Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cllem0 Structured version   Unicode version

Theorem cllem0 38164
Description: The class of all sets with property  ph ( z ) is closed under the binary operation on sets defined in  R ( x ,  y ). (Contributed by Richard Penner, 3-Jan-2020.)
Hypotheses
Ref Expression
cllem0.v  |-  V  =  { z  |  ph }
cllem0.rex  |-  R  e.  U
cllem0.r  |-  ( z  =  R  ->  ( ph 
<->  ps ) )
cllem0.x  |-  ( z  =  x  ->  ( ph 
<->  ch ) )
cllem0.y  |-  ( z  =  y  ->  ( ph 
<->  th ) )
cllem0.closed  |-  ( ( ch  /\  th )  ->  ps )
Assertion
Ref Expression
cllem0  |-  A. x  e.  V  A. y  e.  V  R  e.  V
Distinct variable groups:    ps, z    ch, z    th, z    x, y, z    y, V    z, R
Allowed substitution hints:    ph( x, y, z)    ps( x, y)    ch( x, y)    th( x, y)    R( x, y)    U( x, y, z)    V( x, z)

Proof of Theorem cllem0
StepHypRef Expression
1 cllem0.rex . . . . . . 7  |-  R  e.  U
21elexi 3116 . . . . . 6  |-  R  e. 
_V
3 cllem0.r . . . . . 6  |-  ( z  =  R  ->  ( ph 
<->  ps ) )
4 cllem0.v . . . . . 6  |-  V  =  { z  |  ph }
52, 3, 4elab2 3246 . . . . 5  |-  ( R  e.  V  <->  ps )
65ralbii 2885 . . . 4  |-  ( A. y  e.  V  R  e.  V  <->  A. y  e.  V  ps )
76ralbii 2885 . . 3  |-  ( A. x  e.  V  A. y  e.  V  R  e.  V  <->  A. x  e.  V  A. y  e.  V  ps )
8 df-ral 2809 . . . 4  |-  ( A. y  e.  V  ps  <->  A. y ( y  e.  V  ->  ps )
)
98ralbii 2885 . . 3  |-  ( A. x  e.  V  A. y  e.  V  ps  <->  A. x  e.  V  A. y ( y  e.  V  ->  ps )
)
10 df-ral 2809 . . 3  |-  ( A. x  e.  V  A. y ( y  e.  V  ->  ps )  <->  A. x ( x  e.  V  ->  A. y
( y  e.  V  ->  ps ) ) )
117, 9, 103bitri 271 . 2  |-  ( A. x  e.  V  A. y  e.  V  R  e.  V  <->  A. x ( x  e.  V  ->  A. y
( y  e.  V  ->  ps ) ) )
12 vex 3109 . . . . . 6  |-  x  e. 
_V
13 cllem0.x . . . . . 6  |-  ( z  =  x  ->  ( ph 
<->  ch ) )
1412, 13, 4elab2 3246 . . . . 5  |-  ( x  e.  V  <->  ch )
15 vex 3109 . . . . . 6  |-  y  e. 
_V
16 cllem0.y . . . . . 6  |-  ( z  =  y  ->  ( ph 
<->  th ) )
1715, 16, 4elab2 3246 . . . . 5  |-  ( y  e.  V  <->  th )
18 cllem0.closed . . . . 5  |-  ( ( ch  /\  th )  ->  ps )
1914, 17, 18syl2anb 477 . . . 4  |-  ( ( x  e.  V  /\  y  e.  V )  ->  ps )
2019ex 432 . . 3  |-  ( x  e.  V  ->  (
y  e.  V  ->  ps ) )
2120alrimiv 1724 . 2  |-  ( x  e.  V  ->  A. y
( y  e.  V  ->  ps ) )
2211, 21mpgbir 1627 1  |-  A. x  e.  V  A. y  e.  V  R  e.  V
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1396    = wceq 1398    e. wcel 1823   {cab 2439   A.wral 2804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-v 3108
This theorem is referenced by:  superficl  38165  superuncl  38166  ssficl  38167  ssuncl  38168  ssdifcl  38169  sssymdifcl  38170  trficl  38186
  Copyright terms: Public domain W3C validator