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Theorem dfac3 8498
Description: Equivalence of two versions of the Axiom of Choice. The left-hand side is defined as the Axiom of Choice (first form) of [Enderton] p. 49. The right-hand side is the Axiom of Choice of [TakeutiZaring] p. 83. The proof does not depend on AC. (Contributed by NM, 24-Mar-2004.) (Revised by Stefan O'Rear, 22-Feb-2015.)
Assertion
Ref Expression
dfac3  |-  (CHOICE  <->  A. x E. f A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  z ) )
Distinct variable group:    x, f, z

Proof of Theorem dfac3
Dummy variables  y  w  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ac 8493 . 2  |-  (CHOICE  <->  A. y E. f ( f  C_  y  /\  f  Fn  dom  y ) )
2 vex 3116 . . . . . . . 8  |-  x  e. 
_V
32uniex 6578 . . . . . . . 8  |-  U. x  e.  _V
42, 3xpex 6711 . . . . . . 7  |-  ( x  X.  U. x )  e.  _V
5 simpl 457 . . . . . . . . . 10  |-  ( ( w  e.  x  /\  v  e.  w )  ->  w  e.  x )
6 elunii 4250 . . . . . . . . . . 11  |-  ( ( v  e.  w  /\  w  e.  x )  ->  v  e.  U. x
)
76ancoms 453 . . . . . . . . . 10  |-  ( ( w  e.  x  /\  v  e.  w )  ->  v  e.  U. x
)
85, 7jca 532 . . . . . . . . 9  |-  ( ( w  e.  x  /\  v  e.  w )  ->  ( w  e.  x  /\  v  e.  U. x
) )
98ssopab2i 4775 . . . . . . . 8  |-  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }  C_  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  U. x
) }
10 df-xp 5005 . . . . . . . 8  |-  ( x  X.  U. x )  =  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  U. x
) }
119, 10sseqtr4i 3537 . . . . . . 7  |-  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }  C_  ( x  X.  U. x )
124, 11ssexi 4592 . . . . . 6  |-  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }  e.  _V
13 sseq2 3526 . . . . . . . 8  |-  ( y  =  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }  ->  ( f  C_  y 
<->  f  C_  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) } ) )
14 dmeq 5201 . . . . . . . . 9  |-  ( y  =  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }  ->  dom  y  =  dom  { <. w ,  v
>.  |  ( w  e.  x  /\  v  e.  w ) } )
1514fneq2d 5670 . . . . . . . 8  |-  ( y  =  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }  ->  ( f  Fn 
dom  y  <->  f  Fn  dom  { <. w ,  v
>.  |  ( w  e.  x  /\  v  e.  w ) } ) )
1613, 15anbi12d 710 . . . . . . 7  |-  ( y  =  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }  ->  ( ( f 
C_  y  /\  f  Fn  dom  y )  <->  ( f  C_ 
{ <. w ,  v
>.  |  ( w  e.  x  /\  v  e.  w ) }  /\  f  Fn  dom  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) } ) ) )
1716exbidv 1690 . . . . . 6  |-  ( y  =  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }  ->  ( E. f
( f  C_  y  /\  f  Fn  dom  y )  <->  E. f
( f  C_  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }  /\  f  Fn  dom  {
<. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) } ) ) )
1812, 17spcv 3204 . . . . 5  |-  ( A. y E. f ( f 
C_  y  /\  f  Fn  dom  y )  ->  E. f ( f  C_  {
<. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }  /\  f  Fn  dom  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) } ) )
19 fndm 5678 . . . . . . . . . . . . 13  |-  ( f  Fn  dom  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }  ->  dom  f  =  dom  { <. w ,  v
>.  |  ( w  e.  x  /\  v  e.  w ) } )
20 eleq2 2540 . . . . . . . . . . . . . 14  |-  ( dom  f  =  dom  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }  ->  ( z  e.  dom  f  <->  z  e.  dom  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) } ) )
21 dmopab 5211 . . . . . . . . . . . . . . . 16  |-  dom  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }  =  { w  |  E. v ( w  e.  x  /\  v  e.  w ) }
2221eleq2i 2545 . . . . . . . . . . . . . . 15  |-  ( z  e.  dom  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) } 
<->  z  e.  { w  |  E. v ( w  e.  x  /\  v  e.  w ) } )
23 vex 3116 . . . . . . . . . . . . . . . 16  |-  z  e. 
_V
24 elequ1 1770 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  z  ->  (
w  e.  x  <->  z  e.  x ) )
25 eleq2 2540 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  z  ->  (
v  e.  w  <->  v  e.  z ) )
2624, 25anbi12d 710 . . . . . . . . . . . . . . . . 17  |-  ( w  =  z  ->  (
( w  e.  x  /\  v  e.  w
)  <->  ( z  e.  x  /\  v  e.  z ) ) )
2726exbidv 1690 . . . . . . . . . . . . . . . 16  |-  ( w  =  z  ->  ( E. v ( w  e.  x  /\  v  e.  w )  <->  E. v
( z  e.  x  /\  v  e.  z
) ) )
2823, 27elab 3250 . . . . . . . . . . . . . . 15  |-  ( z  e.  { w  |  E. v ( w  e.  x  /\  v  e.  w ) }  <->  E. v
( z  e.  x  /\  v  e.  z
) )
29 19.42v 1949 . . . . . . . . . . . . . . . 16  |-  ( E. v ( z  e.  x  /\  v  e.  z )  <->  ( z  e.  x  /\  E. v 
v  e.  z ) )
30 n0 3794 . . . . . . . . . . . . . . . . 17  |-  ( z  =/=  (/)  <->  E. v  v  e.  z )
3130anbi2i 694 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  x  /\  z  =/=  (/) )  <->  ( z  e.  x  /\  E. v 
v  e.  z ) )
3229, 31bitr4i 252 . . . . . . . . . . . . . . 15  |-  ( E. v ( z  e.  x  /\  v  e.  z )  <->  ( z  e.  x  /\  z  =/=  (/) ) )
3322, 28, 323bitrri 272 . . . . . . . . . . . . . 14  |-  ( ( z  e.  x  /\  z  =/=  (/) )  <->  z  e.  dom  { <. w ,  v
>.  |  ( w  e.  x  /\  v  e.  w ) } )
3420, 33syl6rbbr 264 . . . . . . . . . . . . 13  |-  ( dom  f  =  dom  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }  ->  ( ( z  e.  x  /\  z  =/=  (/) )  <->  z  e.  dom  f ) )
3519, 34syl 16 . . . . . . . . . . . 12  |-  ( f  Fn  dom  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }  ->  ( ( z  e.  x  /\  z  =/=  (/) )  <->  z  e.  dom  f ) )
3635adantl 466 . . . . . . . . . . 11  |-  ( ( f  C_  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }  /\  f  Fn  dom  {
<. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) } )  ->  ( ( z  e.  x  /\  z  =/=  (/) )  <->  z  e.  dom  f ) )
37 fnfun 5676 . . . . . . . . . . . 12  |-  ( f  Fn  dom  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }  ->  Fun  f )
38 funfvima3 6135 . . . . . . . . . . . . 13  |-  ( ( Fun  f  /\  f  C_ 
{ <. w ,  v
>.  |  ( w  e.  x  /\  v  e.  w ) } )  ->  ( z  e. 
dom  f  ->  (
f `  z )  e.  ( { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) } " { z } ) ) )
3938ancoms 453 . . . . . . . . . . . 12  |-  ( ( f  C_  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }  /\  Fun  f )  ->  ( z  e. 
dom  f  ->  (
f `  z )  e.  ( { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) } " { z } ) ) )
4037, 39sylan2 474 . . . . . . . . . . 11  |-  ( ( f  C_  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }  /\  f  Fn  dom  {
<. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) } )  ->  ( z  e. 
dom  f  ->  (
f `  z )  e.  ( { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) } " { z } ) ) )
4136, 40sylbid 215 . . . . . . . . . 10  |-  ( ( f  C_  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }  /\  f  Fn  dom  {
<. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) } )  ->  ( ( z  e.  x  /\  z  =/=  (/) )  ->  (
f `  z )  e.  ( { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) } " { z } ) ) )
4241imp 429 . . . . . . . . 9  |-  ( ( ( f  C_  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }  /\  f  Fn  dom  {
<. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) } )  /\  ( z  e.  x  /\  z  =/=  (/) ) )  ->  (
f `  z )  e.  ( { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) } " { z } ) )
43 ibar 504 . . . . . . . . . . . . 13  |-  ( z  e.  x  ->  (
u  e.  z  <->  ( z  e.  x  /\  u  e.  z ) ) )
4443abbi2dv 2604 . . . . . . . . . . . 12  |-  ( z  e.  x  ->  z  =  { u  |  ( z  e.  x  /\  u  e.  z ) } )
45 imasng 5357 . . . . . . . . . . . . . 14  |-  ( z  e.  _V  ->  ( { <. w ,  v
>.  |  ( w  e.  x  /\  v  e.  w ) } " { z } )  =  { u  |  z { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) } u } )
4623, 45ax-mp 5 . . . . . . . . . . . . 13  |-  ( {
<. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) } " { z } )  =  { u  |  z { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) } u }
47 vex 3116 . . . . . . . . . . . . . . 15  |-  u  e. 
_V
48 elequ1 1770 . . . . . . . . . . . . . . . 16  |-  ( v  =  u  ->  (
v  e.  z  <->  u  e.  z ) )
4948anbi2d 703 . . . . . . . . . . . . . . 15  |-  ( v  =  u  ->  (
( z  e.  x  /\  v  e.  z
)  <->  ( z  e.  x  /\  u  e.  z ) ) )
50 eqid 2467 . . . . . . . . . . . . . . 15  |-  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }  =  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }
5123, 47, 26, 49, 50brab 4770 . . . . . . . . . . . . . 14  |-  ( z { <. w ,  v
>.  |  ( w  e.  x  /\  v  e.  w ) } u  <->  ( z  e.  x  /\  u  e.  z )
)
5251abbii 2601 . . . . . . . . . . . . 13  |-  { u  |  z { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) } u }  =  { u  |  (
z  e.  x  /\  u  e.  z ) }
5346, 52eqtri 2496 . . . . . . . . . . . 12  |-  ( {
<. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) } " { z } )  =  { u  |  ( z  e.  x  /\  u  e.  z
) }
5444, 53syl6reqr 2527 . . . . . . . . . . 11  |-  ( z  e.  x  ->  ( { <. w ,  v
>.  |  ( w  e.  x  /\  v  e.  w ) } " { z } )  =  z )
5554eleq2d 2537 . . . . . . . . . 10  |-  ( z  e.  x  ->  (
( f `  z
)  e.  ( {
<. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) } " { z } )  <-> 
( f `  z
)  e.  z ) )
5655ad2antrl 727 . . . . . . . . 9  |-  ( ( ( f  C_  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }  /\  f  Fn  dom  {
<. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) } )  /\  ( z  e.  x  /\  z  =/=  (/) ) )  ->  (
( f `  z
)  e.  ( {
<. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) } " { z } )  <-> 
( f `  z
)  e.  z ) )
5742, 56mpbid 210 . . . . . . . 8  |-  ( ( ( f  C_  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }  /\  f  Fn  dom  {
<. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) } )  /\  ( z  e.  x  /\  z  =/=  (/) ) )  ->  (
f `  z )  e.  z )
5857exp32 605 . . . . . . 7  |-  ( ( f  C_  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }  /\  f  Fn  dom  {
<. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) } )  ->  ( z  e.  x  ->  ( z  =/=  (/)  ->  ( f `  z )  e.  z ) ) )
5958ralrimiv 2876 . . . . . 6  |-  ( ( f  C_  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }  /\  f  Fn  dom  {
<. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) } )  ->  A. z  e.  x  ( z  =/=  (/)  ->  (
f `  z )  e.  z ) )
6059eximi 1635 . . . . 5  |-  ( E. f ( f  C_  {
<. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }  /\  f  Fn  dom  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) } )  ->  E. f A. z  e.  x  ( z  =/=  (/)  ->  (
f `  z )  e.  z ) )
6118, 60syl 16 . . . 4  |-  ( A. y E. f ( f 
C_  y  /\  f  Fn  dom  y )  ->  E. f A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  z ) )
6261alrimiv 1695 . . 3  |-  ( A. y E. f ( f 
C_  y  /\  f  Fn  dom  y )  ->  A. x E. f A. z  e.  x  (
z  =/=  (/)  ->  (
f `  z )  e.  z ) )
63 eqid 2467 . . . . 5  |-  ( w  e.  dom  y  |->  ( f `  { u  |  w y u }
) )  =  ( w  e.  dom  y  |->  ( f `  {
u  |  w y u } ) )
6463aceq3lem 8497 . . . 4  |-  ( A. x E. f A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  z )  ->  E. f
( f  C_  y  /\  f  Fn  dom  y ) )
6564alrimiv 1695 . . 3  |-  ( A. x E. f A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  z )  ->  A. y E. f ( f  C_  y  /\  f  Fn  dom  y ) )
6662, 65impbii 188 . 2  |-  ( A. y E. f ( f 
C_  y  /\  f  Fn  dom  y )  <->  A. x E. f A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  z ) )
671, 66bitri 249 1  |-  (CHOICE  <->  A. x E. f A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1377    = wceq 1379   E.wex 1596    e. wcel 1767   {cab 2452    =/= wne 2662   A.wral 2814   _Vcvv 3113    C_ wss 3476   (/)c0 3785   {csn 4027   U.cuni 4245   class class class wbr 4447   {copab 4504    |-> cmpt 4505    X. cxp 4997   dom cdm 4999   "cima 5002   Fun wfun 5580    Fn wfn 5581   ` cfv 5586  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-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  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-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  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-fv 5594  df-ac 8493
This theorem is referenced by:  dfac4  8499  dfac5  8505  dfac2a  8506  dfac2  8507  dfac8  8511  dfac9  8512  ac4  8851  dfac11  30612
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