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Theorem nfdm 4578
 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 𝑥𝐴
Assertion
Ref Expression
nfdm 𝑥dom 𝐴

Proof of Theorem nfdm
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-dm 4355 . 2 dom 𝐴 = {𝑦 ∣ ∃𝑧 𝑦𝐴𝑧}
2 nfcv 2178 . . . . 5 𝑥𝑦
3 nfrn.1 . . . . 5 𝑥𝐴
4 nfcv 2178 . . . . 5 𝑥𝑧
52, 3, 4nfbr 3808 . . . 4 𝑥 𝑦𝐴𝑧
65nfex 1528 . . 3 𝑥𝑧 𝑦𝐴𝑧
76nfab 2182 . 2 𝑥{𝑦 ∣ ∃𝑧 𝑦𝐴𝑧}
81, 7nfcxfr 2175 1 𝑥dom 𝐴
 Colors of variables: wff set class Syntax hints:  ∃wex 1381  {cab 2026  Ⅎwnfc 2165   class class class wbr 3764  dom cdm 4345 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-dm 4355 This theorem is referenced by:  nfrn  4579  dmiin  4580  nffn  4995
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