Theorem nfdm 5820

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 5561	. 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 5100	. . . 4 ⊢ Ⅎ𝑥 𝑦𝐴𝑧
6	5	nfex 2323	. . 3 ⊢ Ⅎ𝑥∃𝑧 𝑦𝐴𝑧
7	6	nfab 2910	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑧 𝑦𝐴𝑧}
8	1, 7	nfcxfr 2902	1 ⊢ Ⅎ𝑥dom 𝐴

Syntax hints:  ∃wex 1787  {cab 2714  Ⅎwnfc 2884   class class class wbr 5053  dom cdm 5551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-dm 5561

This theorem is referenced by:  nfrn  5821  dmiin  5822  nffn  6478  funimass4f  30691  bnj1398  32727  bnj1491  32750  nosupbnd2  33656  noinfbnd2  33671  fnlimcnv  42883  fnlimfvre  42890  fnlimabslt  42895  lmbr3  42963  itgsinexplem1  43170  fourierdlem16  43339  fourierdlem21  43344  fourierdlem22  43345  fourierdlem68  43390  fourierdlem80  43402  fourierdlem103  43425  fourierdlem104  43426  issmff  43942  issmfdf  43945  smfpimltmpt  43954  smfpimltxrmpt  43966  smfpreimagtf  43975  smflim  43984  smfpimgtxr  43987  smfpimgtmpt  43988  smfpimgtxrmpt  43991  smflim2  44011  smfpimcc  44013  smfsup  44019  smfsupmpt  44020  smfsupxr  44021  smfinflem  44022  smfinf  44023  smfinfmpt  44024  smflimsup  44033  smfliminf  44036  nfdfat  44291