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Theorem fvclss 5981
Description: Upper bound for the class of values of a class. (Contributed by NM, 9-Nov-1995.)
Assertion
Ref Expression
fvclss  |-  { y  |  E. x  y  =  ( F `  x ) }  C_  ( ran  F  u.  { (/)
} )
Distinct variable group:    x, y, F

Proof of Theorem fvclss
StepHypRef Expression
1 eqcom 2439 . . . . . . . . . 10  |-  ( y  =  ( F `  x )  <->  ( F `  x )  =  y )
2 tz6.12i 5752 . . . . . . . . . 10  |-  ( y  =/=  (/)  ->  ( ( F `  x )  =  y  ->  x F y ) )
31, 2syl5bi 210 . . . . . . . . 9  |-  ( y  =/=  (/)  ->  ( y  =  ( F `  x )  ->  x F y ) )
43eximdv 1633 . . . . . . . 8  |-  ( y  =/=  (/)  ->  ( E. x  y  =  ( F `  x )  ->  E. x  x F y ) )
5 vex 2960 . . . . . . . . 9  |-  y  e. 
_V
65elrn 5111 . . . . . . . 8  |-  ( y  e.  ran  F  <->  E. x  x F y )
74, 6syl6ibr 220 . . . . . . 7  |-  ( y  =/=  (/)  ->  ( E. x  y  =  ( F `  x )  ->  y  e.  ran  F
) )
87com12 30 . . . . . 6  |-  ( E. x  y  =  ( F `  x )  ->  ( y  =/=  (/)  ->  y  e.  ran  F ) )
98necon1bd 2673 . . . . 5  |-  ( E. x  y  =  ( F `  x )  ->  ( -.  y  e.  ran  F  ->  y  =  (/) ) )
10 elsn 3830 . . . . 5  |-  ( y  e.  { (/) }  <->  y  =  (/) )
119, 10syl6ibr 220 . . . 4  |-  ( E. x  y  =  ( F `  x )  ->  ( -.  y  e.  ran  F  ->  y  e.  { (/) } ) )
1211orrd 369 . . 3  |-  ( E. x  y  =  ( F `  x )  ->  ( y  e. 
ran  F  \/  y  e.  { (/) } ) )
1312ss2abi 3416 . 2  |-  { y  |  E. x  y  =  ( F `  x ) }  C_  { y  |  ( y  e.  ran  F  \/  y  e.  { (/) } ) }
14 df-un 3326 . 2  |-  ( ran 
F  u.  { (/) } )  =  { y  |  ( y  e. 
ran  F  \/  y  e.  { (/) } ) }
1513, 14sseqtr4i 3382 1  |-  { y  |  E. x  y  =  ( F `  x ) }  C_  ( ran  F  u.  { (/)
} )
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 359   E.wex 1551    = wceq 1653    e. wcel 1726   {cab 2423    =/= wne 2600    u. cun 3319    C_ wss 3321   (/)c0 3629   {csn 3815   class class class wbr 4213   ran crn 4880   ` cfv 5455
This theorem is referenced by:  fvclex  5982
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-cnv 4887  df-dm 4889  df-rn 4890  df-iota 5419  df-fv 5463
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