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Theorem dfac11 30612
Description: The right-hand side of this theorem (compare with ac4 8851), sometimes known as the "axiom of multiple choice", is a choice equivalent. Curiously, this statement cannot be proved without ax-reg 8014, despite not mentioning the cumulative hierarchy in any way as most consequences of regularity do.

This is definition (MC) of [Schechter] p. 141. EDITORIAL: the proof is not original with me of course but I lost my reference sometime after writing it.

A multiple choice function allows any total order to be extended to a choice function, which in turn defines a well-ordering. Since a well ordering on a set defines a simple ordering of the power set, this allows the trivial well-ordering of the empty set to be transfinitely bootstrapped up the cumulative hierarchy to any desired level. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Stefan O'Rear, 1-Jun-2015.)

Assertion
Ref Expression
dfac11  |-  (CHOICE  <->  A. x E. f A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  ( ( ~P z  i^i 
Fin )  \  { (/)
} ) ) )
Distinct variable group:    x, z, f

Proof of Theorem dfac11
Dummy variables  a 
b  c  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfac3 8498 . . 3  |-  (CHOICE  <->  A. a E. c A. d  e.  a  ( d  =/=  (/)  ->  ( c `  d )  e.  d ) )
2 raleq 3058 . . . . . 6  |-  ( a  =  x  ->  ( A. d  e.  a 
( d  =/=  (/)  ->  (
c `  d )  e.  d )  <->  A. d  e.  x  ( d  =/=  (/)  ->  ( c `  d )  e.  d ) ) )
32exbidv 1690 . . . . 5  |-  ( a  =  x  ->  ( E. c A. d  e.  a  ( d  =/=  (/)  ->  ( c `  d )  e.  d )  <->  E. c A. d  e.  x  ( d  =/=  (/)  ->  ( c `  d )  e.  d ) ) )
43cbvalv 1996 . . . 4  |-  ( A. a E. c A. d  e.  a  ( d  =/=  (/)  ->  ( c `  d )  e.  d )  <->  A. x E. c A. d  e.  x  ( d  =/=  (/)  ->  (
c `  d )  e.  d ) )
5 neeq1 2748 . . . . . . . . . 10  |-  ( d  =  z  ->  (
d  =/=  (/)  <->  z  =/=  (/) ) )
6 fveq2 5864 . . . . . . . . . . 11  |-  ( d  =  z  ->  (
c `  d )  =  ( c `  z ) )
7 id 22 . . . . . . . . . . 11  |-  ( d  =  z  ->  d  =  z )
86, 7eleq12d 2549 . . . . . . . . . 10  |-  ( d  =  z  ->  (
( c `  d
)  e.  d  <->  ( c `  z )  e.  z ) )
95, 8imbi12d 320 . . . . . . . . 9  |-  ( d  =  z  ->  (
( d  =/=  (/)  ->  (
c `  d )  e.  d )  <->  ( z  =/=  (/)  ->  ( c `  z )  e.  z ) ) )
109cbvralv 3088 . . . . . . . 8  |-  ( A. d  e.  x  (
d  =/=  (/)  ->  (
c `  d )  e.  d )  <->  A. z  e.  x  ( z  =/=  (/)  ->  ( c `  z )  e.  z ) )
11 fveq2 5864 . . . . . . . . . . . . . . 15  |-  ( b  =  z  ->  (
c `  b )  =  ( c `  z ) )
1211sneqd 4039 . . . . . . . . . . . . . 14  |-  ( b  =  z  ->  { ( c `  b ) }  =  { ( c `  z ) } )
13 eqid 2467 . . . . . . . . . . . . . 14  |-  ( b  e.  x  |->  { ( c `  b ) } )  =  ( b  e.  x  |->  { ( c `  b
) } )
14 snex 4688 . . . . . . . . . . . . . 14  |-  { ( c `  z ) }  e.  _V
1512, 13, 14fvmpt 5948 . . . . . . . . . . . . 13  |-  ( z  e.  x  ->  (
( b  e.  x  |->  { ( c `  b ) } ) `
 z )  =  { ( c `  z ) } )
16153ad2ant1 1017 . . . . . . . . . . . 12  |-  ( ( z  e.  x  /\  z  =/=  (/)  /\  ( c `
 z )  e.  z )  ->  (
( b  e.  x  |->  { ( c `  b ) } ) `
 z )  =  { ( c `  z ) } )
17 simp3 998 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  x  /\  z  =/=  (/)  /\  ( c `
 z )  e.  z )  ->  (
c `  z )  e.  z )
1817snssd 4172 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  x  /\  z  =/=  (/)  /\  ( c `
 z )  e.  z )  ->  { ( c `  z ) }  C_  z )
1914elpw 4016 . . . . . . . . . . . . . . 15  |-  ( { ( c `  z
) }  e.  ~P z 
<->  { ( c `  z ) }  C_  z )
2018, 19sylibr 212 . . . . . . . . . . . . . 14  |-  ( ( z  e.  x  /\  z  =/=  (/)  /\  ( c `
 z )  e.  z )  ->  { ( c `  z ) }  e.  ~P z
)
21 snfi 7593 . . . . . . . . . . . . . . 15  |-  { ( c `  z ) }  e.  Fin
2221a1i 11 . . . . . . . . . . . . . 14  |-  ( ( z  e.  x  /\  z  =/=  (/)  /\  ( c `
 z )  e.  z )  ->  { ( c `  z ) }  e.  Fin )
2320, 22elind 3688 . . . . . . . . . . . . 13  |-  ( ( z  e.  x  /\  z  =/=  (/)  /\  ( c `
 z )  e.  z )  ->  { ( c `  z ) }  e.  ( ~P z  i^i  Fin )
)
24 fvex 5874 . . . . . . . . . . . . . . 15  |-  ( c `
 z )  e. 
_V
2524snnz 4145 . . . . . . . . . . . . . 14  |-  { ( c `  z ) }  =/=  (/)
2625a1i 11 . . . . . . . . . . . . 13  |-  ( ( z  e.  x  /\  z  =/=  (/)  /\  ( c `
 z )  e.  z )  ->  { ( c `  z ) }  =/=  (/) )
27 eldifsn 4152 . . . . . . . . . . . . 13  |-  ( { ( c `  z
) }  e.  ( ( ~P z  i^i 
Fin )  \  { (/)
} )  <->  ( {
( c `  z
) }  e.  ( ~P z  i^i  Fin )  /\  { ( c `
 z ) }  =/=  (/) ) )
2823, 26, 27sylanbrc 664 . . . . . . . . . . . 12  |-  ( ( z  e.  x  /\  z  =/=  (/)  /\  ( c `
 z )  e.  z )  ->  { ( c `  z ) }  e.  ( ( ~P z  i^i  Fin )  \  { (/) } ) )
2916, 28eqeltrd 2555 . . . . . . . . . . 11  |-  ( ( z  e.  x  /\  z  =/=  (/)  /\  ( c `
 z )  e.  z )  ->  (
( b  e.  x  |->  { ( c `  b ) } ) `
 z )  e.  ( ( ~P z  i^i  Fin )  \  { (/)
} ) )
30293exp 1195 . . . . . . . . . 10  |-  ( z  e.  x  ->  (
z  =/=  (/)  ->  (
( c `  z
)  e.  z  -> 
( ( b  e.  x  |->  { ( c `
 b ) } ) `  z )  e.  ( ( ~P z  i^i  Fin )  \  { (/) } ) ) ) )
3130a2d 26 . . . . . . . . 9  |-  ( z  e.  x  ->  (
( z  =/=  (/)  ->  (
c `  z )  e.  z )  ->  (
z  =/=  (/)  ->  (
( b  e.  x  |->  { ( c `  b ) } ) `
 z )  e.  ( ( ~P z  i^i  Fin )  \  { (/)
} ) ) ) )
3231ralimia 2855 . . . . . . . 8  |-  ( A. z  e.  x  (
z  =/=  (/)  ->  (
c `  z )  e.  z )  ->  A. z  e.  x  ( z  =/=  (/)  ->  ( (
b  e.  x  |->  { ( c `  b
) } ) `  z )  e.  ( ( ~P z  i^i 
Fin )  \  { (/)
} ) ) )
3310, 32sylbi 195 . . . . . . 7  |-  ( A. d  e.  x  (
d  =/=  (/)  ->  (
c `  d )  e.  d )  ->  A. z  e.  x  ( z  =/=  (/)  ->  ( (
b  e.  x  |->  { ( c `  b
) } ) `  z )  e.  ( ( ~P z  i^i 
Fin )  \  { (/)
} ) ) )
34 vex 3116 . . . . . . . . 9  |-  x  e. 
_V
3534mptex 6129 . . . . . . . 8  |-  ( b  e.  x  |->  { ( c `  b ) } )  e.  _V
36 fveq1 5863 . . . . . . . . . . 11  |-  ( f  =  ( b  e.  x  |->  { ( c `
 b ) } )  ->  ( f `  z )  =  ( ( b  e.  x  |->  { ( c `  b ) } ) `
 z ) )
3736eleq1d 2536 . . . . . . . . . 10  |-  ( f  =  ( b  e.  x  |->  { ( c `
 b ) } )  ->  ( (
f `  z )  e.  ( ( ~P z  i^i  Fin )  \  { (/)
} )  <->  ( (
b  e.  x  |->  { ( c `  b
) } ) `  z )  e.  ( ( ~P z  i^i 
Fin )  \  { (/)
} ) ) )
3837imbi2d 316 . . . . . . . . 9  |-  ( f  =  ( b  e.  x  |->  { ( c `
 b ) } )  ->  ( (
z  =/=  (/)  ->  (
f `  z )  e.  ( ( ~P z  i^i  Fin )  \  { (/)
} ) )  <->  ( z  =/=  (/)  ->  ( (
b  e.  x  |->  { ( c `  b
) } ) `  z )  e.  ( ( ~P z  i^i 
Fin )  \  { (/)
} ) ) ) )
3938ralbidv 2903 . . . . . . . 8  |-  ( f  =  ( b  e.  x  |->  { ( c `
 b ) } )  ->  ( A. z  e.  x  (
z  =/=  (/)  ->  (
f `  z )  e.  ( ( ~P z  i^i  Fin )  \  { (/)
} ) )  <->  A. z  e.  x  ( z  =/=  (/)  ->  ( (
b  e.  x  |->  { ( c `  b
) } ) `  z )  e.  ( ( ~P z  i^i 
Fin )  \  { (/)
} ) ) ) )
4035, 39spcev 3205 . . . . . . 7  |-  ( A. z  e.  x  (
z  =/=  (/)  ->  (
( b  e.  x  |->  { ( c `  b ) } ) `
 z )  e.  ( ( ~P z  i^i  Fin )  \  { (/)
} ) )  ->  E. f A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  ( ( ~P z  i^i 
Fin )  \  { (/)
} ) ) )
4133, 40syl 16 . . . . . 6  |-  ( A. d  e.  x  (
d  =/=  (/)  ->  (
c `  d )  e.  d )  ->  E. f A. z  e.  x  ( z  =/=  (/)  ->  (
f `  z )  e.  ( ( ~P z  i^i  Fin )  \  { (/)
} ) ) )
4241exlimiv 1698 . . . . 5  |-  ( E. c A. d  e.  x  ( d  =/=  (/)  ->  ( c `  d )  e.  d )  ->  E. f A. z  e.  x  ( z  =/=  (/)  ->  (
f `  z )  e.  ( ( ~P z  i^i  Fin )  \  { (/)
} ) ) )
4342alimi 1614 . . . 4  |-  ( A. x E. c A. d  e.  x  ( d  =/=  (/)  ->  ( c `  d )  e.  d )  ->  A. x E. f A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  ( ( ~P z  i^i 
Fin )  \  { (/)
} ) ) )
444, 43sylbi 195 . . 3  |-  ( A. a E. c A. d  e.  a  ( d  =/=  (/)  ->  ( c `  d )  e.  d )  ->  A. x E. f A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  ( ( ~P z  i^i 
Fin )  \  { (/)
} ) ) )
451, 44sylbi 195 . 2  |-  (CHOICE  ->  A. x E. f A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  ( ( ~P z  i^i 
Fin )  \  { (/)
} ) ) )
46 fvex 5874 . . . . . . 7  |-  ( R1
`  ( rank `  a
) )  e.  _V
4746pwex 4630 . . . . . 6  |-  ~P ( R1 `  ( rank `  a
) )  e.  _V
48 raleq 3058 . . . . . . 7  |-  ( x  =  ~P ( R1
`  ( rank `  a
) )  ->  ( A. z  e.  x  ( z  =/=  (/)  ->  (
f `  z )  e.  ( ( ~P z  i^i  Fin )  \  { (/)
} ) )  <->  A. z  e.  ~P  ( R1 `  ( rank `  a )
) ( z  =/=  (/)  ->  ( f `  z )  e.  ( ( ~P z  i^i 
Fin )  \  { (/)
} ) ) ) )
4948exbidv 1690 . . . . . 6  |-  ( x  =  ~P ( R1
`  ( rank `  a
) )  ->  ( E. f A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  ( ( ~P z  i^i 
Fin )  \  { (/)
} ) )  <->  E. f A. z  e.  ~P  ( R1 `  ( rank `  a ) ) ( z  =/=  (/)  ->  (
f `  z )  e.  ( ( ~P z  i^i  Fin )  \  { (/)
} ) ) ) )
5047, 49spcv 3204 . . . . 5  |-  ( A. x E. f A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  ( ( ~P z  i^i 
Fin )  \  { (/)
} ) )  ->  E. f A. z  e. 
~P  ( R1 `  ( rank `  a )
) ( z  =/=  (/)  ->  ( f `  z )  e.  ( ( ~P z  i^i 
Fin )  \  { (/)
} ) ) )
51 rankon 8209 . . . . . . . 8  |-  ( rank `  a )  e.  On
5251a1i 11 . . . . . . 7  |-  ( A. z  e.  ~P  ( R1 `  ( rank `  a
) ) ( z  =/=  (/)  ->  ( f `  z )  e.  ( ( ~P z  i^i 
Fin )  \  { (/)
} ) )  -> 
( rank `  a )  e.  On )
53 id 22 . . . . . . 7  |-  ( A. z  e.  ~P  ( R1 `  ( rank `  a
) ) ( z  =/=  (/)  ->  ( f `  z )  e.  ( ( ~P z  i^i 
Fin )  \  { (/)
} ) )  ->  A. z  e.  ~P  ( R1 `  ( rank `  a ) ) ( z  =/=  (/)  ->  (
f `  z )  e.  ( ( ~P z  i^i  Fin )  \  { (/)
} ) ) )
5452, 53aomclem8 30611 . . . . . 6  |-  ( A. z  e.  ~P  ( R1 `  ( rank `  a
) ) ( z  =/=  (/)  ->  ( f `  z )  e.  ( ( ~P z  i^i 
Fin )  \  { (/)
} ) )  ->  E. b  b  We  ( R1 `  ( rank `  a ) ) )
5554exlimiv 1698 . . . . 5  |-  ( E. f A. z  e. 
~P  ( R1 `  ( rank `  a )
) ( z  =/=  (/)  ->  ( f `  z )  e.  ( ( ~P z  i^i 
Fin )  \  { (/)
} ) )  ->  E. b  b  We  ( R1 `  ( rank `  a ) ) )
56 vex 3116 . . . . . 6  |-  a  e. 
_V
57 r1rankid 8273 . . . . . 6  |-  ( a  e.  _V  ->  a  C_  ( R1 `  ( rank `  a ) ) )
58 wess 4866 . . . . . . 7  |-  ( a 
C_  ( R1 `  ( rank `  a )
)  ->  ( b  We  ( R1 `  ( rank `  a ) )  ->  b  We  a
) )
5958eximdv 1686 . . . . . 6  |-  ( a 
C_  ( R1 `  ( rank `  a )
)  ->  ( E. b  b  We  ( R1 `  ( rank `  a
) )  ->  E. b 
b  We  a ) )
6056, 57, 59mp2b 10 . . . . 5  |-  ( E. b  b  We  ( R1 `  ( rank `  a
) )  ->  E. b 
b  We  a )
6150, 55, 603syl 20 . . . 4  |-  ( A. x E. f A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  ( ( ~P z  i^i 
Fin )  \  { (/)
} ) )  ->  E. b  b  We  a )
6261alrimiv 1695 . . 3  |-  ( A. x E. f A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  ( ( ~P z  i^i 
Fin )  \  { (/)
} ) )  ->  A. a E. b  b  We  a )
63 dfac8 8511 . . 3  |-  (CHOICE  <->  A. a E. b  b  We  a )
6462, 63sylibr 212 . 2  |-  ( A. x E. f A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  ( ( ~P z  i^i 
Fin )  \  { (/)
} ) )  -> CHOICE )
6545, 64impbii 188 1  |-  (CHOICE  <->  A. x E. f A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  ( ( ~P z  i^i 
Fin )  \  { (/)
} ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 973   A.wal 1377    = wceq 1379   E.wex 1596    e. wcel 1767    =/= wne 2662   A.wral 2814   _Vcvv 3113    \ cdif 3473    i^i cin 3475    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   {csn 4027    |-> cmpt 4505    We wwe 4837   Oncon0 4878   ` cfv 5586   Fincfn 7513   R1cr1 8176   rankcrnk 8177  CHOICEwac 8492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-reg 8014  ax-inf2 8054
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-er 7308  df-map 7419  df-en 7514  df-fin 7517  df-sup 7897  df-r1 8178  df-rank 8179  df-card 8316  df-ac 8493
This theorem is referenced by: (None)
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