Theorem nfdm

Description: Bound-variable hypothesis builder for domain. (Contributed by NM, 30-Jan-2004) (Revised by Mario Carneiro, 15-Oct-2016)

	Ref	Expression
Hypothesis	nfrn.1	⊢ Ⅎ 𝑥 𝐴
Assertion	nfdm	⊢ Ⅎ 𝑥 dom 𝐴

Proof

Step	Hyp	Ref	Expression
1		nfrn.1	⊢ Ⅎ 𝑥 𝐴
2		df-dm	⊢ dom 𝐴 = { 𝑦 ∣ ∃ 𝑧 𝑦 𝐴 𝑧 }
3		nfcv	⊢ Ⅎ 𝑥 𝑦
4		nfcv	⊢ Ⅎ 𝑥 𝑧
5	3 1 4	nfbr	⊢ Ⅎ 𝑥 𝑦 𝐴 𝑧
6	5	nfex	⊢ Ⅎ 𝑥 ∃ 𝑧 𝑦 𝐴 𝑧
7	6	nfab	⊢ Ⅎ 𝑥 { 𝑦 ∣ ∃ 𝑧 𝑦 𝐴 𝑧 }
8	2 7	nfcxfr	⊢ Ⅎ 𝑥 dom 𝐴