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Theorem fpwwe 9013
Description: Given any function  F from the powerset of  A to  A, canth2 7663 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 8402. 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 6273 . . . . . 6  |-  ( Y ( F  o.  1st ) R )  =  ( ( F  o.  1st ) `  <. Y ,  R >. )
2 fo1st 6793 . . . . . . . 8  |-  1st : _V -onto-> _V
3 fofn 5779 . . . . . . . 8  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
42, 3ax-mp 5 . . . . . . 7  |-  1st  Fn  _V
5 opex 4701 . . . . . . 7  |-  <. Y ,  R >.  e.  _V
6 fvco2 5923 . . . . . . 7  |-  ( ( 1st  Fn  _V  /\  <. Y ,  R >.  e. 
_V )  ->  (
( F  o.  1st ) `  <. Y ,  R >. )  =  ( F `  ( 1st `  <. Y ,  R >. ) ) )
74, 5, 6mp2an 670 . . . . . 6  |-  ( ( F  o.  1st ) `  <. Y ,  R >. )  =  ( F `
 ( 1st `  <. Y ,  R >. )
)
81, 7eqtri 2483 . . . . 5  |-  ( Y ( F  o.  1st ) R )  =  ( F `  ( 1st `  <. Y ,  R >. ) )
9 fpwwe.1 . . . . . . . 8  |-  W  =  { <. x ,  r
>.  |  ( (
x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  ( F `  ( `' r " { y } ) )  =  y ) ) }
109bropaex12 5062 . . . . . . 7  |-  ( Y W R  ->  ( Y  e.  _V  /\  R  e.  _V ) )
11 op1stg 6785 . . . . . . 7  |-  ( ( Y  e.  _V  /\  R  e.  _V )  ->  ( 1st `  <. Y ,  R >. )  =  Y )
1210, 11syl 16 . . . . . 6  |-  ( Y W R  ->  ( 1st `  <. Y ,  R >. )  =  Y )
1312fveq2d 5852 . . . . 5  |-  ( Y W R  ->  ( F `  ( 1st ` 
<. Y ,  R >. ) )  =  ( F `
 Y ) )
148, 13syl5eq 2507 . . . 4  |-  ( Y W R  ->  ( Y ( F  o.  1st ) R )  =  ( F `  Y
) )
1514eleq1d 2523 . . 3  |-  ( Y W R  ->  (
( Y ( F  o.  1st ) R )  e.  Y  <->  ( F `  Y )  e.  Y
) )
1615pm5.32i 635 . 2  |-  ( ( Y W R  /\  ( Y ( F  o.  1st ) R )  e.  Y )  <->  ( Y W R  /\  ( F `  Y )  e.  Y ) )
17 vex 3109 . . . . . . . . . 10  |-  r  e. 
_V
18 cnvexg 6719 . . . . . . . . . 10  |-  ( r  e.  _V  ->  `' r  e.  _V )
19 imaexg 6710 . . . . . . . . . 10  |-  ( `' r  e.  _V  ->  ( `' r " {
y } )  e. 
_V )
2017, 18, 19mp2b 10 . . . . . . . . 9  |-  ( `' r " { y } )  e.  _V
21 vex 3109 . . . . . . . . . . . 12  |-  u  e. 
_V
2217inex1 4578 . . . . . . . . . . . 12  |-  ( r  i^i  ( u  X.  u ) )  e. 
_V
2321, 22algrflem 6882 . . . . . . . . . . 11  |-  ( u ( F  o.  1st ) ( r  i^i  ( u  X.  u
) ) )  =  ( F `  u
)
24 fveq2 5848 . . . . . . . . . . 11  |-  ( u  =  ( `' r
" { y } )  ->  ( F `  u )  =  ( F `  ( `' r " { y } ) ) )
2523, 24syl5eq 2507 . . . . . . . . . 10  |-  ( u  =  ( `' r
" { y } )  ->  ( u
( F  o.  1st ) ( r  i^i  ( u  X.  u
) ) )  =  ( F `  ( `' r " {
y } ) ) )
2625eqeq1d 2456 . . . . . . . . 9  |-  ( u  =  ( `' r
" { y } )  ->  ( (
u ( F  o.  1st ) ( r  i^i  ( u  X.  u
) ) )  =  y  <->  ( F `  ( `' r " {
y } ) )  =  y ) )
2720, 26sbcie 3359 . . . . . . . 8  |-  ( [. ( `' r " {
y } )  /  u ]. ( u ( F  o.  1st )
( r  i^i  (
u  X.  u ) ) )  =  y  <-> 
( F `  ( `' r " {
y } ) )  =  y )
2827ralbii 2885 . . . . . . 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 )
2928anbi2i 692 . . . . . 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 ) )
3029anbi2i 692 . . . . 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 ) ) )
3130opabbii 4503 . . . 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 ) ) }
329, 31eqtr4i 2486 . . 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 ) ) }
33 fpwwe.2 . . 3  |-  ( ph  ->  A  e.  _V )
34 vex 3109 . . . . 5  |-  x  e. 
_V
3534, 17algrflem 6882 . . . 4  |-  ( x ( F  o.  1st ) r )  =  ( F `  x
)
36 simp1 994 . . . . . . 7  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  x  C_  A )
37 selpw 4006 . . . . . . 7  |-  ( x  e.  ~P A  <->  x  C_  A
)
3836, 37sylibr 212 . . . . . 6  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  x  e.  ~P A )
39 19.8a 1862 . . . . . . . 8  |-  ( r  We  x  ->  E. r 
r  We  x )
40393ad2ant3 1017 . . . . . . 7  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  E. r 
r  We  x )
41 ween 8407 . . . . . . 7  |-  ( x  e.  dom  card  <->  E. r 
r  We  x )
4240, 41sylibr 212 . . . . . 6  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  x  e.  dom  card )
4338, 42elind 3674 . . . . 5  |-  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  ->  x  e.  ( ~P A  i^i  dom 
card ) )
44 fpwwe.3 . . . . 5  |-  ( (
ph  /\  x  e.  ( ~P A  i^i  dom  card ) )  ->  ( F `  x )  e.  A )
4543, 44sylan2 472 . . . 4  |-  ( (
ph  /\  ( x  C_  A  /\  r  C_  ( x  X.  x
)  /\  r  We  x ) )  -> 
( F `  x
)  e.  A )
4635, 45syl5eqel 2546 . . 3  |-  ( (
ph  /\  ( x  C_  A  /\  r  C_  ( x  X.  x
)  /\  r  We  x ) )  -> 
( x ( F  o.  1st ) r )  e.  A )
47 fpwwe.4 . . 3  |-  X  = 
U. dom  W
4832, 33, 46, 47fpwwe2 9010 . 2  |-  ( ph  ->  ( ( Y W R  /\  ( Y ( F  o.  1st ) R )  e.  Y
)  <->  ( Y  =  X  /\  R  =  ( W `  X
) ) ) )
4916, 48syl5bbr 259 1  |-  ( ph  ->  ( ( Y W R  /\  ( F `
 Y )  e.  Y )  <->  ( Y  =  X  /\  R  =  ( W `  X
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398   E.wex 1617    e. wcel 1823   A.wral 2804   _Vcvv 3106   [.wsbc 3324    i^i cin 3460    C_ wss 3461   ~Pcpw 3999   {csn 4016   <.cop 4022   U.cuni 4235   class class class wbr 4439   {copab 4496    We wwe 4826    X. cxp 4986   `'ccnv 4987   dom cdm 4988   "cima 4991    o. ccom 4992    Fn wfn 5565   -onto->wfo 5568   ` cfv 5570  (class class class)co 6270   1stc1st 6771   cardccrd 8307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-1st 6773  df-recs 7034  df-en 7510  df-oi 7927  df-card 8311
This theorem is referenced by:  canth4  9014  canthnumlem  9015  canthp1lem2  9020
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