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Theorem axdc3 8726
Description: Dependent Choice. Axiom DC1 of [Schechter] p. 149, with the addition of an initial value  C. This theorem is weaker than the Axiom of Choice but is stronger than Countable Choice. It shows the existence of a sequence whose values can only be shown to exist (but cannot be constructed explicitly) and also depend on earlier values in the sequence. (Contributed by Mario Carneiro, 27-Jan-2013.)
Hypothesis
Ref Expression
axdc3.1  |-  A  e. 
_V
Assertion
Ref Expression
axdc3  |-  ( ( C  e.  A  /\  F : A --> ( ~P A  \  { (/) } ) )  ->  E. g
( g : om --> A  /\  ( g `  (/) )  =  C  /\  A. k  e.  om  (
g `  suc  k )  e.  ( F `  ( g `  k
) ) ) )
Distinct variable groups:    A, g,
k    C, g, k    g, F, k

Proof of Theorem axdc3
Dummy variables  n  s  t  x  y 
j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 axdc3.1 . 2  |-  A  e. 
_V
2 feq1 5642 . . . . 5  |-  ( t  =  s  ->  (
t : suc  n --> A 
<->  s : suc  n --> A ) )
3 fveq1 5790 . . . . . 6  |-  ( t  =  s  ->  (
t `  (/) )  =  ( s `  (/) ) )
43eqeq1d 2453 . . . . 5  |-  ( t  =  s  ->  (
( t `  (/) )  =  C  <->  ( s `  (/) )  =  C ) )
5 fveq1 5790 . . . . . . . 8  |-  ( t  =  s  ->  (
t `  suc  j )  =  ( s `  suc  j ) )
6 fveq1 5790 . . . . . . . . 9  |-  ( t  =  s  ->  (
t `  j )  =  ( s `  j ) )
76fveq2d 5795 . . . . . . . 8  |-  ( t  =  s  ->  ( F `  ( t `  j ) )  =  ( F `  (
s `  j )
) )
85, 7eleq12d 2533 . . . . . . 7  |-  ( t  =  s  ->  (
( t `  suc  j )  e.  ( F `  ( t `
 j ) )  <-> 
( s `  suc  j )  e.  ( F `  ( s `
 j ) ) ) )
98ralbidv 2838 . . . . . 6  |-  ( t  =  s  ->  ( A. j  e.  n  ( t `  suc  j )  e.  ( F `  ( t `
 j ) )  <->  A. j  e.  n  ( s `  suc  j )  e.  ( F `  ( s `
 j ) ) ) )
10 suceq 4884 . . . . . . . . 9  |-  ( j  =  k  ->  suc  j  =  suc  k )
1110fveq2d 5795 . . . . . . . 8  |-  ( j  =  k  ->  (
s `  suc  j )  =  ( s `  suc  k ) )
12 fveq2 5791 . . . . . . . . 9  |-  ( j  =  k  ->  (
s `  j )  =  ( s `  k ) )
1312fveq2d 5795 . . . . . . . 8  |-  ( j  =  k  ->  ( F `  ( s `  j ) )  =  ( F `  (
s `  k )
) )
1411, 13eleq12d 2533 . . . . . . 7  |-  ( j  =  k  ->  (
( s `  suc  j )  e.  ( F `  ( s `
 j ) )  <-> 
( s `  suc  k )  e.  ( F `  ( s `
 k ) ) ) )
1514cbvralv 3045 . . . . . 6  |-  ( A. j  e.  n  (
s `  suc  j )  e.  ( F `  ( s `  j
) )  <->  A. k  e.  n  ( s `  suc  k )  e.  ( F `  (
s `  k )
) )
169, 15syl6bb 261 . . . . 5  |-  ( t  =  s  ->  ( A. j  e.  n  ( t `  suc  j )  e.  ( F `  ( t `
 j ) )  <->  A. k  e.  n  ( s `  suc  k )  e.  ( F `  ( s `
 k ) ) ) )
172, 4, 163anbi123d 1290 . . . 4  |-  ( t  =  s  ->  (
( t : suc  n
--> A  /\  ( t `
 (/) )  =  C  /\  A. j  e.  n  ( t `  suc  j )  e.  ( F `  ( t `
 j ) ) )  <->  ( s : suc  n --> A  /\  ( s `  (/) )  =  C  /\  A. k  e.  n  ( s `  suc  k )  e.  ( F `  (
s `  k )
) ) ) )
1817rexbidv 2848 . . 3  |-  ( t  =  s  ->  ( E. n  e.  om  ( t : suc  n
--> A  /\  ( t `
 (/) )  =  C  /\  A. j  e.  n  ( t `  suc  j )  e.  ( F `  ( t `
 j ) ) )  <->  E. n  e.  om  ( s : suc  n
--> A  /\  ( s `
 (/) )  =  C  /\  A. k  e.  n  ( s `  suc  k )  e.  ( F `  ( s `
 k ) ) ) ) )
1918cbvabv 2594 . 2  |-  { t  |  E. n  e. 
om  ( t : suc  n --> A  /\  ( t `  (/) )  =  C  /\  A. j  e.  n  ( t `  suc  j )  e.  ( F `  (
t `  j )
) ) }  =  { s  |  E. n  e.  om  (
s : suc  n --> A  /\  ( s `  (/) )  =  C  /\  A. k  e.  n  ( s `  suc  k
)  e.  ( F `
 ( s `  k ) ) ) }
20 eqid 2451 . 2  |-  ( x  e.  { t  |  E. n  e.  om  ( t : suc  n
--> A  /\  ( t `
 (/) )  =  C  /\  A. j  e.  n  ( t `  suc  j )  e.  ( F `  ( t `
 j ) ) ) }  |->  { y  e.  { t  |  E. n  e.  om  ( t : suc  n
--> A  /\  ( t `
 (/) )  =  C  /\  A. j  e.  n  ( t `  suc  j )  e.  ( F `  ( t `
 j ) ) ) }  |  ( dom  y  =  suc  dom  x  /\  ( y  |`  dom  x )  =  x ) } )  =  ( x  e. 
{ t  |  E. n  e.  om  (
t : suc  n --> A  /\  ( t `  (/) )  =  C  /\  A. j  e.  n  ( t `  suc  j
)  e.  ( F `
 ( t `  j ) ) ) }  |->  { y  e. 
{ t  |  E. n  e.  om  (
t : suc  n --> A  /\  ( t `  (/) )  =  C  /\  A. j  e.  n  ( t `  suc  j
)  e.  ( F `
 ( t `  j ) ) ) }  |  ( dom  y  =  suc  dom  x  /\  ( y  |`  dom  x )  =  x ) } )
211, 19, 20axdc3lem4 8725 1  |-  ( ( C  e.  A  /\  F : A --> ( ~P A  \  { (/) } ) )  ->  E. g
( g : om --> A  /\  ( g `  (/) )  =  C  /\  A. k  e.  om  (
g `  suc  k )  e.  ( F `  ( g `  k
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370   E.wex 1587    e. wcel 1758   {cab 2436   A.wral 2795   E.wrex 2796   {crab 2799   _Vcvv 3070    \ cdif 3425   (/)c0 3737   ~Pcpw 3960   {csn 3977    |-> cmpt 4450   suc csuc 4821   dom cdm 4940    |` cres 4942   -->wf 5514   ` cfv 5518   omcom 6578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-dc 8718
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-om 6579  df-1o 7022
This theorem is referenced by:  axdc4lem  8727
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