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Theorem dnnumch2 29541
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 29540 . 2  |-  ( ph  ->  E. x  e.  On  ( F  |`  x ) : x -1-1-onto-> A )
5 f1ofo 5751 . . . . . 6  |-  ( ( F  |`  x ) : x -1-1-onto-> A  ->  ( F  |`  x ) : x
-onto-> A )
6 forn 5726 . . . . . 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 5237 . . . . . 6  |-  ( F  |`  x )  C_  F
9 rnss 5171 . . . . . 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 3494 . . . 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 2944 . 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 1370    e. wcel 1758    =/= wne 2645   A.wral 2796   E.wrex 2797   _Vcvv 3072    \ cdif 3428    C_ wss 3431   (/)c0 3740   ~Pcpw 3963    |-> cmpt 4453   Oncon0 4822   ran crn 4944    |` cres 4945   -onto->wfo 5519   -1-1-onto->wf1o 5520   ` cfv 5521  recscrecs 6936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-recs 6937
This theorem is referenced by:  dnnumch3lem  29542  dnnumch3  29543
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