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Theorem nfdfat 27984
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 27964 . 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 5114 . . . 4  |-  F/_ x dom  F
52, 4nfel 2582 . . 3  |-  F/ x  A  e.  dom  F
62nfsn 3868 . . . . 5  |-  F/_ x { A }
73, 6nfres 5151 . . . 4  |-  F/_ x
( F  |`  { A } )
87nffun 5479 . . 3  |-  F/ x Fun  ( F  |`  { A } )
95, 8nfan 1847 . 2  |-  F/ x
( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) )
101, 9nfxfr 1580 1  |-  F/ x  F defAt  A
Colors of variables: wff set class
Syntax hints:    /\ wa 360   F/wnf 1554    e. wcel 1726   F/_wnfc 2561   {csn 3816   dom cdm 4881    |` cres 4883   Fun wfun 5451   defAt wdfat 27961
This theorem is referenced by:  nfafv  27990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-res 4893  df-fun 5459  df-dfat 27964
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