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Theorem funfvdm2 5579
Description: The value of a function. Definition of function value in [Enderton] p. 43. (Contributed by Jim Kingdon, 1-Jan-2019.)
Assertion
Ref Expression
funfvdm2  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  =  U. {
y  |  A F y } )
Distinct variable groups:    y, A    y, F

Proof of Theorem funfvdm2
StepHypRef Expression
1 funfvdm 5578 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  =  U. ( F " { A }
) )
2 imasng 4992 . . . 4  |-  ( A  e.  dom  F  -> 
( F " { A } )  =  {
y  |  A F y } )
32adantl 277 . . 3  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F " { A } )  =  {
y  |  A F y } )
43unieqd 3820 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  U. ( F " { A } )  =  U. { y  |  A F y } )
51, 4eqtrd 2210 1  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  =  U. {
y  |  A F y } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   {cab 2163   {csn 3592   U.cuni 3809   class class class wbr 4002   dom cdm 4625   "cima 4628   Fun wfun 5209   ` cfv 5215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-opab 4064  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-fv 5223
This theorem is referenced by:  funfvdm2f  5580
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