Theorem nfdm 5965

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 5699	. 2 ⊢ dom 𝐴 = {𝑦 ∣ ∃𝑧 𝑦𝐴𝑧}
2		nfcv 2903	. . . . 5 ⊢ Ⅎ𝑥𝑦
3		nfrn.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
4		nfcv 2903	. . . . 5 ⊢ Ⅎ𝑥𝑧
5	2, 3, 4	nfbr 5195	. . . 4 ⊢ Ⅎ𝑥 𝑦𝐴𝑧
6	5	nfex 2323	. . 3 ⊢ Ⅎ𝑥∃𝑧 𝑦𝐴𝑧
7	6	nfab 2909	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑧 𝑦𝐴𝑧}
8	1, 7	nfcxfr 2901	1 ⊢ Ⅎ𝑥dom 𝐴

Syntax hints:  ∃wex 1776  {cab 2712  Ⅎwnfc 2888   class class class wbr 5148  dom cdm 5689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-dm 5699

This theorem is referenced by:  nfrn  5966  dmiin  5967  nffn  6668  nosupbnd2  27776  noinfbnd2  27791  funimass4f  32654  bnj1398  35027  bnj1491  35050  fnlimcnv  45623  fnlimfvre  45630  fnlimabslt  45635  lmbr3  45703  itgsinexplem1  45910  fourierdlem16  46079  fourierdlem21  46084  fourierdlem22  46085  fourierdlem68  46130  fourierdlem80  46142  fourierdlem103  46165  fourierdlem104  46166  issmff  46690  issmfdf  46693  smfpimltmpt  46702  smfpimltxr  46703  smfpimltxrmptf  46714  smfpreimagtf  46724  smflim  46733  smfpimgtxr  46736  smfpimgtmpt  46737  smfpimgtxrmptf  46740  smflim2  46762  smfpimcc  46764  smfsup  46770  smfsupmpt  46771  smfsupxr  46772  smfinflem  46773  smfinf  46774  smflimsup  46784  smfliminf  46787  adddmmbl2  46790  muldmmbl2  46792  smfpimne2  46796  smfdivdmmbl2  46797  fsupdm  46798  finfdm  46802  nfdfat  47077