Theorem nfdfat 41210

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

Step	Hyp	Ref	Expression
1		df-dfat 41196	. 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
4	3	nfdm 5367	. . . 4 |- F/_ x dom F
5	2, 4	nfel 2777	. . 3 |- F/ x A e. dom F
6	2	nfsn 4242	. . . . 5 |- F/_ x { A }
7	3, 6	nfres 5398	. . . 4 |- F/_ x ( F |` { A } )
8	7	nffun 5911	. . 3 |- F/ x Fun ( F |` { A } )
9	5, 8	nfan 1828	. 2 |- F/ x ( A e. dom F /\ Fun ( F |` { A } ) )
10	1, 9	nfxfr 1779	1 |- F/ x F defAt A

Syntax hints:   /\ wa 384   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   {csn 4177   dom cdm 5114   |` cres 5116   Fun wfun 5882   defAt wdfat 41193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-fun 5890  df-dfat 41196

This theorem is referenced by:  nfafv  41216