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Theorem dnwech 30922
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 30921 . . . 4  |-  ( ph  ->  ( x  e.  A  |-> 
|^| ( `' F " { x } ) ) : A -1-1-> On )
5 f1f1orn 5833 . . . 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 5787 . . . . 5  |-  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) : A -1-1-> On  ->  ( x  e.  A  |-> 
|^| ( `' F " { x } ) ) : A --> On )
8 frn 5743 . . . . 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 6614 . . . 4  |-  _E  We  On
11 wess 4872 . . . 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 2467 . . . 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 6248 . . 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 5882 . . . . . . . . 9  |-  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w )  e.  _V
1716epelc 4799 . . . . . . . 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 30920 . . . . . . . . . 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 30920 . . . . . . . . . 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 2549 . . . . . . . 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 4516 . . . . 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 3696 . . . . . 6  |-  ( H  i^i  ( A  X.  A ) )  =  ( ( A  X.  A )  i^i  H
)
27 df-xp 5011 . . . . . . 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 3703 . . . . . 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 5139 . . . . . 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 2500 . . . . 5  |-  ( H  i^i  ( A  X.  A ) )  =  { <. v ,  w >.  |  ( ( v  e.  A  /\  w  e.  A )  /\  |^| ( `' F " { v } )  e.  |^| ( `' F " { w } ) ) }
32 incom 3696 . . . . . 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 3701 . . . . . 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 5139 . . . . . 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 2500 . . . . 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 2533 . . . 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 4873 . . . 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 5073 . . 3  |-  ( H  We  A  <->  ( H  i^i  ( A  X.  A
) )  We  A
)
40 weinxp 5073 . . 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 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   _Vcvv 3118    \ cdif 3478    i^i cin 3480    C_ wss 3481   (/)c0 3790   ~Pcpw 4016   {csn 4033   |^|cint 4288   class class class wbr 4453   {copab 4510    |-> cmpt 4511    _E cep 4795    We wwe 4843   Oncon0 4884    X. cxp 5003   `'ccnv 5004   ran crn 5006   "cima 5008   -->wf 5590   -1-1->wf1 5591   -1-1-onto->wf1o 5593   ` cfv 5594  recscrecs 7053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-recs 7054
This theorem is referenced by:  aomclem3  30930
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