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Theorem axdc2 8075
Description: An apparent strengthening of ax-dc 8072 (but derived from it) which shows that there is a denumerable sequence  g for any function that maps elements of a set  A to nonempty subsets of 
A such that  g (
x  +  1 )  e.  F ( g ( x ) ) for all  x  e.  om. The finitistic version of this can be proven by induction, but the infinite version requires this new axiom. (Contributed by Mario Carneiro, 25-Jan-2013.)
Hypothesis
Ref Expression
axdc2.1  |-  A  e. 
_V
Assertion
Ref Expression
axdc2  |-  ( ( A  =/=  (/)  /\  F : A --> ( ~P A  \  { (/) } ) )  ->  E. g ( g : om --> A  /\  A. k  e.  om  (
g `  suc  k )  e.  ( F `  ( g `  k
) ) ) )
Distinct variable groups:    A, g,
k    g, F, k

Proof of Theorem axdc2
Dummy variables  h  s  t  x  y  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 axdc2.1 . 2  |-  A  e. 
_V
2 eleq1 2343 . . . . 5  |-  ( s  =  x  ->  (
s  e.  A  <->  x  e.  A ) )
32adantr 451 . . . 4  |-  ( ( s  =  x  /\  t  =  y )  ->  ( s  e.  A  <->  x  e.  A ) )
4 fveq2 5525 . . . . . 6  |-  ( s  =  x  ->  ( F `  s )  =  ( F `  x ) )
54eleq2d 2350 . . . . 5  |-  ( s  =  x  ->  (
t  e.  ( F `
 s )  <->  t  e.  ( F `  x ) ) )
6 eleq1 2343 . . . . 5  |-  ( t  =  y  ->  (
t  e.  ( F `
 x )  <->  y  e.  ( F `  x ) ) )
75, 6sylan9bb 680 . . . 4  |-  ( ( s  =  x  /\  t  =  y )  ->  ( t  e.  ( F `  s )  <-> 
y  e.  ( F `
 x ) ) )
83, 7anbi12d 691 . . 3  |-  ( ( s  =  x  /\  t  =  y )  ->  ( ( s  e.  A  /\  t  e.  ( F `  s
) )  <->  ( x  e.  A  /\  y  e.  ( F `  x
) ) ) )
98cbvopabv 4088 . 2  |-  { <. s ,  t >.  |  ( s  e.  A  /\  t  e.  ( F `  s ) ) }  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( F `  x ) ) }
10 fveq2 5525 . . 3  |-  ( n  =  x  ->  (
h `  n )  =  ( h `  x ) )
1110cbvmptv 4111 . 2  |-  ( n  e.  om  |->  ( h `
 n ) )  =  ( x  e. 
om  |->  ( h `  x ) )
121, 9, 11axdc2lem 8074 1  |-  ( ( A  =/=  (/)  /\  F : A --> ( ~P A  \  { (/) } ) )  ->  E. g ( g : om --> A  /\  A. k  e.  om  (
g `  suc  k )  e.  ( F `  ( g `  k
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    e. wcel 1684    =/= wne 2446   A.wral 2543   _Vcvv 2788    \ cdif 3149   (/)c0 3455   ~Pcpw 3625   {csn 3640   {copab 4076    e. cmpt 4077   suc csuc 4394   omcom 4656   -->wf 5251   ` cfv 5255
This theorem is referenced by:  axdc3lem4  8079
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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-dc 8072
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-rab 2552  df-v 2790  df-sbc 2992  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-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-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-fv 5263  df-1o 6479
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