dom  O ( a  _E  b  ->  ( O `  a
) R ( O `
 b ) ) )
38 soisoi 6040 . 2  |-  ( ( (  _E  Or  dom  O  /\  R  Po  ran  O )  /\  ( O : dom  O -onto-> ran  O  /\  A. a  e. 
dom  O A. b  e.  dom  O ( a  _E  b  ->  ( O `  a ) R ( O `  b ) ) ) )  ->  O  Isom  _E  ,  R  ( dom 
O ,  ran  O
) )
3921, 28, 32, 37, 38syl22anc 1219 1  |-  ( ph  ->  O  Isom  _E  ,  R  ( dom  O ,  ran  O ) )
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    C_ wss 3349   class class class wbr 4313    e. cmpt 4371    _E cep 4651    Po wpo 4660    Or wor 4661   Se wse 4698    We wwe 4699   Ord word 4739   Oncon0 4740   dom cdm 4861   ran crn 4862   "cima 4864   Fun wfun 5433   -->wf 5435   -onto->wfo 5437   ` cfv 5439    Isom wiso 5440   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-isom 5448  df-riota 6073  df-recs 6853  df-oi 7745
This theorem is referenced by:  ordtypelem9  7761  ordtypelem10  7762  oiiso2  7766
  Copyright terms: Public domain W3C validator