ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  caovclg Unicode version

Theorem caovclg 5891
Description: Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 26-May-2014.)
Hypothesis
Ref Expression
caovclg.1  |-  ( (
ph  /\  ( x  e.  C  /\  y  e.  D ) )  -> 
( x F y )  e.  E )
Assertion
Ref Expression
caovclg  |-  ( (
ph  /\  ( A  e.  C  /\  B  e.  D ) )  -> 
( A F B )  e.  E )
Distinct variable groups:    x, y, A   
y, B    x, C, y    x, D, y    x, E, y    ph, x, y   
x, F, y
Allowed substitution hint:    B( x)

Proof of Theorem caovclg
StepHypRef Expression
1 caovclg.1 . . 3  |-  ( (
ph  /\  ( x  e.  C  /\  y  e.  D ) )  -> 
( x F y )  e.  E )
21ralrimivva 2491 . 2  |-  ( ph  ->  A. x  e.  C  A. y  e.  D  ( x F y )  e.  E )
3 oveq1 5749 . . . 4  |-  ( x  =  A  ->  (
x F y )  =  ( A F y ) )
43eleq1d 2186 . . 3  |-  ( x  =  A  ->  (
( x F y )  e.  E  <->  ( A F y )  e.  E ) )
5 oveq2 5750 . . . 4  |-  ( y  =  B  ->  ( A F y )  =  ( A F B ) )
65eleq1d 2186 . . 3  |-  ( y  =  B  ->  (
( A F y )  e.  E  <->  ( A F B )  e.  E
) )
74, 6rspc2v 2776 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A. x  e.  C  A. y  e.  D  ( x F y )  e.  E  ->  ( A F B )  e.  E ) )
82, 7mpan9 279 1  |-  ( (
ph  /\  ( A  e.  C  /\  B  e.  D ) )  -> 
( A F B )  e.  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316    e. wcel 1465   A.wral 2393  (class class class)co 5742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-iota 5058  df-fv 5101  df-ov 5745
This theorem is referenced by:  caovcld  5892  caovcl  5893  caovlem2d  5931  grprinvd  5934  frec2uzrdg  10150  frecuzrdgsuc  10155  iseqovex  10197  seq3val  10199  seqf  10202  seq3caopr  10224
  Copyright terms: Public domain W3C validator