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Theorem dnnumch2 27142
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 27141 . 2  |-  ( ph  ->  E. x  e.  On  ( F  |`  x ) : x -1-1-onto-> A )
5 f1ofo 5479 . . . . . 6  |-  ( ( F  |`  x ) : x -1-1-onto-> A  ->  ( F  |`  x ) : x
-onto-> A )
6 forn 5454 . . . . . 6  |-  ( ( F  |`  x ) : x -onto-> A  ->  ran  ( F  |`  x
)  =  A )
75, 6syl 15 . . . . 5  |-  ( ( F  |`  x ) : x -1-1-onto-> A  ->  ran  ( F  |`  x )  =  A )
8 resss 4979 . . . . . 6  |-  ( F  |`  x )  C_  F
9 rnss 4907 . . . . . 6  |-  ( ( F  |`  x )  C_  F  ->  ran  ( F  |`  x )  C_  ran  F )
108, 9mp1i 11 . . . . 5  |-  ( ( F  |`  x ) : x -1-1-onto-> A  ->  ran  ( F  |`  x )  C_  ran  F )
117, 10eqsstr3d 3213 . . . 4  |-  ( ( F  |`  x ) : x -1-1-onto-> A  ->  A  C_  ran  F )
1211a1i 10 . . 3  |-  ( ph  ->  ( ( F  |`  x ) : x -1-1-onto-> A  ->  A  C_  ran  F ) )
1312rexlimdvw 2670 . 2  |-  ( ph  ->  ( E. x  e.  On  ( F  |`  x ) : x -1-1-onto-> A  ->  A  C_  ran  F ) )
144, 13mpd 14 1  |-  ( ph  ->  A  C_  ran  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   _Vcvv 2788    \ cdif 3149    C_ wss 3152   (/)c0 3455   ~Pcpw 3625    e. cmpt 4077   Oncon0 4392   ran crn 4690    |` cres 4691   -onto->wfo 5253   -1-1-onto->wf1o 5254   ` cfv 5255  recscrecs 6387
This theorem is referenced by:  dnnumch3lem  27143  dnnumch3  27144
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6388
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