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Theorem ac6 8362
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 8366, 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 3430 . . . . . 6  |-  { y  e.  B  |  ph }  C_  B
43rgenw 2775 . . . . 5  |-  A. x  e.  A  { y  e.  B  |  ph }  C_  B
5 iunss 4134 . . . . 5  |-  ( U_ x  e.  A  {
y  e.  B  |  ph }  C_  B  <->  A. x  e.  A  { y  e.  B  |  ph }  C_  B )
64, 5mpbir 202 . . . 4  |-  U_ x  e.  A  { y  e.  B  |  ph }  C_  B
72, 6ssexi 4350 . . 3  |-  U_ x  e.  A  { y  e.  B  |  ph }  e.  _V
8 numth3 8352 . . 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 8361 . 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 1270 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 178    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   {crab 2711   _Vcvv 2958    C_ wss 3322   U_ciun 4095   dom cdm 4880   -->wf 5452   ` cfv 5456   cardccrd 7824
This theorem is referenced by:  ac6c4  8363  ac6s  8366
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-ac2 8345
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-suc 4589  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-riota 6551  df-recs 6635  df-en 7112  df-card 7828  df-ac 7999
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