Theorem nfdm 5950

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 5686	. 2 ⊢ dom 𝐴 = {𝑦 ∣ ∃𝑧 𝑦𝐴𝑧}
2		nfcv 2902	. . . . 5 ⊢ Ⅎ𝑥𝑦
3		nfrn.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
4		nfcv 2902	. . . . 5 ⊢ Ⅎ𝑥𝑧
5	2, 3, 4	nfbr 5195	. . . 4 ⊢ Ⅎ𝑥 𝑦𝐴𝑧
6	5	nfex 2316	. . 3 ⊢ Ⅎ𝑥∃𝑧 𝑦𝐴𝑧
7	6	nfab 2908	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑧 𝑦𝐴𝑧}
8	1, 7	nfcxfr 2900	1 ⊢ Ⅎ𝑥dom 𝐴

Syntax hints:  ∃wex 1780  {cab 2708  Ⅎwnfc 2882   class class class wbr 5148  dom cdm 5676

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-dm 5686

This theorem is referenced by:  nfrn  5951  dmiin  5952  nffn  6648  nosupbnd2  27562  noinfbnd2  27577  funimass4f  32294  bnj1398  34509  bnj1491  34532  fnlimcnv  44842  fnlimfvre  44849  fnlimabslt  44854  lmbr3  44922  itgsinexplem1  45129  fourierdlem16  45298  fourierdlem21  45303  fourierdlem22  45304  fourierdlem68  45349  fourierdlem80  45361  fourierdlem103  45384  fourierdlem104  45385  issmff  45909  issmfdf  45912  smfpimltmpt  45921  smfpimltxr  45922  smfpimltxrmptf  45933  smfpreimagtf  45943  smflim  45952  smfpimgtxr  45955  smfpimgtmpt  45956  smfpimgtxrmptf  45959  smflim2  45981  smfpimcc  45983  smfsup  45989  smfsupmpt  45990  smfsupxr  45991  smfinflem  45992  smfinf  45993  smfinfmpt  45994  smflimsup  46003  smfliminf  46006  adddmmbl2  46009  muldmmbl2  46011  smfpimne2  46015  smfdivdmmbl2  46016  fsupdm  46017  finfdm  46021  nfdfat  46294