Theorem nfdm 4637

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 4411	. 2 |- dom A = { y | E. z y A z }
2		nfcv 2223	. . . . 5 |- F/_ x y
3		nfrn.1	. . . . 5 |- F/_ x A
4		nfcv 2223	. . . . 5 |- F/_ x z
5	2, 3, 4	nfbr 3855	. . . 4 |- F/ x y A z
6	5	nfex 1569	. . 3 |- F/ x E. z y A z
7	6	nfab 2227	. 2 |- F/_ x { y | E. z y A z }
8	1, 7	nfcxfr 2220	1 |- F/_ x dom A

Syntax hints:   E.wex 1422   {cab 2069   F/_wnfc 2210   class class class wbr 3811   dom cdm 4401

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-un 2988  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812  df-dm 4411

This theorem is referenced by:  nfrn  4638  dmiin  4639  nffn  5063