Theorem nfdm 4968

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 4729	. 2 |- dom A = { y | E. z y A z }
2		nfcv 2372	. . . . 5 |- F/_ x y
3		nfrn.1	. . . . 5 |- F/_ x A
4		nfcv 2372	. . . . 5 |- F/_ x z
5	2, 3, 4	nfbr 4130	. . . 4 |- F/ x y A z
6	5	nfex 1683	. . 3 |- F/ x E. z y A z
7	6	nfab 2377	. 2 |- F/_ x { y | E. z y A z }
8	1, 7	nfcxfr 2369	1 |- F/_ x dom A

Syntax hints:   E.wex 1538   {cab 2215   F/_wnfc 2359   class class class wbr 4083   dom cdm 4719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-dm 4729

This theorem is referenced by:  nfrn  4969  dmiin  4970  nffn  5417  ellimc3apf  15334