Theorem nfdm 4791

Description: Bound-variable hypothesis builder for domain. (Contributed by NM, 30-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypothesis

Ref	Expression
nfrn.1	|- F/_ x A

Assertion

Ref	Expression
nfdm	|- F/_ x dom A

Proof of Theorem nfdm

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-dm 4557	. 2 |- dom A = { y | E. z y A z }
2		nfcv 2282	. . . . 5 |- F/_ x y
3		nfrn.1	. . . . 5 |- F/_ x A
4		nfcv 2282	. . . . 5 |- F/_ x z
5	2, 3, 4	nfbr 3982	. . . 4 |- F/ x y A z
6	5	nfex 1617	. . 3 |- F/ x E. z y A z
7	6	nfab 2287	. 2 |- F/_ x { y | E. z y A z }
8	1, 7	nfcxfr 2279	1 |- F/_ x dom A

Syntax hints:   E.wex 1469   {cab 2126   F/_wnfc 2269   class class class wbr 3937   dom cdm 4547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-dm 4557

This theorem is referenced by:  nfrn  4792  dmiin  4793  nffn  5227  ellimc3apf  12837