Theorem nfdm 5945

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 5668	. 2 ⊢ dom 𝐴 = {𝑦 ∣ ∃𝑧 𝑦𝐴𝑧}
2		nfcv 2926	. . . . 5 ⊢ Ⅎ𝑥𝑦
3		nfrn.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
4		nfcv 2926	. . . . 5 ⊢ Ⅎ𝑥𝑧
5	2, 3, 4	nfbr 5158	. . . 4 ⊢ Ⅎ𝑥 𝑦𝐴𝑧
6	5	nfex 2358	. . 3 ⊢ Ⅎ𝑥∃𝑧 𝑦𝐴𝑧
7	6	nfab 2932	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑧 𝑦𝐴𝑧}
8	1, 7	nfcxfr 2924	1 ⊢ Ⅎ𝑥dom 𝐴

This proof depends on syntax axioms:  ∃wex 1812  {cab 2741  Ⅎwnfc 2911   class class class wbr 5108  dom cdm 5658

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-dm 5668

This theorem is used by:  nfrn  5946  dmiin  5947  nffn  6668  nosupbnd2  28130  noinfbnd2  28145  funimass4f  33329  bnj1398  35775  bnj1491  35798  srepdflem1  36126  fnlimcnv  46950  fnlimfvre  46957  fnlimabslt  46962  lmbr3  47030  itgsinexplem1  47237  fourierdlem16  47406  fourierdlem21  47411  fourierdlem22  47412  fourierdlem68  47457  fourierdlem80  47469  fourierdlem103  47492  fourierdlem104  47493  issmff  48017  issmfdf  48020  smfpimltmpt  48029  smfpimltxr  48030  smfpimltxrmptf  48041  smfpreimagtf  48051  smflim  48060  smfpimgtxr  48063  smfpimgtmpt  48064  smfpimgtxrmptf  48067  smflim2  48089  smfpimcc  48091  smfsup  48097  smfsupmpt  48098  smfsupxr  48099  smfinflem  48100  smfinf  48101  smflimsup  48111  smfliminf  48114  adddmmbl2  48117  muldmmbl2  48119  smfpimne2  48123  smfdivdmmbl2  48124  fsupdm  48125  finfdm  48129  nfdfat  48470