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Theorem dfac11 27171
Description: The right-hand side of this theorem (compare with ac4 8104), sometimes known as the "axiom of multiple choice", is a choice equivalent. Curiously, this statement cannot be proved without ax-reg 7308, 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 7750 . . 3  |-  (CHOICE  <->  A. a E. c A. d  e.  a  ( d  =/=  (/)  ->  ( c `  d )  e.  d ) )
2 raleq 2738 . . . . . 6  |-  ( a  =  x  ->  ( A. d  e.  a 
( d  =/=  (/)  ->  (
c `  d )  e.  d )  <->  A. d  e.  x  ( d  =/=  (/)  ->  ( c `  d )  e.  d ) ) )
32exbidv 1614 . . . . 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 1944 . . . 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 2456 . . . . . . . . . 10  |-  ( d  =  z  ->  (
d  =/=  (/)  <->  z  =/=  (/) ) )
6 fveq2 5527 . . . . . . . . . . 11  |-  ( d  =  z  ->  (
c `  d )  =  ( c `  z ) )
7 id 19 . . . . . . . . . . 11  |-  ( d  =  z  ->  d  =  z )
86, 7eleq12d 2353 . . . . . . . . . 10  |-  ( d  =  z  ->  (
( c `  d
)  e.  d  <->  ( c `  z )  e.  z ) )
95, 8imbi12d 311 . . . . . . . . 9  |-  ( d  =  z  ->  (
( d  =/=  (/)  ->  (
c `  d )  e.  d )  <->  ( z  =/=  (/)  ->  ( c `  z )  e.  z ) ) )
109cbvralv 2766 . . . . . . . 8  |-  ( A. d  e.  x  (
d  =/=  (/)  ->  (
c `  d )  e.  d )  <->  A. z  e.  x  ( z  =/=  (/)  ->  ( c `  z )  e.  z ) )
11 fveq2 5527 . . . . . . . . . . . . . . 15  |-  ( b  =  z  ->  (
c `  b )  =  ( c `  z ) )
1211sneqd 3655 . . . . . . . . . . . . . 14  |-  ( b  =  z  ->  { ( c `  b ) }  =  { ( c `  z ) } )
13 eqid 2285 . . . . . . . . . . . . . 14  |-  ( b  e.  x  |->  { ( c `  b ) } )  =  ( b  e.  x  |->  { ( c `  b
) } )
14 snex 4218 . . . . . . . . . . . . . 14  |-  { ( c `  z ) }  e.  _V
1512, 13, 14fvmpt 5604 . . . . . . . . . . . . 13  |-  ( z  e.  x  ->  (
( b  e.  x  |->  { ( c `  b ) } ) `
 z )  =  { ( c `  z ) } )
16153ad2ant1 976 . . . . . . . . . . . 12  |-  ( ( z  e.  x  /\  z  =/=  (/)  /\  ( c `
 z )  e.  z )  ->  (
( b  e.  x  |->  { ( c `  b ) } ) `
 z )  =  { ( c `  z ) } )
17 simp3 957 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  x  /\  z  =/=  (/)  /\  ( c `
 z )  e.  z )  ->  (
c `  z )  e.  z )
1817snssd 3762 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  x  /\  z  =/=  (/)  /\  ( c `
 z )  e.  z )  ->  { ( c `  z ) }  C_  z )
1914elpw 3633 . . . . . . . . . . . . . . 15  |-  ( { ( c `  z
) }  e.  ~P z 
<->  { ( c `  z ) }  C_  z )
2018, 19sylibr 203 . . . . . . . . . . . . . 14  |-  ( ( z  e.  x  /\  z  =/=  (/)  /\  ( c `
 z )  e.  z )  ->  { ( c `  z ) }  e.  ~P z
)
21 snfi 6943 . . . . . . . . . . . . . . 15  |-  { ( c `  z ) }  e.  Fin
2221a1i 10 . . . . . . . . . . . . . 14  |-  ( ( z  e.  x  /\  z  =/=  (/)  /\  ( c `
 z )  e.  z )  ->  { ( c `  z ) }  e.  Fin )
23 elin 3360 . . . . . . . . . . . . . 14  |-  ( { ( c `  z
) }  e.  ( ~P z  i^i  Fin ) 
<->  ( { ( c `
 z ) }  e.  ~P z  /\  { ( c `  z
) }  e.  Fin ) )
2420, 22, 23sylanbrc 645 . . . . . . . . . . . . 13  |-  ( ( z  e.  x  /\  z  =/=  (/)  /\  ( c `
 z )  e.  z )  ->  { ( c `  z ) }  e.  ( ~P z  i^i  Fin )
)
25 fvex 5541 . . . . . . . . . . . . . . 15  |-  ( c `
 z )  e. 
_V
2625snnz 3746 . . . . . . . . . . . . . 14  |-  { ( c `  z ) }  =/=  (/)
2726a1i 10 . . . . . . . . . . . . 13  |-  ( ( z  e.  x  /\  z  =/=  (/)  /\  ( c `
 z )  e.  z )  ->  { ( c `  z ) }  =/=  (/) )
28 eldifsn 3751 . . . . . . . . . . . . 13  |-  ( { ( c `  z
) }  e.  ( ( ~P z  i^i 
Fin )  \  { (/)
} )  <->  ( {
( c `  z
) }  e.  ( ~P z  i^i  Fin )  /\  { ( c `
 z ) }  =/=  (/) ) )
2924, 27, 28sylanbrc 645 . . . . . . . . . . . 12  |-  ( ( z  e.  x  /\  z  =/=  (/)  /\  ( c `
 z )  e.  z )  ->  { ( c `  z ) }  e.  ( ( ~P z  i^i  Fin )  \  { (/) } ) )
3016, 29eqeltrd 2359 . . . . . . . . . . 11  |-  ( ( z  e.  x  /\  z  =/=  (/)  /\  ( c `
 z )  e.  z )  ->  (
( b  e.  x  |->  { ( c `  b ) } ) `
 z )  e.  ( ( ~P z  i^i  Fin )  \  { (/)
} ) )
31303exp 1150 . . . . . . . . . 10  |-  ( z  e.  x  ->  (
z  =/=  (/)  ->  (
( c `  z
)  e.  z  -> 
( ( b  e.  x  |->  { ( c `
 b ) } ) `  z )  e.  ( ( ~P z  i^i  Fin )  \  { (/) } ) ) ) )
3231a2d 23 . . . . . . . . 9  |-  ( z  e.  x  ->  (
( z  =/=  (/)  ->  (
c `  z )  e.  z )  ->  (
z  =/=  (/)  ->  (
( b  e.  x  |->  { ( c `  b ) } ) `
 z )  e.  ( ( ~P z  i^i  Fin )  \  { (/)
} ) ) ) )
3332ralimia 2618 . . . . . . . 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 )  \  { (/)
} ) ) )
3410, 33sylbi 187 . . . . . . 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 )  \  { (/)
} ) ) )
35 vex 2793 . . . . . . . . 9  |-  x  e. 
_V
3635mptex 5748 . . . . . . . 8  |-  ( b  e.  x  |->  { ( c `  b ) } )  e.  _V
37 fveq1 5526 . . . . . . . . . . 11  |-  ( f  =  ( b  e.  x  |->  { ( c `
 b ) } )  ->  ( f `  z )  =  ( ( b  e.  x  |->  { ( c `  b ) } ) `
 z ) )
3837eleq1d 2351 . . . . . . . . . 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 )  \  { (/)
} ) ) )
3938imbi2d 307 . . . . . . . . 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 )  \  { (/)
} ) ) ) )
4039ralbidv 2565 . . . . . . . 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 )  \  { (/)
} ) ) ) )
4136, 40spcev 2877 . . . . . . 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 )  \  { (/)
} ) ) )
4234, 41syl 15 . . . . . 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 )  \  { (/)
} ) ) )
4342exlimiv 1668 . . . . 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 )  \  { (/)
} ) ) )
4443alimi 1548 . . . 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 )  \  { (/)
} ) ) )
454, 44sylbi 187 . . 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 )  \  { (/)
} ) ) )
461, 45sylbi 187 . 2  |-  (CHOICE  ->  A. x E. f A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  ( ( ~P z  i^i 
Fin )  \  { (/)
} ) ) )
47 fvex 5541 . . . . . . 7  |-  ( R1
`  ( rank `  a
) )  e.  _V
4847pwex 4195 . . . . . 6  |-  ~P ( R1 `  ( rank `  a
) )  e.  _V
49 raleq 2738 . . . . . . 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 )  \  { (/)
} ) ) ) )
5049exbidv 1614 . . . . . 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 )  \  { (/)
} ) ) ) )
5148, 50spcv 2876 . . . . 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 )  \  { (/)
} ) ) )
52 rankon 7469 . . . . . . . 8  |-  ( rank `  a )  e.  On
5352a1i 10 . . . . . . 7  |-  ( A. z  e.  ~P  ( R1 `  ( rank `  a
) ) ( z  =/=  (/)  ->  ( f `  z )  e.  ( ( ~P z  i^i 
Fin )  \  { (/)
} ) )  -> 
( rank `  a )  e.  On )
54 id 19 . . . . . . 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 )  \  { (/)
} ) ) )
5553, 54aomclem8 27170 . . . . . 6  |-  ( A. z  e.  ~P  ( R1 `  ( rank `  a
) ) ( z  =/=  (/)  ->  ( f `  z )  e.  ( ( ~P z  i^i 
Fin )  \  { (/)
} ) )  ->  E. b  b  We  ( R1 `  ( rank `  a ) ) )
5655exlimiv 1668 . . . . 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 ) ) )
57 vex 2793 . . . . . 6  |-  a  e. 
_V
58 r1rankid 7533 . . . . . 6  |-  ( a  e.  _V  ->  a  C_  ( R1 `  ( rank `  a ) ) )
59 wess 4382 . . . . . . 7  |-  ( a 
C_  ( R1 `  ( rank `  a )
)  ->  ( b  We  ( R1 `  ( rank `  a ) )  ->  b  We  a
) )
6059eximdv 1610 . . . . . 6  |-  ( a 
C_  ( R1 `  ( rank `  a )
)  ->  ( E. b  b  We  ( R1 `  ( rank `  a
) )  ->  E. b 
b  We  a ) )
6157, 58, 60mp2b 9 . . . . 5  |-  ( E. b  b  We  ( R1 `  ( rank `  a
) )  ->  E. b 
b  We  a )
6251, 56, 613syl 18 . . . 4  |-  ( A. x E. f A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  ( ( ~P z  i^i 
Fin )  \  { (/)
} ) )  ->  E. b  b  We  a )
6362alrimiv 1619 . . 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 )
64 dfac8 7763 . . 3  |-  (CHOICE  <->  A. a E. b  b  We  a )
6563, 64sylibr 203 . 2  |-  ( A. x E. f A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  ( ( ~P z  i^i 
Fin )  \  { (/)
} ) )  -> CHOICE )
6646, 65impbii 180 1  |-  (CHOICE  <->  A. x E. f A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  ( ( ~P z  i^i 
Fin )  \  { (/)
} ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934   A.wal 1529   E.wex 1530    = wceq 1625    e. wcel 1686    =/= wne 2448   A.wral 2545   _Vcvv 2790    \ cdif 3151    i^i cin 3153    C_ wss 3154   (/)c0 3457   ~Pcpw 3627   {csn 3642    e. cmpt 4079    We wwe 4353   Oncon0 4394   ` cfv 5257   Fincfn 6865   R1cr1 7436   rankcrnk 7437  CHOICEwac 7744
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-reg 7308  ax-inf2 7344
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-2o 6482  df-er 6662  df-map 6776  df-en 6866  df-fin 6869  df-sup 7196  df-r1 7438  df-rank 7439  df-card 7574  df-ac 7745
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