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Theorem axdc3 8034
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
StepHypRef Expression
1 axdc3.1 . 2  |-  A  e. 
_V
2 feq1 5299 . . . . 5  |-  ( t  =  s  ->  (
t : suc  n --> A 
<->  s : suc  n --> A ) )
3 fveq1 5443 . . . . . 6  |-  ( t  =  s  ->  (
t `  (/) )  =  ( s `  (/) ) )
43eqeq1d 2264 . . . . 5  |-  ( t  =  s  ->  (
( t `  (/) )  =  C  <->  ( s `  (/) )  =  C ) )
5 fveq1 5443 . . . . . . . 8  |-  ( t  =  s  ->  (
t `  suc  j )  =  ( s `  suc  j ) )
6 fveq1 5443 . . . . . . . . 9  |-  ( t  =  s  ->  (
t `  j )  =  ( s `  j ) )
76fveq2d 5448 . . . . . . . 8  |-  ( t  =  s  ->  ( F `  ( t `  j ) )  =  ( F `  (
s `  j )
) )
85, 7eleq12d 2324 . . . . . . 7  |-  ( t  =  s  ->  (
( t `  suc  j )  e.  ( F `  ( t `
 j ) )  <-> 
( s `  suc  j )  e.  ( F `  ( s `
 j ) ) ) )
98ralbidv 2536 . . . . . 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 4415 . . . . . . . . 9  |-  ( j  =  k  ->  suc  j  =  suc  k )
1110fveq2d 5448 . . . . . . . 8  |-  ( j  =  k  ->  (
s `  suc  j )  =  ( s `  suc  k ) )
12 fveq2 5444 . . . . . . . . 9  |-  ( j  =  k  ->  (
s `  j )  =  ( s `  k ) )
1312fveq2d 5448 . . . . . . . 8  |-  ( j  =  k  ->  ( F `  ( s `  j ) )  =  ( F `  (
s `  k )
) )
1411, 13eleq12d 2324 . . . . . . 7  |-  ( j  =  k  ->  (
( s `  suc  j )  e.  ( F `  ( s `
 j ) )  <-> 
( s `  suc  k )  e.  ( F `  ( s `
 k ) ) ) )
1514cbvralv 2734 . . . . . 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 254 . . . . 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 1257 . . . 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 2537 . . 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 2375 . 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 2256 . 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 8033 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 set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 939   E.wex 1537    = wceq 1619    e. wcel 1621   {cab 2242   A.wral 2516   E.wrex 2517   {crab 2520   _Vcvv 2757    \ cdif 3110   (/)c0 3416   ~Pcpw 3585   {csn 3600    e. cmpt 4037   suc csuc 4352   omcom 4614   dom cdm 4647    |` cres 4649   -->wf 4655   ` cfv 4659
This theorem is referenced by:  axdc4lem  8035
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-dc 8026
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-reu 2523  df-rab 2525  df-v 2759  df-sbc 2953  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-1o 6433
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