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Theorem cnvresima 4840
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 2605 . . . 4  |-  t  e. 
_V
21elima3 4705 . . 3  |-  ( t  e.  ( `' ( F  |`  A ) " B )  <->  E. s
( s  e.  B  /\  <. s ,  t
>.  e.  `' ( F  |`  A ) ) )
31elima3 4705 . . . . 5  |-  ( t  e.  ( `' F " B )  <->  E. s
( s  e.  B  /\  <. s ,  t
>.  e.  `' F ) )
43anbi1i 446 . . . 4  |-  ( ( t  e.  ( `' F " B )  /\  t  e.  A
)  <->  ( E. s
( s  e.  B  /\  <. s ,  t
>.  e.  `' F )  /\  t  e.  A
) )
5 elin 3156 . . . 4  |-  ( t  e.  ( ( `' F " B )  i^i  A )  <->  ( t  e.  ( `' F " B )  /\  t  e.  A ) )
6 vex 2605 . . . . . . . . . 10  |-  s  e. 
_V
76, 1opelcnv 4545 . . . . . . . . 9  |-  ( <.
s ,  t >.  e.  `' ( F  |`  A )  <->  <. t ,  s >.  e.  ( F  |`  A ) )
86opelres 4645 . . . . . . . . . 10  |-  ( <.
t ,  s >.  e.  ( F  |`  A )  <-> 
( <. t ,  s
>.  e.  F  /\  t  e.  A ) )
96, 1opelcnv 4545 . . . . . . . . . . 11  |-  ( <.
s ,  t >.  e.  `' F  <->  <. t ,  s
>.  e.  F )
109anbi1i 446 . . . . . . . . . 10  |-  ( (
<. s ,  t >.  e.  `' F  /\  t  e.  A )  <->  ( <. t ,  s >.  e.  F  /\  t  e.  A
) )
118, 10bitr4i 185 . . . . . . . . 9  |-  ( <.
t ,  s >.  e.  ( F  |`  A )  <-> 
( <. s ,  t
>.  e.  `' F  /\  t  e.  A )
)
127, 11bitri 182 . . . . . . . 8  |-  ( <.
s ,  t >.  e.  `' ( F  |`  A )  <->  ( <. s ,  t >.  e.  `' F  /\  t  e.  A
) )
1312anbi2i 445 . . . . . . 7  |-  ( ( s  e.  B  /\  <.
s ,  t >.  e.  `' ( F  |`  A ) )  <->  ( s  e.  B  /\  ( <. s ,  t >.  e.  `' F  /\  t  e.  A ) ) )
14 anass 393 . . . . . . 7  |-  ( ( ( s  e.  B  /\  <. s ,  t
>.  e.  `' F )  /\  t  e.  A
)  <->  ( s  e.  B  /\  ( <.
s ,  t >.  e.  `' F  /\  t  e.  A ) ) )
1513, 14bitr4i 185 . . . . . 6  |-  ( ( s  e.  B  /\  <.
s ,  t >.  e.  `' ( F  |`  A ) )  <->  ( (
s  e.  B  /\  <.
s ,  t >.  e.  `' F )  /\  t  e.  A ) )
1615exbii 1537 . . . . 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 1824 . . . . 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 182 . . . 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 211 . . 3  |-  ( E. s ( s  e.  B  /\  <. s ,  t >.  e.  `' ( F  |`  A ) )  <->  t  e.  ( ( `' F " B )  i^i  A
) )
202, 19bitri 182 . 2  |-  ( t  e.  ( `' ( F  |`  A ) " B )  <->  t  e.  ( ( `' F " B )  i^i  A
) )
2120eqriv 2079 1  |-  ( `' ( F  |`  A )
" B )  =  ( ( `' F " B )  i^i  A
)
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1285   E.wex 1422    e. wcel 1434    i^i cin 2973   <.cop 3409   `'ccnv 4370    |` cres 4373   "cima 4374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-xp 4377  df-cnv 4379  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384
This theorem is referenced by: (None)
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