Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dnnumch2 Unicode version

Theorem dnnumch2 26811
Description: Define an enumeration (weak dominance version) of a set from a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
Hypotheses
Ref Expression
dnnumch.f  |-  F  = recs ( ( z  e. 
_V  |->  ( G `  ( A  \  ran  z
) ) ) )
dnnumch.a  |-  ( ph  ->  A  e.  V )
dnnumch.g  |-  ( ph  ->  A. y  e.  ~P  A ( y  =/=  (/)  ->  ( G `  y )  e.  y ) )
Assertion
Ref Expression
dnnumch2  |-  ( ph  ->  A  C_  ran  F )
Distinct variable groups:    y, F    y, G, z    y, A, z
Allowed substitution hints:    ph( y, z)    F( z)    V( y, z)

Proof of Theorem dnnumch2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dnnumch.f . . 3  |-  F  = recs ( ( z  e. 
_V  |->  ( G `  ( A  \  ran  z
) ) ) )
2 dnnumch.a . . 3  |-  ( ph  ->  A  e.  V )
3 dnnumch.g . . 3  |-  ( ph  ->  A. y  e.  ~P  A ( y  =/=  (/)  ->  ( G `  y )  e.  y ) )
41, 2, 3dnnumch1 26810 . 2  |-  ( ph  ->  E. x  e.  On  ( F  |`  x ) : x -1-1-onto-> A )
5 f1ofo 5621 . . . . . 6  |-  ( ( F  |`  x ) : x -1-1-onto-> A  ->  ( F  |`  x ) : x
-onto-> A )
6 forn 5596 . . . . . 6  |-  ( ( F  |`  x ) : x -onto-> A  ->  ran  ( F  |`  x
)  =  A )
75, 6syl 16 . . . . 5  |-  ( ( F  |`  x ) : x -1-1-onto-> A  ->  ran  ( F  |`  x )  =  A )
8 resss 5110 . . . . . 6  |-  ( F  |`  x )  C_  F
9 rnss 5038 . . . . . 6  |-  ( ( F  |`  x )  C_  F  ->  ran  ( F  |`  x )  C_  ran  F )
108, 9mp1i 12 . . . . 5  |-  ( ( F  |`  x ) : x -1-1-onto-> A  ->  ran  ( F  |`  x )  C_  ran  F )
117, 10eqsstr3d 3326 . . . 4  |-  ( ( F  |`  x ) : x -1-1-onto-> A  ->  A  C_  ran  F )
1211a1i 11 . . 3  |-  ( ph  ->  ( ( F  |`  x ) : x -1-1-onto-> A  ->  A  C_  ran  F ) )
1312rexlimdvw 2776 . 2  |-  ( ph  ->  ( E. x  e.  On  ( F  |`  x ) : x -1-1-onto-> A  ->  A  C_  ran  F ) )
144, 13mpd 15 1  |-  ( ph  ->  A  C_  ran  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717    =/= wne 2550   A.wral 2649   E.wrex 2650   _Vcvv 2899    \ cdif 3260    C_ wss 3263   (/)c0 3571   ~Pcpw 3742    e. cmpt 4207   Oncon0 4522   ran crn 4819    |` cres 4820   -onto->wfo 5392   -1-1-onto->wf1o 5393   ` cfv 5394  recscrecs 6568
This theorem is referenced by:  dnnumch3lem  26812  dnnumch3  26813
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-suc 4528  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-recs 6569
  Copyright terms: Public domain W3C validator