Theorem nfdm 5890

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 5624	. 2 ⊢ dom 𝐴 = {𝑦 ∣ ∃𝑧 𝑦𝐴𝑧}
2		nfcv 2894	. . . . 5 ⊢ Ⅎ𝑥𝑦
3		nfrn.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
4		nfcv 2894	. . . . 5 ⊢ Ⅎ𝑥𝑧
5	2, 3, 4	nfbr 5136	. . . 4 ⊢ Ⅎ𝑥 𝑦𝐴𝑧
6	5	nfex 2325	. . 3 ⊢ Ⅎ𝑥∃𝑧 𝑦𝐴𝑧
7	6	nfab 2900	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑧 𝑦𝐴𝑧}
8	1, 7	nfcxfr 2892	1 ⊢ Ⅎ𝑥dom 𝐴

Syntax hints:  ∃wex 1780  {cab 2709  Ⅎwnfc 2879   class class class wbr 5089  dom cdm 5614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-dm 5624

This theorem is referenced by:  nfrn  5891  dmiin  5892  nffn  6580  nosupbnd2  27655  noinfbnd2  27670  funimass4f  32619  bnj1398  35046  bnj1491  35069  fnlimcnv  45775  fnlimfvre  45782  fnlimabslt  45787  lmbr3  45855  itgsinexplem1  46062  fourierdlem16  46231  fourierdlem21  46236  fourierdlem22  46237  fourierdlem68  46282  fourierdlem80  46294  fourierdlem103  46317  fourierdlem104  46318  issmff  46842  issmfdf  46845  smfpimltmpt  46854  smfpimltxr  46855  smfpimltxrmptf  46866  smfpreimagtf  46876  smflim  46885  smfpimgtxr  46888  smfpimgtmpt  46889  smfpimgtxrmptf  46892  smflim2  46914  smfpimcc  46916  smfsup  46922  smfsupmpt  46923  smfsupxr  46924  smfinflem  46925  smfinf  46926  smflimsup  46936  smfliminf  46939  adddmmbl2  46942  muldmmbl2  46944  smfpimne2  46948  smfdivdmmbl2  46949  fsupdm  46950  finfdm  46954  nfdfat  47237