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Theorem dnwech 26293
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 26292 . . . 4  |-  ( ph  ->  ( x  e.  A  |-> 
|^| ( `' F " { x } ) ) : A -1-1-> On )
5 f1f1orn 5521 . . . 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 5475 . . . . 5  |-  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) : A -1-1-> On  ->  ( x  e.  A  |-> 
|^| ( `' F " { x } ) ) : A --> On )
8 frn 5433 . . . . 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 4612 . . . 4  |-  _E  We  On
11 wess 4417 . . . 4  |-  ( ran  ( x  e.  A  |-> 
|^| ( `' F " { x } ) )  C_  On  ->  (  _E  We  On  ->  _E  We  ran  ( x  e.  A  |->  |^| ( `' F " { x } ) ) ) )
129, 10, 11ee10 1367 . . 3  |-  ( ph  ->  _E  We  ran  (
x  e.  A  |->  |^| ( `' F " { x } ) ) )
13 eqid 2316 . . . 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 5892 . . 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 5577 . . . . . . . . 9  |-  ( ( x  e.  A  |->  |^| ( `' F " { x } ) ) `  w )  e.  _V
1716epelc 4344 . . . . . . . 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 26291 . . . . . . . . . 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 26291 . . . . . . . . . 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 2384 . . . . . . . 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 4119 . . . . 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 3395 . . . . . 6  |-  ( H  i^i  ( A  X.  A ) )  =  ( ( A  X.  A )  i^i  H
)
27 df-xp 4732 . . . . . . 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 3402 . . . . . 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 4853 . . . . . 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 2340 . . . . 5  |-  ( H  i^i  ( A  X.  A ) )  =  { <. v ,  w >.  |  ( ( v  e.  A  /\  w  e.  A )  /\  |^| ( `' F " { v } )  e.  |^| ( `' F " { w } ) ) }
32 incom 3395 . . . . . 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 3400 . . . . . 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 4853 . . . . . 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 2340 . . . . 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 2373 . . . 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 4418 . . . 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 4794 . . 3  |-  ( H  We  A  <->  ( H  i^i  ( A  X.  A
) )  We  A
)
40 weinxp 4794 . . 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 1633    e. wcel 1701    =/= wne 2479   A.wral 2577   _Vcvv 2822    \ cdif 3183    i^i cin 3185    C_ wss 3186   (/)c0 3489   ~Pcpw 3659   {csn 3674   |^|cint 3899   class class class wbr 4060   {copab 4113    e. cmpt 4114    _E cep 4340    We wwe 4388   Oncon0 4429    X. cxp 4724   `'ccnv 4725   ran crn 4727   "cima 4729   -->wf 5288   -1-1->wf1 5289   -1-1-onto->wf1o 5291   ` cfv 5292  recscrecs 6429
This theorem is referenced by:  aomclem3  26301
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-suc 4435  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-isom 5301  df-recs 6430
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