Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dnwech Structured version   Unicode version

Theorem dnwech 29572
Description: Define a well-ordering from a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
Hypotheses
Ref Expression
dnnumch.f  |-  F  = recs ( ( z  e. 
_V  |->  ( G `  ( A  \  ran  z
) ) ) )
dnnumch.a  |-  ( ph  ->  A  e.  V )
dnnumch.g  |-  ( ph  ->  A. y  e.  ~P  A ( y  =/=  (/)  ->  ( G `  y )  e.  y ) )
dnwech.h  |-  H  =  { <. v ,  w >.  |  |^| ( `' F " { v } )  e.  |^| ( `' F " { w } ) }
Assertion
Ref Expression
dnwech  |-  ( ph  ->  H  We  A )
Distinct variable groups:    v, F, w, y    v, G, w, y, z    v, A, w, y, z    ph, v, w
Allowed substitution hints:    ph( y, z)    F( z)    H( y, z, w, v)    V( y, z, w, v)

Proof of Theorem dnwech
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dnnumch.f . . . . 5  |-  F  = recs ( ( z  e. 
_V  |->  ( G `  ( A  \  ran  z
) ) ) )
2 dnnumch.a . . . . 5  |-  ( ph  ->  A  e.  V )
3 dnnumch.g . . . . 5  |-  ( ph  ->  A. y  e.  ~P  A ( y  =/=  (/)  ->  ( G `  y )  e.  y ) )
41, 2, 3dnnumch3 29571 . . . 4  |-  ( ph  ->  ( x  e.  A  |-> 
|^| ( `' F " { x } ) ) : A -1-1-> On )
5 f1f1orn 5763 . . . 4  |-  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) : A -1-1-> On  ->  ( x  e.  A  |-> 
|^| ( `' F " { x } ) ) : A -1-1-onto-> ran  (
x  e.  A  |->  |^| ( `' F " { x } ) ) )
64, 5syl 16 . . 3  |-  ( ph  ->  ( x  e.  A  |-> 
|^| ( `' F " { x } ) ) : A -1-1-onto-> ran  (
x  e.  A  |->  |^| ( `' F " { x } ) ) )
7 f1f 5717 . . . . 5  |-  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) : A -1-1-> On  ->  ( x  e.  A  |-> 
|^| ( `' F " { x } ) ) : A --> On )
8 frn 5676 . . . . 5  |-  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) : A --> On  ->  ran  ( x  e.  A  |-> 
|^| ( `' F " { x } ) )  C_  On )
94, 7, 83syl 20 . . . 4  |-  ( ph  ->  ran  ( x  e.  A  |->  |^| ( `' F " { x } ) )  C_  On )
10 epweon 6508 . . . 4  |-  _E  We  On
11 wess 4818 . . . 4  |-  ( ran  ( x  e.  A  |-> 
|^| ( `' F " { x } ) )  C_  On  ->  (  _E  We  On  ->  _E  We  ran  ( x  e.  A  |->  |^| ( `' F " { x } ) ) ) )
129, 10, 11mpisyl 18 . . 3  |-  ( ph  ->  _E  We  ran  (
x  e.  A  |->  |^| ( `' F " { x } ) ) )
13 eqid 2454 . . . 4  |-  { <. v ,  w >.  |  ( ( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  v )  _E  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w ) }  =  { <. v ,  w >.  |  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  v )  _E  (
( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  w ) }
1413f1owe 6156 . . 3  |-  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) : A -1-1-onto-> ran  (
x  e.  A  |->  |^| ( `' F " { x } ) )  ->  (  _E  We  ran  ( x  e.  A  |->  |^| ( `' F " { x } ) )  ->  { <. v ,  w >.  |  (
( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  v )  _E  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w ) }  We  A ) )
156, 12, 14sylc 60 . 2  |-  ( ph  ->  { <. v ,  w >.  |  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  v )  _E  (
( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  w ) }  We  A )
16 fvex 5812 . . . . . . . . 9  |-  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w )  e.  _V
1716epelc 4745 . . . . . . . 8  |-  ( ( ( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  v )  _E  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w )  <->  ( (
x  e.  A  |->  |^| ( `' F " { x } ) ) `  v )  e.  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w ) )
181, 2, 3dnnumch3lem 29570 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  A )  ->  (
( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  v )  =  |^| ( `' F " { v } ) )
1918adantrr 716 . . . . . . . . 9  |-  ( (
ph  /\  ( v  e.  A  /\  w  e.  A ) )  -> 
( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  v )  =  |^| ( `' F " { v } ) )
201, 2, 3dnnumch3lem 29570 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  A )  ->  (
( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  w )  =  |^| ( `' F " { w } ) )
2120adantrl 715 . . . . . . . . 9  |-  ( (
ph  /\  ( v  e.  A  /\  w  e.  A ) )  -> 
( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w )  =  |^| ( `' F " { w } ) )
2219, 21eleq12d 2536 . . . . . . . 8  |-  ( (
ph  /\  ( v  e.  A  /\  w  e.  A ) )  -> 
( ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  v )  e.  ( ( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  w )  <->  |^| ( `' F " { v } )  e.  |^| ( `' F " { w } ) ) )
2317, 22syl5rbb 258 . . . . . . 7  |-  ( (
ph  /\  ( v  e.  A  /\  w  e.  A ) )  -> 
( |^| ( `' F " { v } )  e.  |^| ( `' F " { w } )  <-> 
( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  v )  _E  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w ) ) )
2423pm5.32da 641 . . . . . 6  |-  ( ph  ->  ( ( ( v  e.  A  /\  w  e.  A )  /\  |^| ( `' F " { v } )  e.  |^| ( `' F " { w } ) )  <->  ( (
v  e.  A  /\  w  e.  A )  /\  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  v )  _E  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w ) ) ) )
2524opabbidv 4466 . . . . 5  |-  ( ph  ->  { <. v ,  w >.  |  ( ( v  e.  A  /\  w  e.  A )  /\  |^| ( `' F " { v } )  e.  |^| ( `' F " { w } ) ) }  =  { <. v ,  w >.  |  (
( v  e.  A  /\  w  e.  A
)  /\  ( (
x  e.  A  |->  |^| ( `' F " { x } ) ) `  v )  _E  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w ) ) } )
26 incom 3654 . . . . . 6  |-  ( H  i^i  ( A  X.  A ) )  =  ( ( A  X.  A )  i^i  H
)
27 df-xp 4957 . . . . . . 7  |-  ( A  X.  A )  =  { <. v ,  w >.  |  ( v  e.  A  /\  w  e.  A ) }
28 dnwech.h . . . . . . 7  |-  H  =  { <. v ,  w >.  |  |^| ( `' F " { v } )  e.  |^| ( `' F " { w } ) }
2927, 28ineq12i 3661 . . . . . 6  |-  ( ( A  X.  A )  i^i  H )  =  ( { <. v ,  w >.  |  (
v  e.  A  /\  w  e.  A ) }  i^i  { <. v ,  w >.  |  |^| ( `' F " { v } )  e.  |^| ( `' F " { w } ) } )
30 inopab 5081 . . . . . 6  |-  ( {
<. v ,  w >.  |  ( v  e.  A  /\  w  e.  A
) }  i^i  { <. v ,  w >.  | 
|^| ( `' F " { v } )  e.  |^| ( `' F " { w } ) } )  =  { <. v ,  w >.  |  ( ( v  e.  A  /\  w  e.  A )  /\  |^| ( `' F " { v } )  e.  |^| ( `' F " { w } ) ) }
3126, 29, 303eqtri 2487 . . . . 5  |-  ( H  i^i  ( A  X.  A ) )  =  { <. v ,  w >.  |  ( ( v  e.  A  /\  w  e.  A )  /\  |^| ( `' F " { v } )  e.  |^| ( `' F " { w } ) ) }
32 incom 3654 . . . . . 6  |-  ( {
<. v ,  w >.  |  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  v )  _E  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w ) }  i^i  ( A  X.  A
) )  =  ( ( A  X.  A
)  i^i  { <. v ,  w >.  |  (
( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  v )  _E  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w ) } )
3327ineq1i 3659 . . . . . 6  |-  ( ( A  X.  A )  i^i  { <. v ,  w >.  |  (
( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  v )  _E  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w ) } )  =  ( { <. v ,  w >.  |  ( v  e.  A  /\  w  e.  A ) }  i^i  { <. v ,  w >.  |  (
( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  v )  _E  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w ) } )
34 inopab 5081 . . . . . 6  |-  ( {
<. v ,  w >.  |  ( v  e.  A  /\  w  e.  A
) }  i^i  { <. v ,  w >.  |  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  v )  _E  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w ) } )  =  { <. v ,  w >.  |  (
( v  e.  A  /\  w  e.  A
)  /\  ( (
x  e.  A  |->  |^| ( `' F " { x } ) ) `  v )  _E  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w ) ) }
3532, 33, 343eqtri 2487 . . . . 5  |-  ( {
<. v ,  w >.  |  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  v )  _E  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w ) }  i^i  ( A  X.  A
) )  =  { <. v ,  w >.  |  ( ( v  e.  A  /\  w  e.  A )  /\  (
( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  v )  _E  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w ) ) }
3625, 31, 353eqtr4g 2520 . . . 4  |-  ( ph  ->  ( H  i^i  ( A  X.  A ) )  =  ( { <. v ,  w >.  |  ( ( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  v )  _E  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w ) }  i^i  ( A  X.  A
) ) )
37 weeq1 4819 . . . 4  |-  ( ( H  i^i  ( A  X.  A ) )  =  ( { <. v ,  w >.  |  ( ( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  v )  _E  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w ) }  i^i  ( A  X.  A
) )  ->  (
( H  i^i  ( A  X.  A ) )  We  A  <->  ( { <. v ,  w >.  |  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  v )  _E  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w ) }  i^i  ( A  X.  A
) )  We  A
) )
3836, 37syl 16 . . 3  |-  ( ph  ->  ( ( H  i^i  ( A  X.  A
) )  We  A  <->  ( { <. v ,  w >.  |  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  v )  _E  (
( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  w ) }  i^i  ( A  X.  A ) )  We  A ) )
39 weinxp 5017 . . 3  |-  ( H  We  A  <->  ( H  i^i  ( A  X.  A
) )  We  A
)
40 weinxp 5017 . . 3  |-  ( {
<. v ,  w >.  |  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  v )  _E  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w ) }  We  A 
<->  ( { <. v ,  w >.  |  (
( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  v )  _E  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w ) }  i^i  ( A  X.  A
) )  We  A
)
4138, 39, 403bitr4g 288 . 2  |-  ( ph  ->  ( H  We  A  <->  {
<. v ,  w >.  |  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  v )  _E  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w ) }  We  A ) )
4215, 41mpbird 232 1  |-  ( ph  ->  H  We  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   _Vcvv 3078    \ cdif 3436    i^i cin 3438    C_ wss 3439   (/)c0 3748   ~Pcpw 3971   {csn 3988   |^|cint 4239   class class class wbr 4403   {copab 4460    |-> cmpt 4461    _E cep 4741    We wwe 4789   Oncon0 4830    X. cxp 4949   `'ccnv 4950   ran crn 4952   "cima 4954   -->wf 5525   -1-1->wf1 5526   -1-1-onto->wf1o 5528   ` cfv 5529  recscrecs 6944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-recs 6945
This theorem is referenced by:  aomclem3  29580
  Copyright terms: Public domain W3C validator