Theorem nfdm 5976

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 5710	. 2 ⊢ dom 𝐴 = {𝑦 ∣ ∃𝑧 𝑦𝐴𝑧}
2		nfcv 2908	. . . . 5 ⊢ Ⅎ𝑥𝑦
3		nfrn.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
4		nfcv 2908	. . . . 5 ⊢ Ⅎ𝑥𝑧
5	2, 3, 4	nfbr 5213	. . . 4 ⊢ Ⅎ𝑥 𝑦𝐴𝑧
6	5	nfex 2328	. . 3 ⊢ Ⅎ𝑥∃𝑧 𝑦𝐴𝑧
7	6	nfab 2914	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑧 𝑦𝐴𝑧}
8	1, 7	nfcxfr 2906	1 ⊢ Ⅎ𝑥dom 𝐴

Syntax hints:  ∃wex 1777  {cab 2717  Ⅎwnfc 2893   class class class wbr 5166  dom cdm 5700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-dm 5710

This theorem is referenced by:  nfrn  5977  dmiin  5978  nffn  6678  nosupbnd2  27779  noinfbnd2  27794  funimass4f  32656  bnj1398  35010  bnj1491  35033  fnlimcnv  45588  fnlimfvre  45595  fnlimabslt  45600  lmbr3  45668  itgsinexplem1  45875  fourierdlem16  46044  fourierdlem21  46049  fourierdlem22  46050  fourierdlem68  46095  fourierdlem80  46107  fourierdlem103  46130  fourierdlem104  46131  issmff  46655  issmfdf  46658  smfpimltmpt  46667  smfpimltxr  46668  smfpimltxrmptf  46679  smfpreimagtf  46689  smflim  46698  smfpimgtxr  46701  smfpimgtmpt  46702  smfpimgtxrmptf  46705  smflim2  46727  smfpimcc  46729  smfsup  46735  smfsupmpt  46736  smfsupxr  46737  smfinflem  46738  smfinf  46739  smfinfmpt  46740  smflimsup  46749  smfliminf  46752  adddmmbl2  46755  muldmmbl2  46757  smfpimne2  46761  smfdivdmmbl2  46762  fsupdm  46763  finfdm  46767  nfdfat  47042