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Theorem funfv2 5587
Description: The value of a function. Definition of function value in [Enderton] p. 43. (Contributed by NM, 22-May-1998.)
Assertion
Ref Expression
funfv2  |-  ( Fun 
F  ->  ( F `  A )  =  U. { y  |  A F y } )
Distinct variable groups:    y, A    y, F

Proof of Theorem funfv2
StepHypRef Expression
1 funfv 5586 . 2  |-  ( Fun 
F  ->  ( F `  A )  =  U. ( F " { A } ) )
2 funrel 5272 . . . 4  |-  ( Fun 
F  ->  Rel  F )
3 relimasn 5036 . . . 4  |-  ( Rel 
F  ->  ( F " { A } )  =  { y  |  A F y } )
42, 3syl 15 . . 3  |-  ( Fun 
F  ->  ( F " { A } )  =  { y  |  A F y } )
54unieqd 3838 . 2  |-  ( Fun 
F  ->  U. ( F " { A }
)  =  U. {
y  |  A F y } )
61, 5eqtrd 2315 1  |-  ( Fun 
F  ->  ( F `  A )  =  U. { y  |  A F y } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623   {cab 2269   {csn 3640   U.cuni 3827   class class class wbr 4023   "cima 4692   Rel wrel 4694   Fun wfun 5249   ` cfv 5255
This theorem is referenced by:  funfv2f  5588
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-13 1686  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-pow 4188  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-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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