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Theorem fpwwe 8477
Description: Given any function  F from the powerset of  A to  A, canth2 7219 gives that the function is not injective, but we can say rather more than that. There is a unique well-ordered subset  <. X , 
( W `  X
) >. which "agrees" with  F in the sense that each initial segment maps to its upper bound, and such that the entire set maps to an element of the set (so that it cannot be extended without losing the well-ordering). This theorem can be used to prove dfac8a 7867. Theorem 1.1 of [KanamoriPincus] p. 415. (Contributed by Mario Carneiro, 18-May-2015.)
Hypotheses
Ref Expression
fpwwe.1  |-  W  =  { <. x ,  r
>.  |  ( (
x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  ( F `  ( `' r " { y } ) )  =  y ) ) }
fpwwe.2  |-  ( ph  ->  A  e.  _V )
fpwwe.3  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  dom  card ) )  ->  ( F `  x )  e.  A )
fpwwe.4  |-  X  = 
U. dom  W
Assertion
Ref Expression
fpwwe  |-  ( ph  ->  ( ( Y W R  /\  ( F `
 Y )  e.  Y )  <->  ( Y  =  X  /\  R  =  ( W `  X
) ) ) )
Distinct variable groups:    x, r, A    y, r, F, x    ph, r, x, y    R, r, x, y    X, r, x, y    Y, r, x, y    W, r, x, y
Allowed substitution hint:    A( y)

Proof of Theorem fpwwe
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 df-ov 6043 . . . . . 6  |-  ( Y ( F  o.  1st ) R )  =  ( ( F  o.  1st ) `  <. Y ,  R >. )
2 fo1st 6325 . . . . . . . 8  |-  1st : _V -onto-> _V
3 fofn 5614 . . . . . . . 8  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
42, 3ax-mp 8 . . . . . . 7  |-  1st  Fn  _V
5 opex 4387 . . . . . . 7  |-  <. Y ,  R >.  e.  _V
6 fvco2 5757 . . . . . . 7  |-  ( ( 1st  Fn  _V  /\  <. Y ,  R >.  e. 
_V )  ->  (
( F  o.  1st ) `  <. Y ,  R >. )  =  ( F `  ( 1st `  <. Y ,  R >. ) ) )
74, 5, 6mp2an 654 . . . . . 6  |-  ( ( F  o.  1st ) `  <. Y ,  R >. )  =  ( F `
 ( 1st `  <. Y ,  R >. )
)
81, 7eqtri 2424 . . . . 5  |-  ( Y ( F  o.  1st ) R )  =  ( F `  ( 1st `  <. Y ,  R >. ) )
9 fpwwe.1 . . . . . . . . 9  |-  W  =  { <. x ,  r
>.  |  ( (
x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  ( F `  ( `' r " { y } ) )  =  y ) ) }
109relopabi 4959 . . . . . . . 8  |-  Rel  W
11 brrelex12 4874 . . . . . . . 8  |-  ( ( Rel  W  /\  Y W R )  ->  ( Y  e.  _V  /\  R  e.  _V ) )
1210, 11mpan 652 . . . . . . 7  |-  ( Y W R  ->  ( Y  e.  _V  /\  R  e.  _V ) )
13 op1stg 6318 . . . . . . 7  |-  ( ( Y  e.  _V  /\  R  e.  _V )  ->  ( 1st `  <. Y ,  R >. )  =  Y )
1412, 13syl 16 . . . . . 6  |-  ( Y W R  ->  ( 1st `  <. Y ,  R >. )  =  Y )
1514fveq2d 5691 . . . . 5  |-  ( Y W R  ->  ( F `  ( 1st ` 
<. Y ,  R >. ) )  =  ( F `
 Y ) )
168, 15syl5eq 2448 . . . 4  |-  ( Y W R  ->  ( Y ( F  o.  1st ) R )  =  ( F `  Y
) )
1716eleq1d 2470 . . 3  |-  ( Y W R  ->  (
( Y ( F  o.  1st ) R )  e.  Y  <->  ( F `  Y )  e.  Y
) )
1817pm5.32i 619 . 2  |-  ( ( Y W R  /\  ( Y ( F  o.  1st ) R )  e.  Y )  <->  ( Y W R  /\  ( F `  Y )  e.  Y ) )
19 vex 2919 . . . . . . . . . 10  |-  r  e. 
_V
20 cnvexg 5364 . . . . . . . . . 10  |-  ( r  e.  _V  ->  `' r  e.  _V )
21 imaexg 5176 . . . . . . . . . 10  |-  ( `' r  e.  _V  ->  ( `' r " {
y } )  e. 
_V )
2219, 20, 21mp2b 10 . . . . . . . . 9  |-  ( `' r " { y } )  e.  _V
23 vex 2919 . . . . . . . . . . . 12  |-  u  e. 
_V
2419inex1 4304 . . . . . . . . . . . 12  |-  ( r  i^i  ( u  X.  u ) )  e. 
_V
2523, 24algrflem 6414 . . . . . . . . . . 11  |-  ( u ( F  o.  1st ) ( r  i^i  ( u  X.  u
) ) )  =  ( F `  u
)
26 fveq2 5687 . . . . . . . . . . 11  |-  ( u  =  ( `' r
" { y } )  ->  ( F `  u )  =  ( F `  ( `' r " { y } ) ) )
2725, 26syl5eq 2448 . . . . . . . . . 10  |-  ( u  =  ( `' r
" { y } )  ->  ( u
( F  o.  1st ) ( r  i^i  ( u  X.  u
) ) )  =  ( F `  ( `' r " {
y } ) ) )
2827eqeq1d 2412 . . . . . . . . 9  |-  ( u  =  ( `' r
" { y } )  ->  ( (
u ( F  o.  1st ) ( r  i^i  ( u  X.  u
) ) )  =  y  <->  ( F `  ( `' r " {
y } ) )  =  y ) )
2922, 28sbcie 3155 . . . . . . . 8  |-  ( [. ( `' r " {
y } )  /  u ]. ( u ( F  o.  1st )
( r  i^i  (
u  X.  u ) ) )  =  y  <-> 
( F `  ( `' r " {
y } ) )  =  y )
3029ralbii 2690 . . . . . . 7  |-  ( A. y  e.  x  [. ( `' r " {
y } )  /  u ]. ( u ( F  o.  1st )
( r  i^i  (
u  X.  u ) ) )  =  y  <->  A. y  e.  x  ( F `  ( `' r " { y } ) )  =  y )
3130anbi2i 676 . . . . . 6  |-  ( ( r  We  x  /\  A. y  e.  x  [. ( `' r " {
y } )  /  u ]. ( u ( F  o.  1st )
( r  i^i  (
u  X.  u ) ) )  =  y )  <->  ( r  We  x  /\  A. y  e.  x  ( F `  ( `' r " { y } ) )  =  y ) )
3231anbi2i 676 . . . . 5  |-  ( ( ( x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. ( u ( F  o.  1st ) ( r  i^i  ( u  X.  u ) ) )  =  y ) )  <->  ( ( x 
C_  A  /\  r  C_  ( x  X.  x
) )  /\  (
r  We  x  /\  A. y  e.  x  ( F `  ( `' r " { y } ) )  =  y ) ) )
3332opabbii 4232 . . . 4  |-  { <. x ,  r >.  |  ( ( x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. ( u ( F  o.  1st ) ( r  i^i  ( u  X.  u ) ) )  =  y ) ) }  =  { <. x ,  r >.  |  ( ( x 
C_  A  /\  r  C_  ( x  X.  x
) )  /\  (
r  We  x  /\  A. y  e.  x  ( F `  ( `' r " { y } ) )  =  y ) ) }
349, 33eqtr4i 2427 . . 3  |-  W  =  { <. x ,  r
>.  |  ( (
x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. (
u ( F  o.  1st ) ( r  i^i  ( u  X.  u
) ) )  =  y ) ) }
35 fpwwe.2 . . 3  |-  ( ph  ->  A  e.  _V )
36 vex 2919 . . . . 5  |-  x  e. 
_V
3736, 19algrflem 6414 . . . 4  |-  ( x ( F  o.  1st ) r )  =  ( F `  x
)
38 simp1 957 . . . . . . 7  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  x  C_  A )
3936elpw 3765 . . . . . . 7  |-  ( x  e.  ~P A  <->  x  C_  A
)
4038, 39sylibr 204 . . . . . 6  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  x  e.  ~P A )
41 19.8a 1758 . . . . . . . 8  |-  ( r  We  x  ->  E. r 
r  We  x )
42413ad2ant3 980 . . . . . . 7  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  E. r 
r  We  x )
43 ween 7872 . . . . . . 7  |-  ( x  e.  dom  card  <->  E. r 
r  We  x )
4442, 43sylibr 204 . . . . . 6  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  x  e.  dom  card )
45 elin 3490 . . . . . 6  |-  ( x  e.  ( ~P A  i^i  dom  card )  <->  ( x  e.  ~P A  /\  x  e.  dom  card ) )
4640, 44, 45sylanbrc 646 . . . . 5  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  x  e.  ( ~P A  i^i  dom 
card ) )
47 fpwwe.3 . . . . 5  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  dom  card ) )  ->  ( F `  x )  e.  A )
4846, 47sylan2 461 . . . 4  |-  ( (
ph  /\  ( x  C_  A  /\  r  C_  ( x  X.  x
)  /\  r  We  x ) )  -> 
( F `  x
)  e.  A )
4937, 48syl5eqel 2488 . . 3  |-  ( (
ph  /\  ( x  C_  A  /\  r  C_  ( x  X.  x
)  /\  r  We  x ) )  -> 
( x ( F  o.  1st ) r )  e.  A )
50 fpwwe.4 . . 3  |-  X  = 
U. dom  W
5134, 35, 49, 50fpwwe2 8474 . 2  |-  ( ph  ->  ( ( Y W R  /\  ( Y ( F  o.  1st ) R )  e.  Y
)  <->  ( Y  =  X  /\  R  =  ( W `  X
) ) ) )
5218, 51syl5bbr 251 1  |-  ( ph  ->  ( ( Y W R  /\  ( F `
 Y )  e.  Y )  <->  ( Y  =  X  /\  R  =  ( W `  X
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916   [.wsbc 3121    i^i cin 3279    C_ wss 3280   ~Pcpw 3759   {csn 3774   <.cop 3777   U.cuni 3975   class class class wbr 4172   {copab 4225    We wwe 4500    X. cxp 4835   `'ccnv 4836   dom cdm 4837   "cima 4840    o. ccom 4841   Rel wrel 4842    Fn wfn 5408   -onto->wfo 5411   ` cfv 5413  (class class class)co 6040   1stc1st 6306   cardccrd 7778
This theorem is referenced by:  canth4  8478  canthnumlem  8479  canthp1lem2  8484
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-1st 6308  df-riota 6508  df-recs 6592  df-en 7069  df-oi 7435  df-card 7782
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