ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  funfvdm2f Unicode version

Theorem funfvdm2f 5363
Description: The value of a function. Version of funfvdm2 5362 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 5362 . 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 2228 . . . . 5  |-  F/_ y
w
52, 3, 4nfbr 3887 . . . 4  |-  F/ y  A F w
6 nfv 1466 . . . 4  |-  F/ w  A F y
7 breq2 3847 . . . 4  |-  ( w  =  y  ->  ( A F w  <->  A F
y ) )
85, 6, 7cbvab 2210 . . 3  |-  { w  |  A F w }  =  { y  |  A F y }
98unieqi 3661 . 2  |-  U. {
w  |  A F w }  =  U. { y  |  A F y }
101, 9syl6eq 2136 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 102    = wceq 1289    e. wcel 1438   {cab 2074   F/_wnfc 2215   U.cuni 3651   class class class wbr 3843   dom cdm 4436   Fun wfun 5004   ` cfv 5010
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-pow 4007  ax-pr 4034
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-br 3844  df-opab 3898  df-id 4118  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-ima 4449  df-iota 4975  df-fun 5012  df-fn 5013  df-fv 5018
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator