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Theorem funfvdm2f 5454
Description: The value of a function. Version of funfvdm2 5453 using a bound-variable hypotheses instead of distinct variable conditions. (Contributed by Jim Kingdon, 1-Jan-2019.)
Hypotheses
Ref Expression
funfvdm2f.1  |-  F/_ y A
funfvdm2f.2  |-  F/_ y F
Assertion
Ref Expression
funfvdm2f  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  =  U. {
y  |  A F y } )

Proof of Theorem funfvdm2f
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 funfvdm2 5453 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  =  U. {
w  |  A F w } )
2 funfvdm2f.1 . . . . 5  |-  F/_ y A
3 funfvdm2f.2 . . . . 5  |-  F/_ y F
4 nfcv 2258 . . . . 5  |-  F/_ y
w
52, 3, 4nfbr 3944 . . . 4  |-  F/ y  A F w
6 nfv 1493 . . . 4  |-  F/ w  A F y
7 breq2 3903 . . . 4  |-  ( w  =  y  ->  ( A F w  <->  A F
y ) )
85, 6, 7cbvab 2240 . . 3  |-  { w  |  A F w }  =  { y  |  A F y }
98unieqi 3716 . 2  |-  U. {
w  |  A F w }  =  U. { y  |  A F y }
101, 9syl6eq 2166 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 103    = wceq 1316    e. wcel 1465   {cab 2103   F/_wnfc 2245   U.cuni 3706   class class class wbr 3899   dom cdm 4509   Fun wfun 5087   ` cfv 5093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-fv 5101
This theorem is referenced by: (None)
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