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Theorem dfac3 7743
Description: Equivalence of two versions of the Axiom of Choice. The left-hand side is 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
Dummy variables  y  w  v  u are mutually distinct and distinct from all other variables.

Proof of Theorem dfac3
StepHypRef Expression
1 df-ac 7738 . 2  |-  (CHOICE  <->  A. y E. f ( f  C_  y  /\  f  Fn  dom  y ) )
2 vex 2792 . . . . . . . 8  |-  x  e. 
_V
32uniex 4515 . . . . . . . 8  |-  U. x  e.  _V
42, 3xpex 4800 . . . . . . 7  |-  ( x  X.  U. x )  e.  _V
5 simpl 445 . . . . . . . . . 10  |-  ( ( w  e.  x  /\  v  e.  w )  ->  w  e.  x )
6 elunii 3833 . . . . . . . . . . 11  |-  ( ( v  e.  w  /\  w  e.  x )  ->  v  e.  U. x
)
76ancoms 441 . . . . . . . . . 10  |-  ( ( w  e.  x  /\  v  e.  w )  ->  v  e.  U. x
)
85, 7jca 520 . . . . . . . . 9  |-  ( ( w  e.  x  /\  v  e.  w )  ->  ( w  e.  x  /\  v  e.  U. x
) )
98ssopab2i 4291 . . . . . . . 8  |-  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }  C_  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  U. x
) }
10 df-xp 4694 . . . . . . . 8  |-  ( x  X.  U. x )  =  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  U. x
) }
119, 10sseqtr4i 3212 . . . . . . 7  |-  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }  C_  ( x  X.  U. x )
124, 11ssexi 4160 . . . . . 6  |-  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }  e.  _V
13 sseq2 3201 . . . . . . . 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 4878 . . . . . . . . 9  |-  ( y  =  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }  ->  dom  y  =  dom  { <. w ,  v
>.  |  ( w  e.  x  /\  v  e.  w ) } )
1514fneq2d 5301 . . . . . . . 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 693 . . . . . . 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 1613 . . . . . 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 2875 . . . . 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 5308 . . . . . . . . . . . . 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 2345 . . . . . . . . . . . . . 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 4888 . . . . . . . . . . . . . . . 16  |-  dom  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }  =  { w  |  E. v ( w  e.  x  /\  v  e.  w ) }
2221eleq2i 2348 . . . . . . . . . . . . . . 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 2792 . . . . . . . . . . . . . . . 16  |-  z  e. 
_V
24 elequ1 1688 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  z  ->  (
w  e.  x  <->  z  e.  x ) )
25 eleq2 2345 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  z  ->  (
v  e.  w  <->  v  e.  z ) )
2624, 25anbi12d 693 . . . . . . . . . . . . . . . . 17  |-  ( w  =  z  ->  (
( w  e.  x  /\  v  e.  w
)  <->  ( z  e.  x  /\  v  e.  z ) ) )
2726exbidv 1613 . . . . . . . . . . . . . . . 16  |-  ( w  =  z  ->  ( E. v ( w  e.  x  /\  v  e.  w )  <->  E. v
( z  e.  x  /\  v  e.  z
) ) )
2823, 27elab 2915 . . . . . . . . . . . . . . 15  |-  ( z  e.  { w  |  E. v ( w  e.  x  /\  v  e.  w ) }  <->  E. v
( z  e.  x  /\  v  e.  z
) )
29 19.42v 1847 . . . . . . . . . . . . . . . 16  |-  ( E. v ( z  e.  x  /\  v  e.  z )  <->  ( z  e.  x  /\  E. v 
v  e.  z ) )
30 n0 3465 . . . . . . . . . . . . . . . . 17  |-  ( z  =/=  (/)  <->  E. v  v  e.  z )
3130anbi2i 677 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  x  /\  z  =/=  (/) )  <->  ( z  e.  x  /\  E. v 
v  e.  z ) )
3229, 31bitr4i 245 . . . . . . . . . . . . . . 15  |-  ( E. v ( z  e.  x  /\  v  e.  z )  <->  ( z  e.  x  /\  z  =/=  (/) ) )
3322, 28, 323bitrri 265 . . . . . . . . . . . . . 14  |-  ( ( z  e.  x  /\  z  =/=  (/) )  <->  z  e.  dom  { <. w ,  v
>.  |  ( w  e.  x  /\  v  e.  w ) } )
3420, 33syl6rbbr 257 . . . . . . . . . . . . 13  |-  ( dom  f  =  dom  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }  ->  ( ( z  e.  x  /\  z  =/=  (/) )  <->  z  e.  dom  f ) )
3519, 34syl 17 . . . . . . . . . . . 12  |-  ( f  Fn  dom  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }  ->  ( ( z  e.  x  /\  z  =/=  (/) )  <->  z  e.  dom  f ) )
3635adantl 454 . . . . . . . . . . 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 5306 . . . . . . . . . . . 12  |-  ( f  Fn  dom  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }  ->  Fun  f )
38 funfvima3 5716 . . . . . . . . . . . . 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 441 . . . . . . . . . . . 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 462 . . . . . . . . . . 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 208 . . . . . . . . . 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 420 . . . . . . . . 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 492 . . . . . . . . . . . . 13  |-  ( z  e.  x  ->  (
u  e.  z  <->  ( z  e.  x  /\  u  e.  z ) ) )
4443abbi2dv 2399 . . . . . . . . . . . 12  |-  ( z  e.  x  ->  z  =  { u  |  ( z  e.  x  /\  u  e.  z ) } )
45 imasng 5034 . . . . . . . . . . . . . 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 10 . . . . . . . . . . . . 13  |-  ( {
<. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) } " { z } )  =  { u  |  z { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) } u }
47 vex 2792 . . . . . . . . . . . . . . 15  |-  u  e. 
_V
48 elequ1 1688 . . . . . . . . . . . . . . . 16  |-  ( v  =  u  ->  (
v  e.  z  <->  u  e.  z ) )
4948anbi2d 686 . . . . . . . . . . . . . . 15  |-  ( v  =  u  ->  (
( z  e.  x  /\  v  e.  z
)  <->  ( z  e.  x  /\  u  e.  z ) ) )
50 eqid 2284 . . . . . . . . . . . . . . 15  |-  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }  =  { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) }
5123, 47, 26, 49, 50brab 4286 . . . . . . . . . . . . . 14  |-  ( z { <. w ,  v
>.  |  ( w  e.  x  /\  v  e.  w ) } u  <->  ( z  e.  x  /\  u  e.  z )
)
5251abbii 2396 . . . . . . . . . . . . 13  |-  { u  |  z { <. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) } u }  =  { u  |  (
z  e.  x  /\  u  e.  z ) }
5346, 52eqtri 2304 . . . . . . . . . . . 12  |-  ( {
<. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) } " { z } )  =  { u  |  ( z  e.  x  /\  u  e.  z
) }
5444, 53syl6reqr 2335 . . . . . . . . . . 11  |-  ( z  e.  x  ->  ( { <. w ,  v
>.  |  ( w  e.  x  /\  v  e.  w ) } " { z } )  =  z )
5554eleq2d 2351 . . . . . . . . . 10  |-  ( z  e.  x  ->  (
( f `  z
)  e.  ( {
<. w ,  v >.  |  ( w  e.  x  /\  v  e.  w ) } " { z } )  <-> 
( f `  z
)  e.  z ) )
5655ad2antrl 710 . . . . . . . . 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 203 . . . . . . . 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 590 . . . . . . 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 2626 . . . . . 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 1564 . . . . 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 17 . . . 4  |-  ( A. y E. f ( f 
C_  y  /\  f  Fn  dom  y )  ->  E. f A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  z ) )
6261alrimiv 1618 . . 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 2284 . . . . 5  |-  ( w  e.  dom  y  |->  ( f `  { u  |  w y u }
) )  =  ( w  e.  dom  y  |->  ( f `  {
u  |  w y u } ) )
6463aceq3lem 7742 . . . 4  |-  ( A. x E. f A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  z )  ->  E. f
( f  C_  y  /\  f  Fn  dom  y ) )
6564alrimiv 1618 . . 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 182 . 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 242 1  |-  (CHOICE  <->  A. x E. f A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360   A.wal 1528   E.wex 1529    = wceq 1624    e. wcel 1685   {cab 2270    =/= wne 2447   A.wral 2544   _Vcvv 2789    C_ wss 3153   (/)c0 3456   {csn 3641   U.cuni 3828   class class class wbr 4024   {copab 4077    e. cmpt 4078    X. cxp 4686   dom cdm 4688   "cima 4691   Fun wfun 5215    Fn wfn 5216   ` cfv 5221  CHOICEwac 7737
This theorem is referenced by:  dfac4  7744  dfac5  7750  dfac2a  7751  dfac2  7752  dfac8  7756  dfac9  7757  ac4  8097  dfac11  26559
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-fv 5229  df-ac 7738
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