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Theorem dnwech 27123
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 27122 . . . 4  |-  ( ph  ->  ( x  e.  A  |-> 
|^| ( `' F " { x } ) ) : A -1-1-> On )
5 f1f1orn 5685 . . . 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 5639 . . . . 5  |-  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) : A -1-1-> On  ->  ( x  e.  A  |-> 
|^| ( `' F " { x } ) ) : A --> On )
8 frn 5597 . . . . 5  |-  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) : A --> On  ->  ran  ( x  e.  A  |-> 
|^| ( `' F " { x } ) )  C_  On )
94, 7, 83syl 19 . . . 4  |-  ( ph  ->  ran  ( x  e.  A  |->  |^| ( `' F " { x } ) )  C_  On )
10 epweon 4764 . . . 4  |-  _E  We  On
11 wess 4569 . . . 4  |-  ( ran  ( x  e.  A  |-> 
|^| ( `' F " { x } ) )  C_  On  ->  (  _E  We  On  ->  _E  We  ran  ( x  e.  A  |->  |^| ( `' F " { x } ) ) ) )
129, 10, 11ee10 1385 . . 3  |-  ( ph  ->  _E  We  ran  (
x  e.  A  |->  |^| ( `' F " { x } ) ) )
13 eqid 2436 . . . 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 6073 . . 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 58 . 2  |-  ( ph  ->  { <. v ,  w >.  |  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  v )  _E  (
( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  w ) }  We  A )
16 fvex 5742 . . . . . . . . 9  |-  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w )  e.  _V
1716epelc 4496 . . . . . . . 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 27121 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  A )  ->  (
( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  v )  =  |^| ( `' F " { v } ) )
1918adantrr 698 . . . . . . . . 9  |-  ( (
ph  /\  ( v  e.  A  /\  w  e.  A ) )  -> 
( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  v )  =  |^| ( `' F " { v } ) )
201, 2, 3dnnumch3lem 27121 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  A )  ->  (
( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  w )  =  |^| ( `' F " { w } ) )
2120adantrl 697 . . . . . . . . 9  |-  ( (
ph  /\  ( v  e.  A  /\  w  e.  A ) )  -> 
( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w )  =  |^| ( `' F " { w } ) )
2219, 21eleq12d 2504 . . . . . . . 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 250 . . . . . . 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 623 . . . . . 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 4271 . . . . 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 3533 . . . . . 6  |-  ( H  i^i  ( A  X.  A ) )  =  ( ( A  X.  A )  i^i  H
)
27 df-xp 4884 . . . . . . 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 3540 . . . . . 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 5005 . . . . . 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 2460 . . . . 5  |-  ( H  i^i  ( A  X.  A ) )  =  { <. v ,  w >.  |  ( ( v  e.  A  /\  w  e.  A )  /\  |^| ( `' F " { v } )  e.  |^| ( `' F " { w } ) ) }
32 incom 3533 . . . . . 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 3538 . . . . . 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 5005 . . . . . 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 2460 . . . . 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 2493 . . . 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 4570 . . . 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 4945 . . 3  |-  ( H  We  A  <->  ( H  i^i  ( A  X.  A
) )  We  A
)
40 weinxp 4945 . . 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 280 . 2  |-  ( ph  ->  ( H  We  A  <->  {
<. v ,  w >.  |  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  v )  _E  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w ) }  We  A ) )
4215, 41mpbird 224 1  |-  ( ph  ->  H  We  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   _Vcvv 2956    \ cdif 3317    i^i cin 3319    C_ wss 3320   (/)c0 3628   ~Pcpw 3799   {csn 3814   |^|cint 4050   class class class wbr 4212   {copab 4265    e. cmpt 4266    _E cep 4492    We wwe 4540   Oncon0 4581    X. cxp 4876   `'ccnv 4877   ran crn 4879   "cima 4881   -->wf 5450   -1-1->wf1 5451   -1-1-onto->wf1o 5453   ` cfv 5454  recscrecs 6632
This theorem is referenced by:  aomclem3  27131
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-suc 4587  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-recs 6633
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