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Theorem nfdfat 38344
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 38330 . 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 5092 . . . 4  |-  F/_ x dom  F
52, 4nfel 2597 . . 3  |-  F/ x  A  e.  dom  F
62nfsn 4054 . . . . 5  |-  F/_ x { A }
73, 6nfres 5123 . . . 4  |-  F/_ x
( F  |`  { A } )
87nffun 5620 . . 3  |-  F/ x Fun  ( F  |`  { A } )
95, 8nfan 1984 . 2  |-  F/ x
( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) )
101, 9nfxfr 1692 1  |-  F/ x  F defAt  A
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370   F/wnf 1663    e. wcel 1868   F/_wnfc 2570   {csn 3996   dom cdm 4850    |` cres 4852   Fun wfun 5592   defAt wdfat 38327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ral 2780  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-br 4421  df-opab 4480  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-res 4862  df-fun 5600  df-dfat 38330
This theorem is referenced by:  nfafv  38350
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