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Theorem ac6 8872
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 8876, 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 3590 . . . . . 6  |-  { y  e.  B  |  ph }  C_  B
43rgenw 2828 . . . . 5  |-  A. x  e.  A  { y  e.  B  |  ph }  C_  B
5 iunss 4372 . . . . 5  |-  ( U_ x  e.  A  {
y  e.  B  |  ph }  C_  B  <->  A. x  e.  A  { y  e.  B  |  ph }  C_  B )
64, 5mpbir 209 . . . 4  |-  U_ x  e.  A  { y  e.  B  |  ph }  C_  B
72, 6ssexi 4598 . . 3  |-  U_ x  e.  A  { y  e.  B  |  ph }  e.  _V
8 numth3 8862 . . 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 5 . 2  |-  U_ x  e.  A  { y  e.  B  |  ph }  e.  dom  card
10 ac6.3 . . 3  |-  ( y  =  ( f `  x )  ->  ( ph 
<->  ps ) )
1110ac6num 8871 . 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 1314 1  |-  ( A. x  e.  A  E. y  e.  B  ph  ->  E. f ( f : A --> B  /\  A. x  e.  A  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   A.wral 2817   E.wrex 2818   {crab 2821   _Vcvv 3118    C_ wss 3481   U_ciun 4331   dom cdm 5005   -->wf 5590   ` cfv 5594   cardccrd 8328
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-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-ac2 8855
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-recs 7054  df-en 7529  df-card 8332  df-ac 8509
This theorem is referenced by:  ac6c4  8873  ac6s  8876
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