Theorem nfdm 4910

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 4673	. 2 |- dom A = { y | E. z y A z }
2		nfcv 2339	. . . . 5 |- F/_ x y
3		nfrn.1	. . . . 5 |- F/_ x A
4		nfcv 2339	. . . . 5 |- F/_ x z
5	2, 3, 4	nfbr 4079	. . . 4 |- F/ x y A z
6	5	nfex 1651	. . 3 |- F/ x E. z y A z
7	6	nfab 2344	. 2 |- F/_ x { y | E. z y A z }
8	1, 7	nfcxfr 2336	1 |- F/_ x dom A

Syntax hints:   E.wex 1506   {cab 2182   F/_wnfc 2326   class class class wbr 4033   dom cdm 4663

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-dm 4673

This theorem is referenced by:  nfrn  4911  dmiin  4912  nffn  5354  ellimc3apf  14896