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Theorem cnvresima 5178
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 2804 . . . 4  |-  t  e. 
_V
21elima3 5035 . . 3  |-  ( t  e.  ( `' ( F  |`  A ) " B )  <->  E. s
( s  e.  B  /\  <. s ,  t
>.  e.  `' ( F  |`  A ) ) )
31elima3 5035 . . . . 5  |-  ( t  e.  ( `' F " B )  <->  E. s
( s  e.  B  /\  <. s ,  t
>.  e.  `' F ) )
43anbi1i 676 . . . 4  |-  ( ( t  e.  ( `' F " B )  /\  t  e.  A
)  <->  ( E. s
( s  e.  B  /\  <. s ,  t
>.  e.  `' F )  /\  t  e.  A
) )
5 elin 3371 . . . 4  |-  ( t  e.  ( ( `' F " B )  i^i  A )  <->  ( t  e.  ( `' F " B )  /\  t  e.  A ) )
6 vex 2804 . . . . . . . . . 10  |-  s  e. 
_V
76, 1opelcnv 4879 . . . . . . . . 9  |-  ( <.
s ,  t >.  e.  `' ( F  |`  A )  <->  <. t ,  s >.  e.  ( F  |`  A ) )
86opelres 4976 . . . . . . . . . 10  |-  ( <.
t ,  s >.  e.  ( F  |`  A )  <-> 
( <. t ,  s
>.  e.  F  /\  t  e.  A ) )
96, 1opelcnv 4879 . . . . . . . . . . 11  |-  ( <.
s ,  t >.  e.  `' F  <->  <. t ,  s
>.  e.  F )
109anbi1i 676 . . . . . . . . . 10  |-  ( (
<. s ,  t >.  e.  `' F  /\  t  e.  A )  <->  ( <. t ,  s >.  e.  F  /\  t  e.  A
) )
118, 10bitr4i 243 . . . . . . . . 9  |-  ( <.
t ,  s >.  e.  ( F  |`  A )  <-> 
( <. s ,  t
>.  e.  `' F  /\  t  e.  A )
)
127, 11bitri 240 . . . . . . . 8  |-  ( <.
s ,  t >.  e.  `' ( F  |`  A )  <->  ( <. s ,  t >.  e.  `' F  /\  t  e.  A
) )
1312anbi2i 675 . . . . . . 7  |-  ( ( s  e.  B  /\  <.
s ,  t >.  e.  `' ( F  |`  A ) )  <->  ( s  e.  B  /\  ( <. s ,  t >.  e.  `' F  /\  t  e.  A ) ) )
14 anass 630 . . . . . . 7  |-  ( ( ( s  e.  B  /\  <. s ,  t
>.  e.  `' F )  /\  t  e.  A
)  <->  ( s  e.  B  /\  ( <.
s ,  t >.  e.  `' F  /\  t  e.  A ) ) )
1513, 14bitr4i 243 . . . . . 6  |-  ( ( s  e.  B  /\  <.
s ,  t >.  e.  `' ( F  |`  A ) )  <->  ( (
s  e.  B  /\  <.
s ,  t >.  e.  `' F )  /\  t  e.  A ) )
1615exbii 1572 . . . . 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 1854 . . . . 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 240 . . . 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 269 . . 3  |-  ( E. s ( s  e.  B  /\  <. s ,  t >.  e.  `' ( F  |`  A ) )  <->  t  e.  ( ( `' F " B )  i^i  A
) )
202, 19bitri 240 . 2  |-  ( t  e.  ( `' ( F  |`  A ) " B )  <->  t  e.  ( ( `' F " B )  i^i  A
) )
2120eqriv 2293 1  |-  ( `' ( F  |`  A )
" B )  =  ( ( `' F " B )  i^i  A
)
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696    i^i cin 3164   <.cop 3656   `'ccnv 4704    |` cres 4707   "cima 4708
This theorem is referenced by:  ramub2  13077  ramub1lem2  13090  cnrest  17029  kgencn  17267  kgencn3  17269  xkoptsub  17364  qtopres  17405  qtoprest  17424  mbfid  19007  mbfres  19015  fimacnvinrn  23214  cvmsss2  23820  islimrs3  25684  islimrs4  25685  lmhmlnmsplit  27288  frlmsplit2  27346
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718
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