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Theorem funfvdm2f 5486
Description: The value of a function. Version of funfvdm2 5485 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 5485 . 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 2281 . . . . 5  |-  F/_ y
w
52, 3, 4nfbr 3974 . . . 4  |-  F/ y  A F w
6 nfv 1508 . . . 4  |-  F/ w  A F y
7 breq2 3933 . . . 4  |-  ( w  =  y  ->  ( A F w  <->  A F
y ) )
85, 6, 7cbvab 2263 . . 3  |-  { w  |  A F w }  =  { y  |  A F y }
98unieqi 3746 . 2  |-  U. {
w  |  A F w }  =  U. { y  |  A F y }
101, 9syl6eq 2188 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 1331    e. wcel 1480   {cab 2125   F/_wnfc 2268   U.cuni 3736   class class class wbr 3929   dom cdm 4539   Fun wfun 5117   ` cfv 5123
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-fv 5131
This theorem is referenced by: (None)
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