Theorem nfdm 4883

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 4648	. 2 |- dom A = { y | E. z y A z }
2		nfcv 2329	. . . . 5 |- F/_ x y
3		nfrn.1	. . . . 5 |- F/_ x A
4		nfcv 2329	. . . . 5 |- F/_ x z
5	2, 3, 4	nfbr 4061	. . . 4 |- F/ x y A z
6	5	nfex 1647	. . 3 |- F/ x E. z y A z
7	6	nfab 2334	. 2 |- F/_ x { y | E. z y A z }
8	1, 7	nfcxfr 2326	1 |- F/_ x dom A

Syntax hints:   E.wex 1502   {cab 2173   F/_wnfc 2316   class class class wbr 4015   dom cdm 4638

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-dm 4648

This theorem is referenced by:  nfrn  4884  dmiin  4885  nffn  5324  ellimc3apf  14425