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Metamath Proof Explorer

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