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

Theorem cnvresima 5032
Description: An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.)
Assertion
Ref Expression
cnvresima  |-  ( `' ( F  |`  A )
" B )  =  ( ( `' F " B )  i^i  A
)

Proof of Theorem cnvresima
Dummy variables  t  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2690 . . . 4  |-  t  e. 
_V
21elima3 4892 . . 3  |-  ( t  e.  ( `' ( F  |`  A ) " B )  <->  E. s
( s  e.  B  /\  <. s ,  t
>.  e.  `' ( F  |`  A ) ) )
31elima3 4892 . . . . 5  |-  ( t  e.  ( `' F " B )  <->  E. s
( s  e.  B  /\  <. s ,  t
>.  e.  `' F ) )
43anbi1i 454 . . . 4  |-  ( ( t  e.  ( `' F " B )  /\  t  e.  A
)  <->  ( E. s
( s  e.  B  /\  <. s ,  t
>.  e.  `' F )  /\  t  e.  A
) )
5 elin 3260 . . . 4  |-  ( t  e.  ( ( `' F " B )  i^i  A )  <->  ( t  e.  ( `' F " B )  /\  t  e.  A ) )
6 vex 2690 . . . . . . . . . 10  |-  s  e. 
_V
76, 1opelcnv 4725 . . . . . . . . 9  |-  ( <.
s ,  t >.  e.  `' ( F  |`  A )  <->  <. t ,  s >.  e.  ( F  |`  A ) )
86opelres 4828 . . . . . . . . . 10  |-  ( <.
t ,  s >.  e.  ( F  |`  A )  <-> 
( <. t ,  s
>.  e.  F  /\  t  e.  A ) )
96, 1opelcnv 4725 . . . . . . . . . . 11  |-  ( <.
s ,  t >.  e.  `' F  <->  <. t ,  s
>.  e.  F )
109anbi1i 454 . . . . . . . . . 10  |-  ( (
<. s ,  t >.  e.  `' F  /\  t  e.  A )  <->  ( <. t ,  s >.  e.  F  /\  t  e.  A
) )
118, 10bitr4i 186 . . . . . . . . 9  |-  ( <.
t ,  s >.  e.  ( F  |`  A )  <-> 
( <. s ,  t
>.  e.  `' F  /\  t  e.  A )
)
127, 11bitri 183 . . . . . . . 8  |-  ( <.
s ,  t >.  e.  `' ( F  |`  A )  <->  ( <. s ,  t >.  e.  `' F  /\  t  e.  A
) )
1312anbi2i 453 . . . . . . 7  |-  ( ( s  e.  B  /\  <.
s ,  t >.  e.  `' ( F  |`  A ) )  <->  ( s  e.  B  /\  ( <. s ,  t >.  e.  `' F  /\  t  e.  A ) ) )
14 anass 399 . . . . . . 7  |-  ( ( ( s  e.  B  /\  <. s ,  t
>.  e.  `' F )  /\  t  e.  A
)  <->  ( s  e.  B  /\  ( <.
s ,  t >.  e.  `' F  /\  t  e.  A ) ) )
1513, 14bitr4i 186 . . . . . 6  |-  ( ( s  e.  B  /\  <.
s ,  t >.  e.  `' ( F  |`  A ) )  <->  ( (
s  e.  B  /\  <.
s ,  t >.  e.  `' F )  /\  t  e.  A ) )
1615exbii 1585 . . . . 5  |-  ( E. s ( s  e.  B  /\  <. s ,  t >.  e.  `' ( F  |`  A ) )  <->  E. s ( ( s  e.  B  /\  <.
s ,  t >.  e.  `' F )  /\  t  e.  A ) )
17 19.41v 1875 . . . . 5  |-  ( E. s ( ( s  e.  B  /\  <. s ,  t >.  e.  `' F )  /\  t  e.  A )  <->  ( E. s ( s  e.  B  /\  <. s ,  t >.  e.  `' F )  /\  t  e.  A ) )
1816, 17bitri 183 . . . 4  |-  ( E. s ( s  e.  B  /\  <. s ,  t >.  e.  `' ( F  |`  A ) )  <->  ( E. s
( s  e.  B  /\  <. s ,  t
>.  e.  `' F )  /\  t  e.  A
) )
194, 5, 183bitr4ri 212 . . 3  |-  ( E. s ( s  e.  B  /\  <. s ,  t >.  e.  `' ( F  |`  A ) )  <->  t  e.  ( ( `' F " B )  i^i  A
) )
202, 19bitri 183 . 2  |-  ( t  e.  ( `' ( F  |`  A ) " B )  <->  t  e.  ( ( `' F " B )  i^i  A
) )
2120eqriv 2137 1  |-  ( `' ( F  |`  A )
" B )  =  ( ( `' F " B )  i^i  A
)
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1332   E.wex 1469    e. wcel 1481    i^i cin 3071   <.cop 3531   `'ccnv 4542    |` cres 4545   "cima 4546
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-pow 4102  ax-pr 4135
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2689  df-un 3076  df-in 3078  df-ss 3085  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-br 3934  df-opab 3994  df-xp 4549  df-cnv 4551  df-dm 4553  df-rn 4554  df-res 4555  df-ima 4556
This theorem is referenced by:  cnrest  12434
  Copyright terms: Public domain W3C validator