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Theorem nfdfat 37564
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 37550 . 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 5064 . . . 4  |-  F/_ x dom  F
52, 4nfel 2577 . . 3  |-  F/ x  A  e.  dom  F
62nfsn 4028 . . . . 5  |-  F/_ x { A }
73, 6nfres 5095 . . . 4  |-  F/_ x
( F  |`  { A } )
87nffun 5590 . . 3  |-  F/ x Fun  ( F  |`  { A } )
95, 8nfan 1956 . 2  |-  F/ x
( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) )
101, 9nfxfr 1666 1  |-  F/ x  F defAt  A
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367   F/wnf 1637    e. wcel 1842   F/_wnfc 2550   {csn 3971   dom cdm 4822    |` cres 4824   Fun wfun 5562   defAt wdfat 37547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2758  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-opab 4453  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-res 4834  df-fun 5570  df-dfat 37550
This theorem is referenced by:  nfafv  37570
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