Theorem nfdm 5931

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 5664	. 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 5166	. . . 4 ⊢ Ⅎ𝑥 𝑦𝐴𝑧
6	5	nfex 2324	. . 3 ⊢ Ⅎ𝑥∃𝑧 𝑦𝐴𝑧
7	6	nfab 2904	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑧 𝑦𝐴𝑧}
8	1, 7	nfcxfr 2896	1 ⊢ Ⅎ𝑥dom 𝐴

Syntax hints:  ∃wex 1779  {cab 2713  Ⅎwnfc 2883   class class class wbr 5119  dom cdm 5654

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707

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 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-dm 5664

This theorem is referenced by:  nfrn  5932  dmiin  5933  nffn  6637  nosupbnd2  27680  noinfbnd2  27695  funimass4f  32615  bnj1398  35065  bnj1491  35088  fnlimcnv  45696  fnlimfvre  45703  fnlimabslt  45708  lmbr3  45776  itgsinexplem1  45983  fourierdlem16  46152  fourierdlem21  46157  fourierdlem22  46158  fourierdlem68  46203  fourierdlem80  46215  fourierdlem103  46238  fourierdlem104  46239  issmff  46763  issmfdf  46766  smfpimltmpt  46775  smfpimltxr  46776  smfpimltxrmptf  46787  smfpreimagtf  46797  smflim  46806  smfpimgtxr  46809  smfpimgtmpt  46810  smfpimgtxrmptf  46813  smflim2  46835  smfpimcc  46837  smfsup  46843  smfsupmpt  46844  smfsupxr  46845  smfinflem  46846  smfinf  46847  smflimsup  46857  smfliminf  46860  adddmmbl2  46863  muldmmbl2  46865  smfpimne2  46869  smfdivdmmbl2  46870  fsupdm  46871  finfdm  46875  nfdfat  47156