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Theorem fpwwe 8148
Description: Given any function  F from the powerset of  A to  A, canth2 6899 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 7541. 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
StepHypRef Expression
1 df-ov 5713 . . . . . 6  |-  ( Y ( F  o.  1st ) R )  =  ( ( F  o.  1st ) `  <. Y ,  R >. )
2 fo1st 5991 . . . . . . . 8  |-  1st : _V -onto-> _V
3 fofn 5310 . . . . . . . 8  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
42, 3ax-mp 10 . . . . . . 7  |-  1st  Fn  _V
5 opex 4130 . . . . . . 7  |-  <. Y ,  R >.  e.  _V
6 fvco2 5446 . . . . . . 7  |-  ( ( 1st  Fn  _V  /\  <. Y ,  R >.  e. 
_V )  ->  (
( F  o.  1st ) `  <. Y ,  R >. )  =  ( F `  ( 1st `  <. Y ,  R >. ) ) )
74, 5, 6mp2an 656 . . . . . 6  |-  ( ( F  o.  1st ) `  <. Y ,  R >. )  =  ( F `
 ( 1st `  <. Y ,  R >. )
)
81, 7eqtri 2273 . . . . 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 4718 . . . . . . . 8  |-  Rel  W
11 brrelex12 4633 . . . . . . . 8  |-  ( ( Rel  W  /\  Y W R )  ->  ( Y  e.  _V  /\  R  e.  _V ) )
1210, 11mpan 654 . . . . . . 7  |-  ( Y W R  ->  ( Y  e.  _V  /\  R  e.  _V ) )
13 op1stg 5984 . . . . . . 7  |-  ( ( Y  e.  _V  /\  R  e.  _V )  ->  ( 1st `  <. Y ,  R >. )  =  Y )
1412, 13syl 17 . . . . . 6  |-  ( Y W R  ->  ( 1st `  <. Y ,  R >. )  =  Y )
1514fveq2d 5381 . . . . 5  |-  ( Y W R  ->  ( F `  ( 1st ` 
<. Y ,  R >. ) )  =  ( F `
 Y ) )
168, 15syl5eq 2297 . . . 4  |-  ( Y W R  ->  ( Y ( F  o.  1st ) R )  =  ( F `  Y
) )
1716eleq1d 2319 . . 3  |-  ( Y W R  ->  (
( Y ( F  o.  1st ) R )  e.  Y  <->  ( F `  Y )  e.  Y
) )
1817pm5.32i 621 . 2  |-  ( ( Y W R  /\  ( Y ( F  o.  1st ) R )  e.  Y )  <->  ( Y W R  /\  ( F `  Y )  e.  Y ) )
19 vex 2730 . . . . . . . . . 10  |-  r  e. 
_V
20 cnvexg 5114 . . . . . . . . . 10  |-  ( r  e.  _V  ->  `' r  e.  _V )
21 imaexg 4933 . . . . . . . . . 10  |-  ( `' r  e.  _V  ->  ( `' r " {
y } )  e. 
_V )
2219, 20, 21mp2b 11 . . . . . . . . 9  |-  ( `' r " { y } )  e.  _V
23 vex 2730 . . . . . . . . . . . 12  |-  u  e. 
_V
2419inex1 4052 . . . . . . . . . . . 12  |-  ( r  i^i  ( u  X.  u ) )  e. 
_V
2523, 24algrflem 6076 . . . . . . . . . . 11  |-  ( u ( F  o.  1st ) ( r  i^i  ( u  X.  u
) ) )  =  ( F `  u
)
26 fveq2 5377 . . . . . . . . . . 11  |-  ( u  =  ( `' r
" { y } )  ->  ( F `  u )  =  ( F `  ( `' r " { y } ) ) )
2725, 26syl5eq 2297 . . . . . . . . . 10  |-  ( u  =  ( `' r
" { y } )  ->  ( u
( F  o.  1st ) ( r  i^i  ( u  X.  u
) ) )  =  ( F `  ( `' r " {
y } ) ) )
2827eqeq1d 2261 . . . . . . . . 9  |-  ( u  =  ( `' r
" { y } )  ->  ( (
u ( F  o.  1st ) ( r  i^i  ( u  X.  u
) ) )  =  y  <->  ( F `  ( `' r " {
y } ) )  =  y ) )
2922, 28sbcie 2955 . . . . . . . 8  |-  ( [. ( `' r " {
y } )  /  u ]. ( u ( F  o.  1st )
( r  i^i  (
u  X.  u ) ) )  =  y  <-> 
( F `  ( `' r " {
y } ) )  =  y )
3029ralbii 2531 . . . . . . 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 678 . . . . . 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 678 . . . . 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 3980 . . . 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 2276 . . 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 2730 . . . . 5  |-  x  e. 
_V
3736, 19algrflem 6076 . . . 4  |-  ( x ( F  o.  1st ) r )  =  ( F `  x
)
38 simp1 960 . . . . . . 7  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  x  C_  A )
3936elpw 3536 . . . . . . 7  |-  ( x  e.  ~P A  <->  x  C_  A
)
4038, 39sylibr 205 . . . . . 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 983 . . . . . . 7  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  E. r 
r  We  x )
43 ween 7546 . . . . . . 7  |-  ( x  e.  dom  card  <->  E. r 
r  We  x )
4442, 43sylibr 205 . . . . . 6  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  x  e.  dom  card )
45 elin 3266 . . . . . 6  |-  ( x  e.  ( ~P A  i^i  dom  card )  <->  ( x  e.  ~P A  /\  x  e.  dom  card ) )
4640, 44, 45sylanbrc 648 . . . . 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 462 . . . 4  |-  ( (
ph  /\  ( x  C_  A  /\  r  C_  ( x  X.  x
)  /\  r  We  x ) )  -> 
( F `  x
)  e.  A )
4937, 48syl5eqel 2337 . . 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 8145 . 2  |-  ( ph  ->  ( ( Y W R  /\  ( Y ( F  o.  1st ) R )  e.  Y
)  <->  ( Y  =  X  /\  R  =  ( W `  X
) ) ) )
5218, 51syl5bbr 252 1  |-  ( ph  ->  ( ( Y W R  /\  ( F `
 Y )  e.  Y )  <->  ( Y  =  X  /\  R  =  ( W `  X
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939   E.wex 1537    = wceq 1619    e. wcel 1621   A.wral 2509   _Vcvv 2727   [.wsbc 2921    i^i cin 3077    C_ wss 3078   ~Pcpw 3530   {csn 3544   <.cop 3547   U.cuni 3727   class class class wbr 3920   {copab 3973    We wwe 4244    X. cxp 4578   `'ccnv 4579   dom cdm 4580   "cima 4583    o. ccom 4584   Rel wrel 4585    Fn wfn 4587   -onto->wfo 4590   ` cfv 4592  (class class class)co 5710   1stc1st 5972   cardccrd 7452
This theorem is referenced by:  canth4  8149  canthnumlem  8150  canthp1lem2  8155
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-1st 5974  df-iota 6143  df-riota 6190  df-recs 6274  df-en 6750  df-oi 7109  df-card 7456
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