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Theorem fvclss 5972
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 2437 . . . . . . . . . 10  |-  ( y  =  ( F `  x )  <->  ( F `  x )  =  y )
2 tz6.12i 5743 . . . . . . . . . 10  |-  ( y  =/=  (/)  ->  ( ( F `  x )  =  y  ->  x F y ) )
31, 2syl5bi 209 . . . . . . . . 9  |-  ( y  =/=  (/)  ->  ( y  =  ( F `  x )  ->  x F y ) )
43eximdv 1632 . . . . . . . 8  |-  ( y  =/=  (/)  ->  ( E. x  y  =  ( F `  x )  ->  E. x  x F y ) )
5 vex 2951 . . . . . . . . 9  |-  y  e. 
_V
65elrn 5102 . . . . . . . 8  |-  ( y  e.  ran  F  <->  E. x  x F y )
74, 6syl6ibr 219 . . . . . . 7  |-  ( y  =/=  (/)  ->  ( E. x  y  =  ( F `  x )  ->  y  e.  ran  F
) )
87com12 29 . . . . . 6  |-  ( E. x  y  =  ( F `  x )  ->  ( y  =/=  (/)  ->  y  e.  ran  F ) )
98necon1bd 2666 . . . . 5  |-  ( E. x  y  =  ( F `  x )  ->  ( -.  y  e.  ran  F  ->  y  =  (/) ) )
10 elsn 3821 . . . . 5  |-  ( y  e.  { (/) }  <->  y  =  (/) )
119, 10syl6ibr 219 . . . 4  |-  ( E. x  y  =  ( F `  x )  ->  ( -.  y  e.  ran  F  ->  y  e.  { (/) } ) )
1211orrd 368 . . 3  |-  ( E. x  y  =  ( F `  x )  ->  ( y  e. 
ran  F  \/  y  e.  { (/) } ) )
1312ss2abi 3407 . 2  |-  { y  |  E. x  y  =  ( F `  x ) }  C_  { y  |  ( y  e.  ran  F  \/  y  e.  { (/) } ) }
14 df-un 3317 . 2  |-  ( ran 
F  u.  { (/) } )  =  { y  |  ( y  e. 
ran  F  \/  y  e.  { (/) } ) }
1513, 14sseqtr4i 3373 1  |-  { y  |  E. x  y  =  ( F `  x ) }  C_  ( ran  F  u.  { (/)
} )
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 358   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2421    =/= wne 2598    u. cun 3310    C_ wss 3312   (/)c0 3620   {csn 3806   class class class wbr 4204   ran crn 4871   ` cfv 5446
This theorem is referenced by:  fvclex  5973
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-cnv 4878  df-dm 4880  df-rn 4881  df-iota 5410  df-fv 5454
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