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Theorem axdc3 8096
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 5391 . . . . 5  |-  ( t  =  s  ->  (
t : suc  n --> A 
<->  s : suc  n --> A ) )
3 fveq1 5540 . . . . . 6  |-  ( t  =  s  ->  (
t `  (/) )  =  ( s `  (/) ) )
43eqeq1d 2304 . . . . 5  |-  ( t  =  s  ->  (
( t `  (/) )  =  C  <->  ( s `  (/) )  =  C ) )
5 fveq1 5540 . . . . . . . 8  |-  ( t  =  s  ->  (
t `  suc  j )  =  ( s `  suc  j ) )
6 fveq1 5540 . . . . . . . . 9  |-  ( t  =  s  ->  (
t `  j )  =  ( s `  j ) )
76fveq2d 5545 . . . . . . . 8  |-  ( t  =  s  ->  ( F `  ( t `  j ) )  =  ( F `  (
s `  j )
) )
85, 7eleq12d 2364 . . . . . . 7  |-  ( t  =  s  ->  (
( t `  suc  j )  e.  ( F `  ( t `
 j ) )  <-> 
( s `  suc  j )  e.  ( F `  ( s `
 j ) ) ) )
98ralbidv 2576 . . . . . 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 4473 . . . . . . . . 9  |-  ( j  =  k  ->  suc  j  =  suc  k )
1110fveq2d 5545 . . . . . . . 8  |-  ( j  =  k  ->  (
s `  suc  j )  =  ( s `  suc  k ) )
12 fveq2 5541 . . . . . . . . 9  |-  ( j  =  k  ->  (
s `  j )  =  ( s `  k ) )
1312fveq2d 5545 . . . . . . . 8  |-  ( j  =  k  ->  ( F `  ( s `  j ) )  =  ( F `  (
s `  k )
) )
1411, 13eleq12d 2364 . . . . . . 7  |-  ( j  =  k  ->  (
( s `  suc  j )  e.  ( F `  ( s `
 j ) )  <-> 
( s `  suc  k )  e.  ( F `  ( s `
 k ) ) ) )
1514cbvralv 2777 . . . . . 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 252 . . . . 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 1252 . . . 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 2577 . . 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 2415 . 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 2296 . 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 8095 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 4    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   E.wrex 2557   {crab 2560   _Vcvv 2801    \ cdif 3162   (/)c0 3468   ~Pcpw 3638   {csn 3653    e. cmpt 4093   suc csuc 4410   omcom 4672   dom cdm 4705    |` cres 4707   -->wf 5267   ` cfv 5271
This theorem is referenced by:  axdc4lem  8097
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-dc 8088
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6495
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