Theorem nfdm 5021

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 4779	. 2 |- dom A = { y | E. z y A z }
2		nfcv 2392	. . . . 5 |- F/_ x y
3		nfrn.1	. . . . 5 |- F/_ x A
4		nfcv 2392	. . . . 5 |- F/_ x z
5	2, 3, 4	nfbr 4172	. . . 4 |- F/ x y A z
6	5	nfex 1690	. . 3 |- F/ x E. z y A z
7	6	nfab 2397	. 2 |- F/_ x { y | E. z y A z }
8	1, 7	nfcxfr 2389	1 |- F/_ x dom A

Syntax hints:   E.wex 1545   {cab 2224   F/_wnfc 2379   class class class wbr 4125   dom cdm 4769

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-dm 4779

This theorem is referenced by:  nfrn  5022  dmiin  5023  nffn  5472  funimass4f  6349  ellimc3apf  15684