MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfse Structured version   Visualization version   GIF version

Theorem nfse 5494
Description: Bound-variable hypothesis builder for set-like relations. (Contributed by Mario Carneiro, 24-Jun-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
Hypotheses
Ref Expression
nffr.r 𝑥𝑅
nffr.a 𝑥𝐴
Assertion
Ref Expression
nfse 𝑥 𝑅 Se 𝐴

Proof of Theorem nfse
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-se 5479 . 2 (𝑅 Se 𝐴 ↔ ∀𝑏𝐴 {𝑎𝐴𝑎𝑅𝑏} ∈ V)
2 nffr.a . . 3 𝑥𝐴
3 nfcv 2955 . . . . . 6 𝑥𝑎
4 nffr.r . . . . . 6 𝑥𝑅
5 nfcv 2955 . . . . . 6 𝑥𝑏
63, 4, 5nfbr 5077 . . . . 5 𝑥 𝑎𝑅𝑏
76, 2nfrabw 3338 . . . 4 𝑥{𝑎𝐴𝑎𝑅𝑏}
87nfel1 2971 . . 3 𝑥{𝑎𝐴𝑎𝑅𝑏} ∈ V
92, 8nfralw 3189 . 2 𝑥𝑏𝐴 {𝑎𝐴𝑎𝑅𝑏} ∈ V
101, 9nfxfr 1854 1 𝑥 𝑅 Se 𝐴
Colors of variables: wff setvar class
Syntax hints:  wnf 1785  wcel 2111  wnfc 2936  wral 3106  {crab 3110  Vcvv 3441   class class class wbr 5030   Se wse 5476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-se 5479
This theorem is referenced by:  nfoi  8962
  Copyright terms: Public domain W3C validator