# 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 ${⊢}\underset{_}{Ⅎ}{x}\phantom{\rule{.4em}{0ex}}{A}$
Assertion nfdm ${⊢}\underset{_}{Ⅎ}{x}\phantom{\rule{.4em}{0ex}}\mathrm{dom}{A}$

### Proof

Step Hyp Ref Expression
1 nfrn.1 ${⊢}\underset{_}{Ⅎ}{x}\phantom{\rule{.4em}{0ex}}{A}$
2 df-dm ${⊢}\mathrm{dom}{A}=\left\{{y}|\exists {z}\phantom{\rule{.4em}{0ex}}{y}{A}{z}\right\}$
3 nfcv ${⊢}\underset{_}{Ⅎ}{x}\phantom{\rule{.4em}{0ex}}{y}$
4 nfcv ${⊢}\underset{_}{Ⅎ}{x}\phantom{\rule{.4em}{0ex}}{z}$
5 3 1 4 nfbr ${⊢}Ⅎ{x}\phantom{\rule{.4em}{0ex}}{y}{A}{z}$
6 5 nfex ${⊢}Ⅎ{x}\phantom{\rule{.4em}{0ex}}\exists {z}\phantom{\rule{.4em}{0ex}}{y}{A}{z}$
7 6 nfab ${⊢}\underset{_}{Ⅎ}{x}\phantom{\rule{.4em}{0ex}}\left\{{y}|\exists {z}\phantom{\rule{.4em}{0ex}}{y}{A}{z}\right\}$
8 2 7 nfcxfr ${⊢}\underset{_}{Ⅎ}{x}\phantom{\rule{.4em}{0ex}}\mathrm{dom}{A}$