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Theorem dfac11 31210
Description: The right-hand side of this theorem (compare with ac4 8786), sometimes known as the "axiom of multiple choice", is a choice equivalent. Curiously, this statement cannot be proved without ax-reg 7951, 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 8433 . . 3  |-  (CHOICE  <->  A. a E. c A. d  e.  a  ( d  =/=  (/)  ->  ( c `  d )  e.  d ) )
2 raleq 2992 . . . . . 6  |-  ( a  =  x  ->  ( A. d  e.  a 
( d  =/=  (/)  ->  (
c `  d )  e.  d )  <->  A. d  e.  x  ( d  =/=  (/)  ->  ( c `  d )  e.  d ) ) )
32exbidv 1729 . . . . 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 2040 . . . 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 2673 . . . . . . . . . 10  |-  ( d  =  z  ->  (
d  =/=  (/)  <->  z  =/=  (/) ) )
6 fveq2 5787 . . . . . . . . . . 11  |-  ( d  =  z  ->  (
c `  d )  =  ( c `  z ) )
7 id 22 . . . . . . . . . . 11  |-  ( d  =  z  ->  d  =  z )
86, 7eleq12d 2474 . . . . . . . . . 10  |-  ( d  =  z  ->  (
( c `  d
)  e.  d  <->  ( c `  z )  e.  z ) )
95, 8imbi12d 318 . . . . . . . . 9  |-  ( d  =  z  ->  (
( d  =/=  (/)  ->  (
c `  d )  e.  d )  <->  ( z  =/=  (/)  ->  ( c `  z )  e.  z ) ) )
109cbvralv 3022 . . . . . . . 8  |-  ( A. d  e.  x  (
d  =/=  (/)  ->  (
c `  d )  e.  d )  <->  A. z  e.  x  ( z  =/=  (/)  ->  ( c `  z )  e.  z ) )
11 fveq2 5787 . . . . . . . . . . . . . . 15  |-  ( b  =  z  ->  (
c `  b )  =  ( c `  z ) )
1211sneqd 3969 . . . . . . . . . . . . . 14  |-  ( b  =  z  ->  { ( c `  b ) }  =  { ( c `  z ) } )
13 eqid 2392 . . . . . . . . . . . . . 14  |-  ( b  e.  x  |->  { ( c `  b ) } )  =  ( b  e.  x  |->  { ( c `  b
) } )
14 snex 4616 . . . . . . . . . . . . . 14  |-  { ( c `  z ) }  e.  _V
1512, 13, 14fvmpt 5870 . . . . . . . . . . . . 13  |-  ( z  e.  x  ->  (
( b  e.  x  |->  { ( c `  b ) } ) `
 z )  =  { ( c `  z ) } )
16153ad2ant1 1015 . . . . . . . . . . . 12  |-  ( ( z  e.  x  /\  z  =/=  (/)  /\  ( c `
 z )  e.  z )  ->  (
( b  e.  x  |->  { ( c `  b ) } ) `
 z )  =  { ( c `  z ) } )
17 simp3 996 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  x  /\  z  =/=  (/)  /\  ( c `
 z )  e.  z )  ->  (
c `  z )  e.  z )
1817snssd 4102 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  x  /\  z  =/=  (/)  /\  ( c `
 z )  e.  z )  ->  { ( c `  z ) }  C_  z )
1914elpw 3946 . . . . . . . . . . . . . . 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 7533 . . . . . . . . . . . . . . 15  |-  { ( c `  z ) }  e.  Fin
2221a1i 11 . . . . . . . . . . . . . 14  |-  ( ( z  e.  x  /\  z  =/=  (/)  /\  ( c `
 z )  e.  z )  ->  { ( c `  z ) }  e.  Fin )
2320, 22elind 3615 . . . . . . . . . . . . 13  |-  ( ( z  e.  x  /\  z  =/=  (/)  /\  ( c `
 z )  e.  z )  ->  { ( c `  z ) }  e.  ( ~P z  i^i  Fin )
)
24 fvex 5797 . . . . . . . . . . . . . . 15  |-  ( c `
 z )  e. 
_V
2524snnz 4075 . . . . . . . . . . . . . 14  |-  { ( c `  z ) }  =/=  (/)
2625a1i 11 . . . . . . . . . . . . 13  |-  ( ( z  e.  x  /\  z  =/=  (/)  /\  ( c `
 z )  e.  z )  ->  { ( c `  z ) }  =/=  (/) )
27 eldifsn 4082 . . . . . . . . . . . . 13  |-  ( { ( c `  z
) }  e.  ( ( ~P z  i^i 
Fin )  \  { (/)
} )  <->  ( {
( c `  z
) }  e.  ( ~P z  i^i  Fin )  /\  { ( c `
 z ) }  =/=  (/) ) )
2823, 26, 27sylanbrc 662 . . . . . . . . . . . 12  |-  ( ( z  e.  x  /\  z  =/=  (/)  /\  ( c `
 z )  e.  z )  ->  { ( c `  z ) }  e.  ( ( ~P z  i^i  Fin )  \  { (/) } ) )
2916, 28eqeltrd 2480 . . . . . . . . . . 11  |-  ( ( z  e.  x  /\  z  =/=  (/)  /\  ( c `
 z )  e.  z )  ->  (
( b  e.  x  |->  { ( c `  b ) } ) `
 z )  e.  ( ( ~P z  i^i  Fin )  \  { (/)
} ) )
30293exp 1193 . . . . . . . . . 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 2783 . . . . . . . 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 3050 . . . . . . . . 9  |-  x  e. 
_V
3534mptex 6060 . . . . . . . 8  |-  ( b  e.  x  |->  { ( c `  b ) } )  e.  _V
36 fveq1 5786 . . . . . . . . . . 11  |-  ( f  =  ( b  e.  x  |->  { ( c `
 b ) } )  ->  ( f `  z )  =  ( ( b  e.  x  |->  { ( c `  b ) } ) `
 z ) )
3736eleq1d 2461 . . . . . . . . . 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 314 . . . . . . . . 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 2831 . . . . . . . 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 3139 . . . . . . 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 1737 . . . . 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 1648 . . . 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 5797 . . . . . . 7  |-  ( R1
`  ( rank `  a
) )  e.  _V
4746pwex 4561 . . . . . 6  |-  ~P ( R1 `  ( rank `  a
) )  e.  _V
48 raleq 2992 . . . . . . 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 1729 . . . . . 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 3138 . . . . 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 8144 . . . . . . . 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 31209 . . . . . 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 1737 . . . . 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 3050 . . . . . 6  |-  a  e. 
_V
57 r1rankid 8208 . . . . . 6  |-  ( a  e.  _V  ->  a  C_  ( R1 `  ( rank `  a ) ) )
58 wess 4793 . . . . . . 7  |-  ( a 
C_  ( R1 `  ( rank `  a )
)  ->  ( b  We  ( R1 `  ( rank `  a ) )  ->  b  We  a
) )
5958eximdv 1725 . . . . . 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 1734 . . 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 8446 . . 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 971   A.wal 1397    = wceq 1399   E.wex 1627    e. wcel 1836    =/= wne 2587   A.wral 2742   _Vcvv 3047    \ cdif 3399    i^i cin 3401    C_ wss 3402   (/)c0 3724   ~Pcpw 3940   {csn 3957    |-> cmpt 4438    We wwe 4764   Oncon0 4805   ` cfv 5509   Fincfn 7453   R1cr1 8111   rankcrnk 8112  CHOICEwac 8427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370  ax-rep 4491  ax-sep 4501  ax-nul 4509  ax-pow 4556  ax-pr 4614  ax-un 6509  ax-reg 7951  ax-inf2 7990
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2232  df-mo 2233  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ne 2589  df-ral 2747  df-rex 2748  df-reu 2749  df-rmo 2750  df-rab 2751  df-v 3049  df-sbc 3266  df-csb 3362  df-dif 3405  df-un 3407  df-in 3409  df-ss 3416  df-pss 3418  df-nul 3725  df-if 3871  df-pw 3942  df-sn 3958  df-pr 3960  df-tp 3962  df-op 3964  df-uni 4177  df-int 4213  df-iun 4258  df-br 4381  df-opab 4439  df-mpt 4440  df-tr 4474  df-eprel 4718  df-id 4722  df-po 4727  df-so 4728  df-fr 4765  df-se 4766  df-we 4767  df-ord 4808  df-on 4809  df-lim 4810  df-suc 4811  df-xp 4932  df-rel 4933  df-cnv 4934  df-co 4935  df-dm 4936  df-rn 4937  df-res 4938  df-ima 4939  df-iota 5473  df-fun 5511  df-fn 5512  df-f 5513  df-f1 5514  df-fo 5515  df-f1o 5516  df-fv 5517  df-isom 5518  df-riota 6176  df-ov 6217  df-oprab 6218  df-mpt2 6219  df-om 6618  df-1st 6717  df-2nd 6718  df-recs 6978  df-rdg 7012  df-1o 7066  df-2o 7067  df-er 7247  df-map 7358  df-en 7454  df-fin 7457  df-sup 7834  df-r1 8113  df-rank 8114  df-card 8251  df-ac 8428
This theorem is referenced by: (None)
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