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Theorem fpwwe 8201
Description: Given any function  F from the powerset of  A to  A, canth2 6947 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 7590. 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 5760 . . . . . 6  |-  ( Y ( F  o.  1st ) R )  =  ( ( F  o.  1st ) `  <. Y ,  R >. )
2 fo1st 6038 . . . . . . . 8  |-  1st : _V -onto-> _V
3 fofn 5356 . . . . . . . 8  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
42, 3ax-mp 10 . . . . . . 7  |-  1st  Fn  _V
5 opex 4174 . . . . . . 7  |-  <. Y ,  R >.  e.  _V
6 fvco2 5493 . . . . . . 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 2276 . . . . 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 4764 . . . . . . . 8  |-  Rel  W
11 brrelex12 4679 . . . . . . . 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 6031 . . . . . . 7  |-  ( ( Y  e.  _V  /\  R  e.  _V )  ->  ( 1st `  <. Y ,  R >. )  =  Y )
1412, 13syl 17 . . . . . 6  |-  ( Y W R  ->  ( 1st `  <. Y ,  R >. )  =  Y )
1514fveq2d 5427 . . . . 5  |-  ( Y W R  ->  ( F `  ( 1st ` 
<. Y ,  R >. ) )  =  ( F `
 Y ) )
168, 15syl5eq 2300 . . . 4  |-  ( Y W R  ->  ( Y ( F  o.  1st ) R )  =  ( F `  Y
) )
1716eleq1d 2322 . . 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 2743 . . . . . . . . . 10  |-  r  e. 
_V
20 cnvexg 5160 . . . . . . . . . 10  |-  ( r  e.  _V  ->  `' r  e.  _V )
21 imaexg 4979 . . . . . . . . . 10  |-  ( `' r  e.  _V  ->  ( `' r " {
y } )  e. 
_V )
2219, 20, 21mp2b 11 . . . . . . . . 9  |-  ( `' r " { y } )  e.  _V
23 vex 2743 . . . . . . . . . . . 12  |-  u  e. 
_V
2419inex1 4095 . . . . . . . . . . . 12  |-  ( r  i^i  ( u  X.  u ) )  e. 
_V
2523, 24algrflem 6123 . . . . . . . . . . 11  |-  ( u ( F  o.  1st ) ( r  i^i  ( u  X.  u
) ) )  =  ( F `  u
)
26 fveq2 5423 . . . . . . . . . . 11  |-  ( u  =  ( `' r
" { y } )  ->  ( F `  u )  =  ( F `  ( `' r " { y } ) ) )
2725, 26syl5eq 2300 . . . . . . . . . 10  |-  ( u  =  ( `' r
" { y } )  ->  ( u
( F  o.  1st ) ( r  i^i  ( u  X.  u
) ) )  =  ( F `  ( `' r " {
y } ) ) )
2827eqeq1d 2264 . . . . . . . . 9  |-  ( u  =  ( `' r
" { y } )  ->  ( (
u ( F  o.  1st ) ( r  i^i  ( u  X.  u
) ) )  =  y  <->  ( F `  ( `' r " {
y } ) )  =  y ) )
2922, 28sbcie 2969 . . . . . . . 8  |-  ( [. ( `' r " {
y } )  /  u ]. ( u ( F  o.  1st )
( r  i^i  (
u  X.  u ) ) )  =  y  <-> 
( F `  ( `' r " {
y } ) )  =  y )
3029ralbii 2538 . . . . . . 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 4023 . . . 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 2279 . . 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 2743 . . . . 5  |-  x  e. 
_V
3736, 19algrflem 6123 . . . 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 3572 . . . . . . 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 7595 . . . . . . 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 3300 . . . . . 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 2340 . . 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 8198 . 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 2516   _Vcvv 2740   [.wsbc 2935    i^i cin 3093    C_ wss 3094   ~Pcpw 3566   {csn 3581   <.cop 3584   U.cuni 3768   class class class wbr 3963   {copab 4016    We wwe 4288    X. cxp 4624   `'ccnv 4625   dom cdm 4626   "cima 4629    o. ccom 4630   Rel wrel 4631    Fn wfn 4633   -onto->wfo 4636   ` cfv 4638  (class class class)co 5757   1stc1st 6019   cardccrd 7501
This theorem is referenced by:  canth4  8202  canthnumlem  8203  canthp1lem2  8208
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 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-ov 5760  df-1st 6021  df-iota 6190  df-riota 6237  df-recs 6321  df-en 6797  df-oi 7158  df-card 7505
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