Theorem nfdm 5906

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 5641	. 2 ⊢ dom 𝐴 = {𝑦 ∣ ∃𝑧 𝑦𝐴𝑧}
2		nfcv 2898	. . . . 5 ⊢ Ⅎ𝑥𝑦
3		nfrn.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
4		nfcv 2898	. . . . 5 ⊢ Ⅎ𝑥𝑧
5	2, 3, 4	nfbr 5132	. . . 4 ⊢ Ⅎ𝑥 𝑦𝐴𝑧
6	5	nfex 2329	. . 3 ⊢ Ⅎ𝑥∃𝑧 𝑦𝐴𝑧
7	6	nfab 2904	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑧 𝑦𝐴𝑧}
8	1, 7	nfcxfr 2896	1 ⊢ Ⅎ𝑥dom 𝐴

Syntax hints:  ∃wex 1781  {cab 2714  Ⅎwnfc 2883   class class class wbr 5085  dom cdm 5631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-dm 5641

This theorem is referenced by:  nfrn  5907  dmiin  5908  nffn  6597  nosupbnd2  27680  noinfbnd2  27695  funimass4f  32710  bnj1398  35176  bnj1491  35199  fnlimcnv  46095  fnlimfvre  46102  fnlimabslt  46107  lmbr3  46175  itgsinexplem1  46382  fourierdlem16  46551  fourierdlem21  46556  fourierdlem22  46557  fourierdlem68  46602  fourierdlem80  46614  fourierdlem103  46637  fourierdlem104  46638  issmff  47162  issmfdf  47165  smfpimltmpt  47174  smfpimltxr  47175  smfpimltxrmptf  47186  smfpreimagtf  47196  smflim  47205  smfpimgtxr  47208  smfpimgtmpt  47209  smfpimgtxrmptf  47212  smflim2  47234  smfpimcc  47236  smfsup  47242  smfsupmpt  47243  smfsupxr  47244  smfinflem  47245  smfinf  47246  smflimsup  47256  smfliminf  47259  adddmmbl2  47262  muldmmbl2  47264  smfpimne2  47268  smfdivdmmbl2  47269  fsupdm  47270  finfdm  47274  nfdfat  47575