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Theorem fvclss 5760
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 2285 . . . . . . . . . 10  |-  ( y  =  ( F `  x )  <->  ( F `  x )  =  y )
2 tz6.12i 5548 . . . . . . . . . 10  |-  ( y  =/=  (/)  ->  ( ( F `  x )  =  y  ->  x F y ) )
31, 2syl5bi 208 . . . . . . . . 9  |-  ( y  =/=  (/)  ->  ( y  =  ( F `  x )  ->  x F y ) )
43eximdv 1608 . . . . . . . 8  |-  ( y  =/=  (/)  ->  ( E. x  y  =  ( F `  x )  ->  E. x  x F y ) )
5 vex 2791 . . . . . . . . 9  |-  y  e. 
_V
65elrn 4919 . . . . . . . 8  |-  ( y  e.  ran  F  <->  E. x  x F y )
74, 6syl6ibr 218 . . . . . . 7  |-  ( y  =/=  (/)  ->  ( E. x  y  =  ( F `  x )  ->  y  e.  ran  F
) )
87com12 27 . . . . . 6  |-  ( E. x  y  =  ( F `  x )  ->  ( y  =/=  (/)  ->  y  e.  ran  F ) )
98necon1bd 2514 . . . . 5  |-  ( E. x  y  =  ( F `  x )  ->  ( -.  y  e.  ran  F  ->  y  =  (/) ) )
10 elsn 3655 . . . . 5  |-  ( y  e.  { (/) }  <->  y  =  (/) )
119, 10syl6ibr 218 . . . 4  |-  ( E. x  y  =  ( F `  x )  ->  ( -.  y  e.  ran  F  ->  y  e.  { (/) } ) )
1211orrd 367 . . 3  |-  ( E. x  y  =  ( F `  x )  ->  ( y  e. 
ran  F  \/  y  e.  { (/) } ) )
1312ss2abi 3245 . 2  |-  { y  |  E. x  y  =  ( F `  x ) }  C_  { y  |  ( y  e.  ran  F  \/  y  e.  { (/) } ) }
14 df-un 3157 . 2  |-  ( ran 
F  u.  { (/) } )  =  { y  |  ( y  e. 
ran  F  \/  y  e.  { (/) } ) }
1513, 14sseqtr4i 3211 1  |-  { y  |  E. x  y  =  ( F `  x ) }  C_  ( ran  F  u.  { (/)
} )
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 357   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446    u. cun 3150    C_ wss 3152   (/)c0 3455   {csn 3640   class class class wbr 4023   ran crn 4690   ` cfv 5255
This theorem is referenced by:  fvclex  5761
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-cnv 4697  df-dm 4699  df-rn 4700  df-iota 5219  df-fv 5263
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