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Theorem cnvresima 5162
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 2791 . . . 4  |-  t  e. 
_V
21elima3 5019 . . 3  |-  ( t  e.  ( `' ( F  |`  A ) " B )  <->  E. s
( s  e.  B  /\  <. s ,  t
>.  e.  `' ( F  |`  A ) ) )
31elima3 5019 . . . . 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 3358 . . . 4  |-  ( t  e.  ( ( `' F " B )  i^i  A )  <->  ( t  e.  ( `' F " B )  /\  t  e.  A ) )
6 vex 2791 . . . . . . . . . 10  |-  s  e. 
_V
76, 1opelcnv 4863 . . . . . . . . 9  |-  ( <.
s ,  t >.  e.  `' ( F  |`  A )  <->  <. t ,  s >.  e.  ( F  |`  A ) )
86opelres 4960 . . . . . . . . . 10  |-  ( <.
t ,  s >.  e.  ( F  |`  A )  <-> 
( <. t ,  s
>.  e.  F  /\  t  e.  A ) )
96, 1opelcnv 4863 . . . . . . . . . . 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 1569 . . . . 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 1842 . . . . 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 2280 1  |-  ( `' ( F  |`  A )
" B )  =  ( ( `' F " B )  i^i  A
)
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684    i^i cin 3151   <.cop 3643   `'ccnv 4688    |` cres 4691   "cima 4692
This theorem is referenced by:  ramub2  13061  ramub1lem2  13074  cnrest  17013  kgencn  17251  kgencn3  17253  xkoptsub  17348  qtopres  17389  qtoprest  17408  mbfid  18991  mbfres  18999  fimacnvinrn  23199  cvmsss2  23805  islimrs3  25581  islimrs4  25582  lmhmlnmsplit  27185  frlmsplit2  27243
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702
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