MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  axdc2 Structured version   Unicode version

Theorem axdc2 8331
Description: An apparent strengthening of ax-dc 8328 (but derived from it) which shows that there is a denumerable sequence  g for any function that maps elements of a set  A to nonempty subsets of 
A such that  g (
x  +  1 )  e.  F ( g ( x ) ) for all  x  e.  om. The finitistic version of this can be proven by induction, but the infinite version requires this new axiom. (Contributed by Mario Carneiro, 25-Jan-2013.)
Hypothesis
Ref Expression
axdc2.1  |-  A  e. 
_V
Assertion
Ref Expression
axdc2  |-  ( ( A  =/=  (/)  /\  F : A --> ( ~P A  \  { (/) } ) )  ->  E. g ( g : om --> A  /\  A. k  e.  om  (
g `  suc  k )  e.  ( F `  ( g `  k
) ) ) )
Distinct variable groups:    A, g,
k    g, F, k

Proof of Theorem axdc2
Dummy variables  h  s  t  x  y  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 axdc2.1 . 2  |-  A  e. 
_V
2 eleq1 2498 . . . . 5  |-  ( s  =  x  ->  (
s  e.  A  <->  x  e.  A ) )
32adantr 453 . . . 4  |-  ( ( s  =  x  /\  t  =  y )  ->  ( s  e.  A  <->  x  e.  A ) )
4 fveq2 5730 . . . . . 6  |-  ( s  =  x  ->  ( F `  s )  =  ( F `  x ) )
54eleq2d 2505 . . . . 5  |-  ( s  =  x  ->  (
t  e.  ( F `
 s )  <->  t  e.  ( F `  x ) ) )
6 eleq1 2498 . . . . 5  |-  ( t  =  y  ->  (
t  e.  ( F `
 x )  <->  y  e.  ( F `  x ) ) )
75, 6sylan9bb 682 . . . 4  |-  ( ( s  =  x  /\  t  =  y )  ->  ( t  e.  ( F `  s )  <-> 
y  e.  ( F `
 x ) ) )
83, 7anbi12d 693 . . 3  |-  ( ( s  =  x  /\  t  =  y )  ->  ( ( s  e.  A  /\  t  e.  ( F `  s
) )  <->  ( x  e.  A  /\  y  e.  ( F `  x
) ) ) )
98cbvopabv 4279 . 2  |-  { <. s ,  t >.  |  ( s  e.  A  /\  t  e.  ( F `  s ) ) }  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( F `  x ) ) }
10 fveq2 5730 . . 3  |-  ( n  =  x  ->  (
h `  n )  =  ( h `  x ) )
1110cbvmptv 4302 . 2  |-  ( n  e.  om  |->  ( h `
 n ) )  =  ( x  e. 
om  |->  ( h `  x ) )
121, 9, 11axdc2lem 8330 1  |-  ( ( A  =/=  (/)  /\  F : A --> ( ~P A  \  { (/) } ) )  ->  E. g ( g : om --> A  /\  A. k  e.  om  (
g `  suc  k )  e.  ( F `  ( g `  k
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   E.wex 1551    e. wcel 1726    =/= wne 2601   A.wral 2707   _Vcvv 2958    \ cdif 3319   (/)c0 3630   ~Pcpw 3801   {csn 3816   {copab 4267    e. cmpt 4268   suc csuc 4585   omcom 4847   -->wf 5452   ` cfv 5456
This theorem is referenced by:  axdc3lem4  8335
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-dc 8328
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464  df-1o 6726
  Copyright terms: Public domain W3C validator