MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfdm Unicode version

Theorem nfdm 4920
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  |-  F/_ x A
Assertion
Ref Expression
nfdm  |-  F/_ x dom  A

Proof of Theorem nfdm
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-dm 4699 . 2  |-  dom  A  =  { y  |  E. z  y A z }
2 nfcv 2419 . . . . 5  |-  F/_ x
y
3 nfrn.1 . . . . 5  |-  F/_ x A
4 nfcv 2419 . . . . 5  |-  F/_ x
z
52, 3, 4nfbr 4067 . . . 4  |-  F/ x  y A z
65nfex 1767 . . 3  |-  F/ x E. z  y A
z
76nfab 2423 . 2  |-  F/_ x { y  |  E. z  y A z }
81, 7nfcxfr 2416 1  |-  F/_ x dom  A
Colors of variables: wff set class
Syntax hints:   E.wex 1528   {cab 2269   F/_wnfc 2406   class class class wbr 4023   dom cdm 4689
This theorem is referenced by:  nfrn  4921  dmiin  4922  nffn  5340  funimass4f  23042  itgsinexplem1  27748  nfdfat  27993  bnj1398  29064  bnj1491  29087
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-dm 4699
  Copyright terms: Public domain W3C validator