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

Theorem nfse 4549
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  |-  F/_ x R
nffr.a  |-  F/_ x A
Assertion
Ref Expression
nfse  |-  F/ 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 4534 . 2  |-  ( R Se  A  <->  A. b  e.  A  { a  e.  A  |  a R b }  e.  _V )
2 nffr.a . . 3  |-  F/_ x A
3 nfcv 2571 . . . . . 6  |-  F/_ x
a
4 nffr.r . . . . . 6  |-  F/_ x R
5 nfcv 2571 . . . . . 6  |-  F/_ x
b
63, 4, 5nfbr 4248 . . . . 5  |-  F/ x  a R b
76, 2nfrab 2881 . . . 4  |-  F/_ x { a  e.  A  |  a R b }
87nfel1 2581 . . 3  |-  F/ x { a  e.  A  |  a R b }  e.  _V
92, 8nfral 2751 . 2  |-  F/ x A. b  e.  A  { a  e.  A  |  a R b }  e.  _V
101, 9nfxfr 1579 1  |-  F/ x  R Se  A
Colors of variables: wff set class
Syntax hints:   F/wnf 1553    e. wcel 1725   F/_wnfc 2558   A.wral 2697   {crab 2701   _Vcvv 2948   class class class wbr 4204   Se wse 4531
This theorem is referenced by:  nfoi  7475
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-se 4534
  Copyright terms: Public domain W3C validator