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Theorem fvclss 6162
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 2438 . . . . . . . . . 10  |-  ( y  =  ( F `  x )  <->  ( F `  x )  =  y )
2 tz6.12i 5901 . . . . . . . . . 10  |-  ( y  =/=  (/)  ->  ( ( F `  x )  =  y  ->  x F y ) )
31, 2syl5bi 220 . . . . . . . . 9  |-  ( y  =/=  (/)  ->  ( y  =  ( F `  x )  ->  x F y ) )
43eximdv 1757 . . . . . . . 8  |-  ( y  =/=  (/)  ->  ( E. x  y  =  ( F `  x )  ->  E. x  x F y ) )
5 vex 3090 . . . . . . . . 9  |-  y  e. 
_V
65elrn 5095 . . . . . . . 8  |-  ( y  e.  ran  F  <->  E. x  x F y )
74, 6syl6ibr 230 . . . . . . 7  |-  ( y  =/=  (/)  ->  ( E. x  y  =  ( F `  x )  ->  y  e.  ran  F
) )
87com12 32 . . . . . 6  |-  ( E. x  y  =  ( F `  x )  ->  ( y  =/=  (/)  ->  y  e.  ran  F ) )
98necon1bd 2649 . . . . 5  |-  ( E. x  y  =  ( F `  x )  ->  ( -.  y  e.  ran  F  ->  y  =  (/) ) )
10 elsn 4016 . . . . 5  |-  ( y  e.  { (/) }  <->  y  =  (/) )
119, 10syl6ibr 230 . . . 4  |-  ( E. x  y  =  ( F `  x )  ->  ( -.  y  e.  ran  F  ->  y  e.  { (/) } ) )
1211orrd 379 . . 3  |-  ( E. x  y  =  ( F `  x )  ->  ( y  e. 
ran  F  \/  y  e.  { (/) } ) )
1312ss2abi 3539 . 2  |-  { y  |  E. x  y  =  ( F `  x ) }  C_  { y  |  ( y  e.  ran  F  \/  y  e.  { (/) } ) }
14 df-un 3447 . 2  |-  ( ran 
F  u.  { (/) } )  =  { y  |  ( y  e. 
ran  F  \/  y  e.  { (/) } ) }
1513, 14sseqtr4i 3503 1  |-  { y  |  E. x  y  =  ( F `  x ) }  C_  ( ran  F  u.  { (/)
} )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 369    = wceq 1437   E.wex 1659    e. wcel 1870   {cab 2414    =/= wne 2625    u. cun 3440    C_ wss 3442   (/)c0 3767   {csn 4002   class class class wbr 4426   ran crn 4855   ` cfv 5601
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-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  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-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-cnv 4862  df-dm 4864  df-rn 4865  df-iota 5565  df-fv 5609
This theorem is referenced by:  fvclex  6779
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