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Theorem dnwech 27145
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 27144 . . . 4  |-  ( ph  ->  ( x  e.  A  |-> 
|^| ( `' F " { x } ) ) : A -1-1-> On )
5 f1f1orn 5483 . . . 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 15 . . 3  |-  ( ph  ->  ( x  e.  A  |-> 
|^| ( `' F " { x } ) ) : A -1-1-onto-> ran  (
x  e.  A  |->  |^| ( `' F " { x } ) ) )
7 f1f 5437 . . . . 5  |-  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) : A -1-1-> On  ->  ( x  e.  A  |-> 
|^| ( `' F " { x } ) ) : A --> On )
8 frn 5395 . . . . 5  |-  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) : A --> On  ->  ran  ( x  e.  A  |-> 
|^| ( `' F " { x } ) )  C_  On )
94, 7, 83syl 18 . . . 4  |-  ( ph  ->  ran  ( x  e.  A  |->  |^| ( `' F " { x } ) )  C_  On )
10 epweon 4575 . . . 4  |-  _E  We  On
11 wess 4380 . . . 4  |-  ( ran  ( x  e.  A  |-> 
|^| ( `' F " { x } ) )  C_  On  ->  (  _E  We  On  ->  _E  We  ran  ( x  e.  A  |->  |^| ( `' F " { x } ) ) ) )
129, 10, 11ee10 1366 . . 3  |-  ( ph  ->  _E  We  ran  (
x  e.  A  |->  |^| ( `' F " { x } ) ) )
13 eqid 2283 . . . 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 5850 . . 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 56 . 2  |-  ( ph  ->  { <. v ,  w >.  |  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  v )  _E  (
( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  w ) }  We  A )
16 fvex 5539 . . . . . . . . 9  |-  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w )  e.  _V
1716epelc 4307 . . . . . . . 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 27143 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  A )  ->  (
( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  v )  =  |^| ( `' F " { v } ) )
1918adantrr 697 . . . . . . . . 9  |-  ( (
ph  /\  ( v  e.  A  /\  w  e.  A ) )  -> 
( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  v )  =  |^| ( `' F " { v } ) )
201, 2, 3dnnumch3lem 27143 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  A )  ->  (
( x  e.  A  |-> 
|^| ( `' F " { x } ) ) `  w )  =  |^| ( `' F " { w } ) )
2120adantrl 696 . . . . . . . . 9  |-  ( (
ph  /\  ( v  e.  A  /\  w  e.  A ) )  -> 
( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w )  =  |^| ( `' F " { w } ) )
2219, 21eleq12d 2351 . . . . . . . 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 249 . . . . . . 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 622 . . . . . 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 4082 . . . . 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 3361 . . . . . 6  |-  ( H  i^i  ( A  X.  A ) )  =  ( ( A  X.  A )  i^i  H
)
27 df-xp 4695 . . . . . . 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 3368 . . . . . 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 4816 . . . . . 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 2307 . . . . 5  |-  ( H  i^i  ( A  X.  A ) )  =  { <. v ,  w >.  |  ( ( v  e.  A  /\  w  e.  A )  /\  |^| ( `' F " { v } )  e.  |^| ( `' F " { w } ) ) }
32 incom 3361 . . . . . 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 3366 . . . . . 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 4816 . . . . . 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 2307 . . . . 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 2340 . . . 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 4381 . . . 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 15 . . 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 4757 . . 3  |-  ( H  We  A  <->  ( H  i^i  ( A  X.  A
) )  We  A
)
40 weinxp 4757 . . 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 279 . 2  |-  ( ph  ->  ( H  We  A  <->  {
<. v ,  w >.  |  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  v )  _E  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w ) }  We  A ) )
4215, 41mpbird 223 1  |-  ( ph  ->  H  We  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   _Vcvv 2788    \ cdif 3149    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   {csn 3640   |^|cint 3862   class class class wbr 4023   {copab 4076    e. cmpt 4077    _E cep 4303    We wwe 4351   Oncon0 4392    X. cxp 4687   `'ccnv 4688   ran crn 4690   "cima 4692   -->wf 5251   -1-1->wf1 5252   -1-1-onto->wf1o 5254   ` cfv 5255  recscrecs 6387
This theorem is referenced by:  aomclem3  27153
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-recs 6388
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