Theorem nfdm 4870

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 4635	. 2 |- dom A = { y | E. z y A z }
2		nfcv 2319	. . . . 5 |- F/_ x y
3		nfrn.1	. . . . 5 |- F/_ x A
4		nfcv 2319	. . . . 5 |- F/_ x z
5	2, 3, 4	nfbr 4048	. . . 4 |- F/ x y A z
6	5	nfex 1637	. . 3 |- F/ x E. z y A z
7	6	nfab 2324	. 2 |- F/_ x { y | E. z y A z }
8	1, 7	nfcxfr 2316	1 |- F/_ x dom A

Syntax hints:   E.wex 1492   {cab 2163   F/_wnfc 2306   class class class wbr 4002   dom cdm 4625

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-br 4003  df-dm 4635

This theorem is referenced by:  nfrn  4871  dmiin  4872  nffn  5311  ellimc3apf  14000