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Theorem dfac11 30984
Description: The right-hand side of this theorem (compare with ac4 8858), sometimes known as the "axiom of multiple choice", is a choice equivalent. Curiously, this statement cannot be proved without ax-reg 8021, 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 8505 . . 3  |-  (CHOICE  <->  A. a E. c A. d  e.  a  ( d  =/=  (/)  ->  ( c `  d )  e.  d ) )
2 raleq 3040 . . . . . 6  |-  ( a  =  x  ->  ( A. d  e.  a 
( d  =/=  (/)  ->  (
c `  d )  e.  d )  <->  A. d  e.  x  ( d  =/=  (/)  ->  ( c `  d )  e.  d ) ) )
32exbidv 1701 . . . . 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 2009 . . . 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 2724 . . . . . . . . . 10  |-  ( d  =  z  ->  (
d  =/=  (/)  <->  z  =/=  (/) ) )
6 fveq2 5856 . . . . . . . . . . 11  |-  ( d  =  z  ->  (
c `  d )  =  ( c `  z ) )
7 id 22 . . . . . . . . . . 11  |-  ( d  =  z  ->  d  =  z )
86, 7eleq12d 2525 . . . . . . . . . 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 3070 . . . . . . . 8  |-  ( A. d  e.  x  (
d  =/=  (/)  ->  (
c `  d )  e.  d )  <->  A. z  e.  x  ( z  =/=  (/)  ->  ( c `  z )  e.  z ) )
11 fveq2 5856 . . . . . . . . . . . . . . 15  |-  ( b  =  z  ->  (
c `  b )  =  ( c `  z ) )
1211sneqd 4026 . . . . . . . . . . . . . 14  |-  ( b  =  z  ->  { ( c `  b ) }  =  { ( c `  z ) } )
13 eqid 2443 . . . . . . . . . . . . . 14  |-  ( b  e.  x  |->  { ( c `  b ) } )  =  ( b  e.  x  |->  { ( c `  b
) } )
14 snex 4678 . . . . . . . . . . . . . 14  |-  { ( c `  z ) }  e.  _V
1512, 13, 14fvmpt 5941 . . . . . . . . . . . . 13  |-  ( z  e.  x  ->  (
( b  e.  x  |->  { ( c `  b ) } ) `
 z )  =  { ( c `  z ) } )
16153ad2ant1 1018 . . . . . . . . . . . 12  |-  ( ( z  e.  x  /\  z  =/=  (/)  /\  ( c `
 z )  e.  z )  ->  (
( b  e.  x  |->  { ( c `  b ) } ) `
 z )  =  { ( c `  z ) } )
17 simp3 999 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  x  /\  z  =/=  (/)  /\  ( c `
 z )  e.  z )  ->  (
c `  z )  e.  z )
1817snssd 4160 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  x  /\  z  =/=  (/)  /\  ( c `
 z )  e.  z )  ->  { ( c `  z ) }  C_  z )
1914elpw 4003 . . . . . . . . . . . . . . 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 7598 . . . . . . . . . . . . . . 15  |-  { ( c `  z ) }  e.  Fin
2221a1i 11 . . . . . . . . . . . . . 14  |-  ( ( z  e.  x  /\  z  =/=  (/)  /\  ( c `
 z )  e.  z )  ->  { ( c `  z ) }  e.  Fin )
2320, 22elind 3673 . . . . . . . . . . . . 13  |-  ( ( z  e.  x  /\  z  =/=  (/)  /\  ( c `
 z )  e.  z )  ->  { ( c `  z ) }  e.  ( ~P z  i^i  Fin )
)
24 fvex 5866 . . . . . . . . . . . . . . 15  |-  ( c `
 z )  e. 
_V
2524snnz 4133 . . . . . . . . . . . . . 14  |-  { ( c `  z ) }  =/=  (/)
2625a1i 11 . . . . . . . . . . . . 13  |-  ( ( z  e.  x  /\  z  =/=  (/)  /\  ( c `
 z )  e.  z )  ->  { ( c `  z ) }  =/=  (/) )
27 eldifsn 4140 . . . . . . . . . . . . 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 2531 . . . . . . . . . . 11  |-  ( ( z  e.  x  /\  z  =/=  (/)  /\  ( c `
 z )  e.  z )  ->  (
( b  e.  x  |->  { ( c `  b ) } ) `
 z )  e.  ( ( ~P z  i^i  Fin )  \  { (/)
} ) )
30293exp 1196 . . . . . . . . . 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 2834 . . . . . . . 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 3098 . . . . . . . . 9  |-  x  e. 
_V
3534mptex 6128 . . . . . . . 8  |-  ( b  e.  x  |->  { ( c `  b ) } )  e.  _V
36 fveq1 5855 . . . . . . . . . . 11  |-  ( f  =  ( b  e.  x  |->  { ( c `
 b ) } )  ->  ( f `  z )  =  ( ( b  e.  x  |->  { ( c `  b ) } ) `
 z ) )
3736eleq1d 2512 . . . . . . . . . 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 2882 . . . . . . . 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 3187 . . . . . . 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 1709 . . . . 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 1620 . . . 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 5866 . . . . . . 7  |-  ( R1
`  ( rank `  a
) )  e.  _V
4746pwex 4620 . . . . . 6  |-  ~P ( R1 `  ( rank `  a
) )  e.  _V
48 raleq 3040 . . . . . . 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 1701 . . . . . 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 3186 . . . . 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 8216 . . . . . . . 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 30983 . . . . . 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 1709 . . . . 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 3098 . . . . . 6  |-  a  e. 
_V
57 r1rankid 8280 . . . . . 6  |-  ( a  e.  _V  ->  a  C_  ( R1 `  ( rank `  a ) ) )
58 wess 4856 . . . . . . 7  |-  ( a 
C_  ( R1 `  ( rank `  a )
)  ->  ( b  We  ( R1 `  ( rank `  a ) )  ->  b  We  a
) )
5958eximdv 1697 . . . . . 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 1706 . . 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 8518 . . 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 974   A.wal 1381    = wceq 1383   E.wex 1599    e. wcel 1804    =/= wne 2638   A.wral 2793   _Vcvv 3095    \ cdif 3458    i^i cin 3460    C_ wss 3461   (/)c0 3770   ~Pcpw 3997   {csn 4014    |-> cmpt 4495    We wwe 4827   Oncon0 4868   ` cfv 5578   Fincfn 7518   R1cr1 8183   rankcrnk 8184  CHOICEwac 8499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-reg 8021  ax-inf2 8061
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-er 7313  df-map 7424  df-en 7519  df-fin 7522  df-sup 7903  df-r1 8185  df-rank 8186  df-card 8323  df-ac 8500
This theorem is referenced by: (None)
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