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Theorem nfdfat 27318
Description: Bound-variable hypothesis builder for "defined at". To prove a deduction version of this theorem is not easily possible because many deduction versions for bound-variable hypothesis builder for constructs the definition of "defined at" is based on are not available (e.g. for Fun/Rel, dom, C_, etc.). (Contributed by Alexander van der Vekens, 26-May-2017.)
Hypotheses
Ref Expression
nfdfat.1  |-  F/_ x F
nfdfat.2  |-  F/_ x A
Assertion
Ref Expression
nfdfat  |-  F/ x  F defAt  A

Proof of Theorem nfdfat
StepHypRef Expression
1 df-dfat 27297 . 2  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) ) )
2 nfdfat.2 . . . 4  |-  F/_ x A
3 nfdfat.1 . . . . 5  |-  F/_ x F
43nfdm 5002 . . . 4  |-  F/_ x dom  F
52, 4nfel 2502 . . 3  |-  F/ x  A  e.  dom  F
62nfsn 3767 . . . . 5  |-  F/_ x { A }
73, 6nfres 5039 . . . 4  |-  F/_ x
( F  |`  { A } )
87nffun 5359 . . 3  |-  F/ x Fun  ( F  |`  { A } )
95, 8nfan 1829 . 2  |-  F/ x
( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) )
101, 9nfxfr 1570 1  |-  F/ x  F defAt  A
Colors of variables: wff set class
Syntax hints:    /\ wa 358   F/wnf 1544    e. wcel 1710   F/_wnfc 2481   {csn 3716   dom cdm 4771    |` cres 4773   Fun wfun 5331   defAt wdfat 27294
This theorem is referenced by:  nfafv  27324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4105  df-opab 4159  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-res 4783  df-fun 5339  df-dfat 27297
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