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Theorem axdc3 8611
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 5530 . . . . 5  |-  ( t  =  s  ->  (
t : suc  n --> A 
<->  s : suc  n --> A ) )
3 fveq1 5678 . . . . . 6  |-  ( t  =  s  ->  (
t `  (/) )  =  ( s `  (/) ) )
43eqeq1d 2441 . . . . 5  |-  ( t  =  s  ->  (
( t `  (/) )  =  C  <->  ( s `  (/) )  =  C ) )
5 fveq1 5678 . . . . . . . 8  |-  ( t  =  s  ->  (
t `  suc  j )  =  ( s `  suc  j ) )
6 fveq1 5678 . . . . . . . . 9  |-  ( t  =  s  ->  (
t `  j )  =  ( s `  j ) )
76fveq2d 5683 . . . . . . . 8  |-  ( t  =  s  ->  ( F `  ( t `  j ) )  =  ( F `  (
s `  j )
) )
85, 7eleq12d 2501 . . . . . . 7  |-  ( t  =  s  ->  (
( t `  suc  j )  e.  ( F `  ( t `
 j ) )  <-> 
( s `  suc  j )  e.  ( F `  ( s `
 j ) ) ) )
98ralbidv 2725 . . . . . 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 4771 . . . . . . . . 9  |-  ( j  =  k  ->  suc  j  =  suc  k )
1110fveq2d 5683 . . . . . . . 8  |-  ( j  =  k  ->  (
s `  suc  j )  =  ( s `  suc  k ) )
12 fveq2 5679 . . . . . . . . 9  |-  ( j  =  k  ->  (
s `  j )  =  ( s `  k ) )
1312fveq2d 5683 . . . . . . . 8  |-  ( j  =  k  ->  ( F `  ( s `  j ) )  =  ( F `  (
s `  k )
) )
1411, 13eleq12d 2501 . . . . . . 7  |-  ( j  =  k  ->  (
( s `  suc  j )  e.  ( F `  ( s `
 j ) )  <-> 
( s `  suc  k )  e.  ( F `  ( s `
 k ) ) ) )
1514cbvralv 2937 . . . . . 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 1282 . . . 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 2726 . . 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 2552 . 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 2433 . 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 8610 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 958    = wceq 1362   E.wex 1589    e. wcel 1755   {cab 2419   A.wral 2705   E.wrex 2706   {crab 2709   _Vcvv 2962    \ cdif 3313   (/)c0 3625   ~Pcpw 3848   {csn 3865    e. cmpt 4338   suc csuc 4708   dom cdm 4827    |` cres 4829   -->wf 5402   ` cfv 5406   omcom 6465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-dc 8603
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-om 6466  df-1o 6908
This theorem is referenced by:  axdc4lem  8612
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