Theorem nfdm 5915

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 5648	. 2 ⊢ dom 𝐴 = {𝑦 ∣ ∃𝑧 𝑦𝐴𝑧}
2		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑥𝑦
3		nfrn.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
4		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑥𝑧
5	2, 3, 4	nfbr 5154	. . . 4 ⊢ Ⅎ𝑥 𝑦𝐴𝑧
6	5	nfex 2323	. . 3 ⊢ Ⅎ𝑥∃𝑧 𝑦𝐴𝑧
7	6	nfab 2897	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑧 𝑦𝐴𝑧}
8	1, 7	nfcxfr 2889	1 ⊢ Ⅎ𝑥dom 𝐴

Syntax hints:  ∃wex 1779  {cab 2707  Ⅎwnfc 2876   class class class wbr 5107  dom cdm 5638

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-dm 5648

This theorem is referenced by:  nfrn  5916  dmiin  5917  nffn  6617  nosupbnd2  27628  noinfbnd2  27643  funimass4f  32561  bnj1398  35024  bnj1491  35047  fnlimcnv  45665  fnlimfvre  45672  fnlimabslt  45677  lmbr3  45745  itgsinexplem1  45952  fourierdlem16  46121  fourierdlem21  46126  fourierdlem22  46127  fourierdlem68  46172  fourierdlem80  46184  fourierdlem103  46207  fourierdlem104  46208  issmff  46732  issmfdf  46735  smfpimltmpt  46744  smfpimltxr  46745  smfpimltxrmptf  46756  smfpreimagtf  46766  smflim  46775  smfpimgtxr  46778  smfpimgtmpt  46779  smfpimgtxrmptf  46782  smflim2  46804  smfpimcc  46806  smfsup  46812  smfsupmpt  46813  smfsupxr  46814  smfinflem  46815  smfinf  46816  smflimsup  46826  smfliminf  46829  adddmmbl2  46832  muldmmbl2  46834  smfpimne2  46838  smfdivdmmbl2  46839  fsupdm  46840  finfdm  46844  nfdfat  47128