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Theorem fvclss 5964
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 2445 . . . . . . . . . 10  |-  ( y  =  ( F `  x )  <->  ( F `  x )  =  y )
2 tz6.12i 5715 . . . . . . . . . 10  |-  ( y  =/=  (/)  ->  ( ( F `  x )  =  y  ->  x F y ) )
31, 2syl5bi 217 . . . . . . . . 9  |-  ( y  =/=  (/)  ->  ( y  =  ( F `  x )  ->  x F y ) )
43eximdv 1676 . . . . . . . 8  |-  ( y  =/=  (/)  ->  ( E. x  y  =  ( F `  x )  ->  E. x  x F y ) )
5 vex 2980 . . . . . . . . 9  |-  y  e. 
_V
65elrn 5085 . . . . . . . 8  |-  ( y  e.  ran  F  <->  E. x  x F y )
74, 6syl6ibr 227 . . . . . . 7  |-  ( y  =/=  (/)  ->  ( E. x  y  =  ( F `  x )  ->  y  e.  ran  F
) )
87com12 31 . . . . . 6  |-  ( E. x  y  =  ( F `  x )  ->  ( y  =/=  (/)  ->  y  e.  ran  F ) )
98necon1bd 2684 . . . . 5  |-  ( E. x  y  =  ( F `  x )  ->  ( -.  y  e.  ran  F  ->  y  =  (/) ) )
10 elsn 3896 . . . . 5  |-  ( y  e.  { (/) }  <->  y  =  (/) )
119, 10syl6ibr 227 . . . 4  |-  ( E. x  y  =  ( F `  x )  ->  ( -.  y  e.  ran  F  ->  y  e.  { (/) } ) )
1211orrd 378 . . 3  |-  ( E. x  y  =  ( F `  x )  ->  ( y  e. 
ran  F  \/  y  e.  { (/) } ) )
1312ss2abi 3429 . 2  |-  { y  |  E. x  y  =  ( F `  x ) }  C_  { y  |  ( y  e.  ran  F  \/  y  e.  { (/) } ) }
14 df-un 3338 . 2  |-  ( ran 
F  u.  { (/) } )  =  { y  |  ( y  e. 
ran  F  \/  y  e.  { (/) } ) }
1513, 14sseqtr4i 3394 1  |-  { y  |  E. x  y  =  ( F `  x ) }  C_  ( ran  F  u.  { (/)
} )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 368    = wceq 1369   E.wex 1586    e. wcel 1756   {cab 2429    =/= wne 2611    u. cun 3331    C_ wss 3333   (/)c0 3642   {csn 3882   class class class wbr 4297   ran crn 4846   ` cfv 5423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-cnv 4853  df-dm 4855  df-rn 4856  df-iota 5386  df-fv 5431
This theorem is referenced by:  fvclex  6554
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