Theorem nfdm 5937

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 2925	. . . . 5 ⊢ Ⅎ𝑥𝑦
3		nfrn.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
4		nfcv 2925	. . . . 5 ⊢ Ⅎ𝑥𝑧
5	2, 3, 4	nfbr 5154	. . . 4 ⊢ Ⅎ𝑥 𝑦𝐴𝑧
6	5	nfex 2357	. . 3 ⊢ Ⅎ𝑥∃𝑧 𝑦𝐴𝑧
7	6	nfab 2931	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑧 𝑦𝐴𝑧}
8	1, 7	nfcxfr 2923	1 ⊢ Ⅎ𝑥dom 𝐴

This proof depends on syntax axioms:  ∃wex 1811  {cab 2741  Ⅎwnfc 2910   class class class wbr 5105  dom cdm 5654

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-br 5106  df-dm 5664

This theorem is used by:  nfrn  5938  dmiin  5939  nffn  6646  nosupbnd2  28085  noinfbnd2  28100  funimass4f  33285  bnj1398  35717  bnj1491  35740  srepdflem1  36039  fnlimcnv  46816  fnlimfvre  46823  fnlimabslt  46828  lmbr3  46896  itgsinexplem1  47103  fourierdlem16  47272  fourierdlem21  47277  fourierdlem22  47278  fourierdlem68  47323  fourierdlem80  47335  fourierdlem103  47358  fourierdlem104  47359  issmff  47883  issmfdf  47886  smfpimltmpt  47895  smfpimltxr  47896  smfpimltxrmptf  47907  smfpreimagtf  47917  smflim  47926  smfpimgtxr  47929  smfpimgtmpt  47930  smfpimgtxrmptf  47933  smflim2  47955  smfpimcc  47957  smfsup  47963  smfsupmpt  47964  smfsupxr  47965  smfinflem  47966  smfinf  47967  smflimsup  47977  smfliminf  47980  adddmmbl2  47983  muldmmbl2  47985  smfpimne2  47989  smfdivdmmbl2  47990  fsupdm  47991  finfdm  47995  nfdfat  48336