Theorem nfdm 5927

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 5657	. 2 ⊢ dom 𝐴 = {𝑦 ∣ ∃𝑧 𝑦𝐴𝑧}
2		nfcv 2924	. . . . 5 ⊢ Ⅎ𝑥𝑦
3		nfrn.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
4		nfcv 2924	. . . . 5 ⊢ Ⅎ𝑥𝑧
5	2, 3, 4	nfbr 5147	. . . 4 ⊢ Ⅎ𝑥 𝑦𝐴𝑧
6	5	nfex 2356	. . 3 ⊢ Ⅎ𝑥∃𝑧 𝑦𝐴𝑧
7	6	nfab 2930	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑧 𝑦𝐴𝑧}
8	1, 7	nfcxfr 2922	1 ⊢ Ⅎ𝑥dom 𝐴

Syntax hints:  ∃wex 1799  {cab 2740  Ⅎwnfc 2909   class class class wbr 5100  dom cdm 5647

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-dm 5657

This theorem is referenced by:  nfrn  5928  dmiin  5929  nffn  6620  nosupbnd2  27780  noinfbnd2  27795  funimass4f  32839  bnj1398  35329  bnj1491  35352  fnlimcnv  46241  fnlimfvre  46248  fnlimabslt  46253  lmbr3  46321  itgsinexplem1  46528  fourierdlem16  46697  fourierdlem21  46702  fourierdlem22  46703  fourierdlem68  46748  fourierdlem80  46760  fourierdlem103  46783  fourierdlem104  46784  issmff  47308  issmfdf  47311  smfpimltmpt  47320  smfpimltxr  47321  smfpimltxrmptf  47332  smfpreimagtf  47342  smflim  47351  smfpimgtxr  47354  smfpimgtmpt  47355  smfpimgtxrmptf  47358  smflim2  47380  smfpimcc  47382  smfsup  47388  smfsupmpt  47389  smfsupxr  47390  smfinflem  47391  smfinf  47392  smflimsup  47402  smfliminf  47405  adddmmbl2  47408  muldmmbl2  47410  smfpimne2  47414  smfdivdmmbl2  47415  fsupdm  47416  finfdm  47420  nfdfat  47721