Theorem nfdm 4855

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 4621	. 2 |- dom A = { y | E. z y A z }
2		nfcv 2312	. . . . 5 |- F/_ x y
3		nfrn.1	. . . . 5 |- F/_ x A
4		nfcv 2312	. . . . 5 |- F/_ x z
5	2, 3, 4	nfbr 4035	. . . 4 |- F/ x y A z
6	5	nfex 1630	. . 3 |- F/ x E. z y A z
7	6	nfab 2317	. 2 |- F/_ x { y | E. z y A z }
8	1, 7	nfcxfr 2309	1 |- F/_ x dom A

Syntax hints:   E.wex 1485   {cab 2156   F/_wnfc 2299   class class class wbr 3989   dom cdm 4611

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-dm 4621

This theorem is referenced by:  nfrn  4856  dmiin  4857  nffn  5294  ellimc3apf  13423