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Theorem dnnumch2 30623
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 30622 . 2  |-  ( ph  ->  E. x  e.  On  ( F  |`  x ) : x -1-1-onto-> A )
5 f1ofo 5823 . . . . . 6  |-  ( ( F  |`  x ) : x -1-1-onto-> A  ->  ( F  |`  x ) : x
-onto-> A )
6 forn 5798 . . . . . 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 5297 . . . . . 6  |-  ( F  |`  x )  C_  F
9 rnss 5231 . . . . . 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 3539 . . . 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 2958 . 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 setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   _Vcvv 3113    \ cdif 3473    C_ wss 3476   (/)c0 3785   ~Pcpw 4010    |-> cmpt 4505   Oncon0 4878   ran crn 5000    |` cres 5001   -onto->wfo 5586   -1-1-onto->wf1o 5587   ` cfv 5588  recscrecs 7041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-recs 7042
This theorem is referenced by:  dnnumch3lem  30624  dnnumch3  30625
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