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Theorem fpwwe 8236
Description: Given any function  F from the powerset of  A to  A, canth2 6982 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 7625. 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 5795 . . . . . 6  |-  ( Y ( F  o.  1st ) R )  =  ( ( F  o.  1st ) `  <. Y ,  R >. )
2 fo1st 6073 . . . . . . . 8  |-  1st : _V -onto-> _V
3 fofn 5391 . . . . . . . 8  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
42, 3ax-mp 10 . . . . . . 7  |-  1st  Fn  _V
5 opex 4209 . . . . . . 7  |-  <. Y ,  R >.  e.  _V
6 fvco2 5528 . . . . . . 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 2278 . . . . 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 4799 . . . . . . . 8  |-  Rel  W
11 brrelex12 4714 . . . . . . . 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 6066 . . . . . . 7  |-  ( ( Y  e.  _V  /\  R  e.  _V )  ->  ( 1st `  <. Y ,  R >. )  =  Y )
1412, 13syl 17 . . . . . 6  |-  ( Y W R  ->  ( 1st `  <. Y ,  R >. )  =  Y )
1514fveq2d 5462 . . . . 5  |-  ( Y W R  ->  ( F `  ( 1st ` 
<. Y ,  R >. ) )  =  ( F `
 Y ) )
168, 15syl5eq 2302 . . . 4  |-  ( Y W R  ->  ( Y ( F  o.  1st ) R )  =  ( F `  Y
) )
1716eleq1d 2324 . . 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 2766 . . . . . . . . . 10  |-  r  e. 
_V
20 cnvexg 5195 . . . . . . . . . 10  |-  ( r  e.  _V  ->  `' r  e.  _V )
21 imaexg 5014 . . . . . . . . . 10  |-  ( `' r  e.  _V  ->  ( `' r " {
y } )  e. 
_V )
2219, 20, 21mp2b 11 . . . . . . . . 9  |-  ( `' r " { y } )  e.  _V
23 vex 2766 . . . . . . . . . . . 12  |-  u  e. 
_V
2419inex1 4129 . . . . . . . . . . . 12  |-  ( r  i^i  ( u  X.  u ) )  e. 
_V
2523, 24algrflem 6158 . . . . . . . . . . 11  |-  ( u ( F  o.  1st ) ( r  i^i  ( u  X.  u
) ) )  =  ( F `  u
)
26 fveq2 5458 . . . . . . . . . . 11  |-  ( u  =  ( `' r
" { y } )  ->  ( F `  u )  =  ( F `  ( `' r " { y } ) ) )
2725, 26syl5eq 2302 . . . . . . . . . 10  |-  ( u  =  ( `' r
" { y } )  ->  ( u
( F  o.  1st ) ( r  i^i  ( u  X.  u
) ) )  =  ( F `  ( `' r " {
y } ) ) )
2827eqeq1d 2266 . . . . . . . . 9  |-  ( u  =  ( `' r
" { y } )  ->  ( (
u ( F  o.  1st ) ( r  i^i  ( u  X.  u
) ) )  =  y  <->  ( F `  ( `' r " {
y } ) )  =  y ) )
2922, 28sbcie 3000 . . . . . . . 8  |-  ( [. ( `' r " {
y } )  /  u ]. ( u ( F  o.  1st )
( r  i^i  (
u  X.  u ) ) )  =  y  <-> 
( F `  ( `' r " {
y } ) )  =  y )
3029ralbii 2542 . . . . . . 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 4057 . . . 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 2281 . . 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 2766 . . . . 5  |-  x  e. 
_V
3736, 19algrflem 6158 . . . 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 3605 . . . . . . 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 7630 . . . . . . 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 3333 . . . . . 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 2342 . . 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 8233 . 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 2518   _Vcvv 2763   [.wsbc 2966    i^i cin 3126    C_ wss 3127   ~Pcpw 3599   {csn 3614   <.cop 3617   U.cuni 3801   class class class wbr 3997   {copab 4050    We wwe 4323    X. cxp 4659   `'ccnv 4660   dom cdm 4661   "cima 4664    o. ccom 4665   Rel wrel 4666    Fn wfn 4668   -onto->wfo 4671   ` cfv 4673  (class class class)co 5792   1stc1st 6054   cardccrd 7536
This theorem is referenced by:  canth4  8237  canthnumlem  8238  canthp1lem2  8243
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 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-ov 5795  df-1st 6056  df-iota 6225  df-riota 6272  df-recs 6356  df-en 6832  df-oi 7193  df-card 7540
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