Theorem nfdm 5946

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 5676	. 2 ⊢ dom 𝐴 = {𝑦 ∣ ∃𝑧 𝑦𝐴𝑧}
2		nfcv 2928	. . . . 5 ⊢ Ⅎ𝑥𝑦
3		nfrn.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
4		nfcv 2928	. . . . 5 ⊢ Ⅎ𝑥𝑧
5	2, 3, 4	nfbr 5163	. . . 4 ⊢ Ⅎ𝑥 𝑦𝐴𝑧
6	5	nfex 2360	. . 3 ⊢ Ⅎ𝑥∃𝑧 𝑦𝐴𝑧
7	6	nfab 2934	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑧 𝑦𝐴𝑧}
8	1, 7	nfcxfr 2926	1 ⊢ Ⅎ𝑥dom 𝐴

This proof depends on syntax axioms:  ∃wex 1812  {cab 2744  Ⅎwnfc 2913   class class class wbr 5114  dom cdm 5666

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 2738

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 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-dm 5676

This theorem is used by:  nfrn  5947  dmiin  5948  nffn  6641  nosupbnd2  27917  noinfbnd2  27932  funimass4f  33019  bnj1398  35453  bnj1491  35476  fnlimcnv  46421  fnlimfvre  46428  fnlimabslt  46433  lmbr3  46501  itgsinexplem1  46708  fourierdlem16  46877  fourierdlem21  46882  fourierdlem22  46883  fourierdlem68  46928  fourierdlem80  46940  fourierdlem103  46963  fourierdlem104  46964  issmff  47488  issmfdf  47491  smfpimltmpt  47500  smfpimltxr  47501  smfpimltxrmptf  47512  smfpreimagtf  47522  smflim  47531  smfpimgtxr  47534  smfpimgtmpt  47535  smfpimgtxrmptf  47538  smflim2  47560  smfpimcc  47562  smfsup  47568  smfsupmpt  47569  smfsupxr  47570  smfinflem  47571  smfinf  47572  smflimsup  47582  smfliminf  47585  adddmmbl2  47588  muldmmbl2  47590  smfpimne2  47594  smfdivdmmbl2  47595  fsupdm  47596  finfdm  47600  nfdfat  47904