Theorem nfdm 4667

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 4438	. 2 |- dom A = { y | E. z y A z }
2		nfcv 2228	. . . . 5 |- F/_ x y
3		nfrn.1	. . . . 5 |- F/_ x A
4		nfcv 2228	. . . . 5 |- F/_ x z
5	2, 3, 4	nfbr 3881	. . . 4 |- F/ x y A z
6	5	nfex 1573	. . 3 |- F/ x E. z y A z
7	6	nfab 2233	. 2 |- F/_ x { y | E. z y A z }
8	1, 7	nfcxfr 2225	1 |- F/_ x dom A

Syntax hints:   E.wex 1426   {cab 2074   F/_wnfc 2215   class class class wbr 3837   dom cdm 4428

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-dm 4438

This theorem is referenced by:  nfrn  4668  dmiin  4669  nffn  5096