Theorem nfdm 5951

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 5687	. 2 ⊢ dom 𝐴 = {𝑦 ∣ ∃𝑧 𝑦𝐴𝑧}
2		nfcv 2904	. . . . 5 ⊢ Ⅎ𝑥𝑦
3		nfrn.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
4		nfcv 2904	. . . . 5 ⊢ Ⅎ𝑥𝑧
5	2, 3, 4	nfbr 5196	. . . 4 ⊢ Ⅎ𝑥 𝑦𝐴𝑧
6	5	nfex 2318	. . 3 ⊢ Ⅎ𝑥∃𝑧 𝑦𝐴𝑧
7	6	nfab 2910	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑧 𝑦𝐴𝑧}
8	1, 7	nfcxfr 2902	1 ⊢ Ⅎ𝑥dom 𝐴

Syntax hints:  ∃wex 1782  {cab 2710  Ⅎwnfc 2884   class class class wbr 5149  dom cdm 5677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-dm 5687

This theorem is referenced by:  nfrn  5952  dmiin  5953  nffn  6649  nosupbnd2  27219  noinfbnd2  27234  funimass4f  31861  bnj1398  34045  bnj1491  34068  fnlimcnv  44383  fnlimfvre  44390  fnlimabslt  44395  lmbr3  44463  itgsinexplem1  44670  fourierdlem16  44839  fourierdlem21  44844  fourierdlem22  44845  fourierdlem68  44890  fourierdlem80  44902  fourierdlem103  44925  fourierdlem104  44926  issmff  45450  issmfdf  45453  smfpimltmpt  45462  smfpimltxr  45463  smfpimltxrmptf  45474  smfpreimagtf  45484  smflim  45493  smfpimgtxr  45496  smfpimgtmpt  45497  smfpimgtxrmptf  45500  smflim2  45522  smfpimcc  45524  smfsup  45530  smfsupmpt  45531  smfsupxr  45532  smfinflem  45533  smfinf  45534  smfinfmpt  45535  smflimsup  45544  smfliminf  45547  adddmmbl2  45550  muldmmbl2  45552  smfpimne2  45556  smfdivdmmbl2  45557  fsupdm  45558  finfdm  45562  nfdfat  45835