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Theorem axdc3 8823
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 5704 . . . . 5  |-  ( t  =  s  ->  (
t : suc  n --> A 
<->  s : suc  n --> A ) )
3 fveq1 5856 . . . . . 6  |-  ( t  =  s  ->  (
t `  (/) )  =  ( s `  (/) ) )
43eqeq1d 2462 . . . . 5  |-  ( t  =  s  ->  (
( t `  (/) )  =  C  <->  ( s `  (/) )  =  C ) )
5 fveq1 5856 . . . . . . . 8  |-  ( t  =  s  ->  (
t `  suc  j )  =  ( s `  suc  j ) )
6 fveq1 5856 . . . . . . . . 9  |-  ( t  =  s  ->  (
t `  j )  =  ( s `  j ) )
76fveq2d 5861 . . . . . . . 8  |-  ( t  =  s  ->  ( F `  ( t `  j ) )  =  ( F `  (
s `  j )
) )
85, 7eleq12d 2542 . . . . . . 7  |-  ( t  =  s  ->  (
( t `  suc  j )  e.  ( F `  ( t `
 j ) )  <-> 
( s `  suc  j )  e.  ( F `  ( s `
 j ) ) ) )
98ralbidv 2896 . . . . . 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 4936 . . . . . . . . 9  |-  ( j  =  k  ->  suc  j  =  suc  k )
1110fveq2d 5861 . . . . . . . 8  |-  ( j  =  k  ->  (
s `  suc  j )  =  ( s `  suc  k ) )
12 fveq2 5857 . . . . . . . . 9  |-  ( j  =  k  ->  (
s `  j )  =  ( s `  k ) )
1312fveq2d 5861 . . . . . . . 8  |-  ( j  =  k  ->  ( F `  ( s `  j ) )  =  ( F `  (
s `  k )
) )
1411, 13eleq12d 2542 . . . . . . 7  |-  ( j  =  k  ->  (
( s `  suc  j )  e.  ( F `  ( s `
 j ) )  <-> 
( s `  suc  k )  e.  ( F `  ( s `
 k ) ) ) )
1514cbvralv 3081 . . . . . 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 1294 . . . 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 2966 . . 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 2603 . 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 2460 . 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 8822 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 968    = wceq 1374   E.wex 1591    e. wcel 1762   {cab 2445   A.wral 2807   E.wrex 2808   {crab 2811   _Vcvv 3106    \ cdif 3466   (/)c0 3778   ~Pcpw 4003   {csn 4020    |-> cmpt 4498   suc csuc 4873   dom cdm 4992    |` cres 4994   -->wf 5575   ` cfv 5579   omcom 6671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-dc 8815
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-om 6672  df-1o 7120
This theorem is referenced by:  axdc4lem  8824
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