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Theorem cnvresima 5344
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 3090 . . . 4  |-  t  e. 
_V
21elima3 5195 . . 3  |-  ( t  e.  ( `' ( F  |`  A ) " B )  <->  E. s
( s  e.  B  /\  <. s ,  t
>.  e.  `' ( F  |`  A ) ) )
31elima3 5195 . . . . 5  |-  ( t  e.  ( `' F " B )  <->  E. s
( s  e.  B  /\  <. s ,  t
>.  e.  `' F ) )
43anbi1i 699 . . . 4  |-  ( ( t  e.  ( `' F " B )  /\  t  e.  A
)  <->  ( E. s
( s  e.  B  /\  <. s ,  t
>.  e.  `' F )  /\  t  e.  A
) )
5 elin 3655 . . . 4  |-  ( t  e.  ( ( `' F " B )  i^i  A )  <->  ( t  e.  ( `' F " B )  /\  t  e.  A ) )
6 vex 3090 . . . . . . . . . 10  |-  s  e. 
_V
76, 1opelcnv 5036 . . . . . . . . 9  |-  ( <.
s ,  t >.  e.  `' ( F  |`  A )  <->  <. t ,  s >.  e.  ( F  |`  A ) )
86opelres 5130 . . . . . . . . . 10  |-  ( <.
t ,  s >.  e.  ( F  |`  A )  <-> 
( <. t ,  s
>.  e.  F  /\  t  e.  A ) )
96, 1opelcnv 5036 . . . . . . . . . . 11  |-  ( <.
s ,  t >.  e.  `' F  <->  <. t ,  s
>.  e.  F )
109anbi1i 699 . . . . . . . . . 10  |-  ( (
<. s ,  t >.  e.  `' F  /\  t  e.  A )  <->  ( <. t ,  s >.  e.  F  /\  t  e.  A
) )
118, 10bitr4i 255 . . . . . . . . 9  |-  ( <.
t ,  s >.  e.  ( F  |`  A )  <-> 
( <. s ,  t
>.  e.  `' F  /\  t  e.  A )
)
127, 11bitri 252 . . . . . . . 8  |-  ( <.
s ,  t >.  e.  `' ( F  |`  A )  <->  ( <. s ,  t >.  e.  `' F  /\  t  e.  A
) )
1312anbi2i 698 . . . . . . 7  |-  ( ( s  e.  B  /\  <.
s ,  t >.  e.  `' ( F  |`  A ) )  <->  ( s  e.  B  /\  ( <. s ,  t >.  e.  `' F  /\  t  e.  A ) ) )
14 anass 653 . . . . . . 7  |-  ( ( ( s  e.  B  /\  <. s ,  t
>.  e.  `' F )  /\  t  e.  A
)  <->  ( s  e.  B  /\  ( <.
s ,  t >.  e.  `' F  /\  t  e.  A ) ) )
1513, 14bitr4i 255 . . . . . 6  |-  ( ( s  e.  B  /\  <.
s ,  t >.  e.  `' ( F  |`  A ) )  <->  ( (
s  e.  B  /\  <.
s ,  t >.  e.  `' F )  /\  t  e.  A ) )
1615exbii 1714 . . . . 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 1822 . . . . 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 252 . . . 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 281 . . 3  |-  ( E. s ( s  e.  B  /\  <. s ,  t >.  e.  `' ( F  |`  A ) )  <->  t  e.  ( ( `' F " B )  i^i  A
) )
202, 19bitri 252 . 2  |-  ( t  e.  ( `' ( F  |`  A ) " B )  <->  t  e.  ( ( `' F " B )  i^i  A
) )
2120eqriv 2425 1  |-  ( `' ( F  |`  A )
" B )  =  ( ( `' F " B )  i^i  A
)
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370    = wceq 1437   E.wex 1659    e. wcel 1870    i^i cin 3441   <.cop 4008   `'ccnv 4853    |` cres 4856   "cima 4857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-xp 4860  df-cnv 4862  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867
This theorem is referenced by:  ramub2  14934  ramub1lem2  14948  cnrest  20232  kgencn  20502  kgencn3  20504  xkoptsub  20600  qtopres  20644  qtoprest  20663  mbfid  22469  mbfres  22477  fimacnvinrn  28075  1stpreima  28127  2ndpreima  28128  cvmsss2  29785  lmhmlnmsplit  35650
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