Theorem nfdm 5943

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 5673	. 2 ⊢ dom 𝐴 = {𝑦 ∣ ∃𝑧 𝑦𝐴𝑧}
2		nfcv 2927	. . . . 5 ⊢ Ⅎ𝑥𝑦
3		nfrn.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
4		nfcv 2927	. . . . 5 ⊢ Ⅎ𝑥𝑧
5	2, 3, 4	nfbr 5160	. . . 4 ⊢ Ⅎ𝑥 𝑦𝐴𝑧
6	5	nfex 2359	. . 3 ⊢ Ⅎ𝑥∃𝑧 𝑦𝐴𝑧
7	6	nfab 2933	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑧 𝑦𝐴𝑧}
8	1, 7	nfcxfr 2925	1 ⊢ Ⅎ𝑥dom 𝐴

This proof depends on syntax axioms:  ∃wex 1812  {cab 2743  Ⅎwnfc 2912   class class class wbr 5111  dom cdm 5663

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-dm 5673

This theorem is used by:  nfrn  5944  dmiin  5945  nffn  6638  nosupbnd2  27933  noinfbnd2  27948  funimass4f  33055  bnj1398  35489  bnj1491  35512  fnlimcnv  46441  fnlimfvre  46448  fnlimabslt  46453  lmbr3  46521  itgsinexplem1  46728  fourierdlem16  46897  fourierdlem21  46902  fourierdlem22  46903  fourierdlem68  46948  fourierdlem80  46960  fourierdlem103  46983  fourierdlem104  46984  issmff  47508  issmfdf  47511  smfpimltmpt  47520  smfpimltxr  47521  smfpimltxrmptf  47532  smfpreimagtf  47542  smflim  47551  smfpimgtxr  47554  smfpimgtmpt  47555  smfpimgtxrmptf  47558  smflim2  47580  smfpimcc  47582  smfsup  47588  smfsupmpt  47589  smfsupxr  47590  smfinflem  47591  smfinf  47592  smflimsup  47602  smfliminf  47605  adddmmbl2  47608  muldmmbl2  47610  smfpimne2  47614  smfdivdmmbl2  47615  fsupdm  47616  finfdm  47620  nfdfat  47924