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Theorem funfv2 5694
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 5693 . 2  |-  ( Fun 
F  ->  ( F `  A )  =  U. ( F " { A } ) )
2 funrel 5375 . . . 4  |-  ( Fun 
F  ->  Rel  F )
3 relimasn 5139 . . . 4  |-  ( Rel 
F  ->  ( F " { A } )  =  { y  |  A F y } )
42, 3syl 15 . . 3  |-  ( Fun 
F  ->  ( F " { A } )  =  { y  |  A F y } )
54unieqd 3940 . 2  |-  ( Fun 
F  ->  U. ( F " { A }
)  =  U. {
y  |  A F y } )
61, 5eqtrd 2398 1  |-  ( Fun 
F  ->  ( F `  A )  =  U. { y  |  A F y } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1647   {cab 2352   {csn 3729   U.cuni 3929   class class class wbr 4125   "cima 4795   Rel wrel 4797   Fun wfun 5352   ` cfv 5358
This theorem is referenced by:  funfv2f  5695
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-fv 5366
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