Theorem nfdm 4976

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 4735	. 2 |- dom A = { y | E. z y A z }
2		nfcv 2374	. . . . 5 |- F/_ x y
3		nfrn.1	. . . . 5 |- F/_ x A
4		nfcv 2374	. . . . 5 |- F/_ x z
5	2, 3, 4	nfbr 4135	. . . 4 |- F/ x y A z
6	5	nfex 1685	. . 3 |- F/ x E. z y A z
7	6	nfab 2379	. 2 |- F/_ x { y | E. z y A z }
8	1, 7	nfcxfr 2371	1 |- F/_ x dom A

Syntax hints:   E.wex 1540   {cab 2217   F/_wnfc 2361   class class class wbr 4088   dom cdm 4725

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-dm 4735

This theorem is referenced by:  nfrn  4977  dmiin  4978  nffn  5426  ellimc3apf  15383