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Theorem ac6 8107
Description: Equivalent of Axiom of Choice. This is useful for proving that there exists, for example, a sequence mapping natural numbers to members of a larger set  B, where  ph depends on  x (the natural number) and  y (to specify a member of  B). A stronger version of this theorem, ac6s 8111, allows  B to be a proper class. (Contributed by NM, 18-Oct-1999.) (Revised by Mario Carneiro, 27-Aug-2015.)
Hypotheses
Ref Expression
ac6.1  |-  A  e. 
_V
ac6.2  |-  B  e. 
_V
ac6.3  |-  ( y  =  ( f `  x )  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ac6  |-  ( A. x  e.  A  E. y  e.  B  ph  ->  E. f ( f : A --> B  /\  A. x  e.  A  ps ) )
Distinct variable groups:    x, f, A    y, f, B, x    ph, f    ps, y
Allowed substitution hints:    ph( x, y)    ps( x, f)    A( y)

Proof of Theorem ac6
StepHypRef Expression
1 ac6.1 . 2  |-  A  e. 
_V
2 ac6.2 . . . 4  |-  B  e. 
_V
3 ssrab2 3258 . . . . . 6  |-  { y  e.  B  |  ph }  C_  B
43rgenw 2610 . . . . 5  |-  A. x  e.  A  { y  e.  B  |  ph }  C_  B
5 iunss 3943 . . . . 5  |-  ( U_ x  e.  A  {
y  e.  B  |  ph }  C_  B  <->  A. x  e.  A  { y  e.  B  |  ph }  C_  B )
64, 5mpbir 200 . . . 4  |-  U_ x  e.  A  { y  e.  B  |  ph }  C_  B
72, 6ssexi 4159 . . 3  |-  U_ x  e.  A  { y  e.  B  |  ph }  e.  _V
8 numth3 8097 . . 3  |-  ( U_ x  e.  A  {
y  e.  B  |  ph }  e.  _V  ->  U_ x  e.  A  {
y  e.  B  |  ph }  e.  dom  card )
97, 8ax-mp 8 . 2  |-  U_ x  e.  A  { y  e.  B  |  ph }  e.  dom  card
10 ac6.3 . . 3  |-  ( y  =  ( f `  x )  ->  ( ph 
<->  ps ) )
1110ac6num 8106 . 2  |-  ( ( A  e.  _V  /\  U_ x  e.  A  {
y  e.  B  |  ph }  e.  dom  card  /\ 
A. x  e.  A  E. y  e.  B  ph )  ->  E. f
( f : A --> B  /\  A. x  e.  A  ps ) )
121, 9, 11mp3an12 1267 1  |-  ( A. x  e.  A  E. y  e.  B  ph  ->  E. f ( f : A --> B  /\  A. x  e.  A  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788    C_ wss 3152   U_ciun 3905   dom cdm 4689   -->wf 5251   ` cfv 5255   cardccrd 7568
This theorem is referenced by:  ac6c4  8108  ac6s  8111
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-ac2 8089
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 6304  df-recs 6388  df-en 6864  df-card 7572  df-ac 7743
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