Theorem nfdm 5663

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	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfdm	⊢ Ⅎ𝑥dom 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 5413	. 2 ⊢ dom 𝐴 = {𝑦 ∣ ∃𝑧 𝑦𝐴𝑧}
2		nfcv 2926	. . . . 5 ⊢ Ⅎ𝑥𝑦
3		nfrn.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
4		nfcv 2926	. . . . 5 ⊢ Ⅎ𝑥𝑧
5	2, 3, 4	nfbr 4972	. . . 4 ⊢ Ⅎ𝑥 𝑦𝐴𝑧
6	5	nfex 2264	. . 3 ⊢ Ⅎ𝑥∃𝑧 𝑦𝐴𝑧
7	6	nfab 2932	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑧 𝑦𝐴𝑧}
8	1, 7	nfcxfr 2924	1 ⊢ Ⅎ𝑥dom 𝐴

Syntax hints:  ∃wex 1742  {cab 2752  Ⅎwnfc 2910   class class class wbr 4925  dom cdm 5403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-rab 3091  df-v 3411  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4926  df-dm 5413

This theorem is referenced by:  nfrn  5664  dmiin  5665  nffn  6282  funimass4f  30158  bnj1398  31980  bnj1491  32003  nosupbnd2  32766  fnlimcnv  41404  fnlimfvre  41411  fnlimabslt  41416  lmbr3  41484  itgsinexplem1  41694  fourierdlem16  41864  fourierdlem21  41869  fourierdlem22  41870  fourierdlem68  41915  fourierdlem80  41927  fourierdlem103  41950  fourierdlem104  41951  issmff  42467  issmfdf  42470  smfpimltmpt  42479  smfpimltxrmpt  42491  smfpreimagtf  42500  smflim  42509  smfpimgtxr  42512  smfpimgtmpt  42513  smfpimgtxrmpt  42516  smflim2  42536  smfpimcc  42538  smfsup  42544  smfsupmpt  42545  smfsupxr  42546  smfinflem  42547  smfinf  42548  smfinfmpt  42549  smflimsup  42558  smfliminf  42561  nfdfat  42757