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Theorem dnnumch2 35609
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 35608 . 2  |-  ( ph  ->  E. x  e.  On  ( F  |`  x ) : x -1-1-onto-> A )
5 f1ofo 5838 . . . . . 6  |-  ( ( F  |`  x ) : x -1-1-onto-> A  ->  ( F  |`  x ) : x
-onto-> A )
6 forn 5813 . . . . . 6  |-  ( ( F  |`  x ) : x -onto-> A  ->  ran  ( F  |`  x
)  =  A )
75, 6syl 17 . . . . 5  |-  ( ( F  |`  x ) : x -1-1-onto-> A  ->  ran  ( F  |`  x )  =  A )
8 resss 5148 . . . . . 6  |-  ( F  |`  x )  C_  F
9 rnss 5083 . . . . . 6  |-  ( ( F  |`  x )  C_  F  ->  ran  ( F  |`  x )  C_  ran  F )
108, 9mp1i 13 . . . . 5  |-  ( ( F  |`  x ) : x -1-1-onto-> A  ->  ran  ( F  |`  x )  C_  ran  F )
117, 10eqsstr3d 3505 . . . 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 2927 . 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 1437    e. wcel 1870    =/= wne 2625   A.wral 2782   E.wrex 2783   _Vcvv 3087    \ cdif 3439    C_ wss 3442   (/)c0 3767   ~Pcpw 3985    |-> cmpt 4484   ran crn 4855    |` cres 4856   Oncon0 5442   -onto->wfo 5599   -1-1-onto->wf1o 5600   ` cfv 5601  recscrecs 7097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-wrecs 7036  df-recs 7098
This theorem is referenced by:  dnnumch3lem  35610  dnnumch3  35611
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