Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nfdfat Structured version   Unicode version

Theorem nfdfat 31682
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 31668 . 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 5242 . . . 4  |-  F/_ x dom  F
52, 4nfel 2642 . . 3  |-  F/ x  A  e.  dom  F
62nfsn 4085 . . . . 5  |-  F/_ x { A }
73, 6nfres 5273 . . . 4  |-  F/_ x
( F  |`  { A } )
87nffun 5608 . . 3  |-  F/ x Fun  ( F  |`  { A } )
95, 8nfan 1875 . 2  |-  F/ x
( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) )
101, 9nfxfr 1625 1  |-  F/ x  F defAt  A
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369   F/wnf 1599    e. wcel 1767   F/_wnfc 2615   {csn 4027   dom cdm 4999    |` cres 5001   Fun wfun 5580   defAt wdfat 31665
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-res 5011  df-fun 5588  df-dfat 31668
This theorem is referenced by:  nfafv  31688
  Copyright terms: Public domain W3C validator