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Theorem respreima 6024
Description: The preimage of a restricted function. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
respreima  |-  ( Fun 
F  ->  ( `' ( F  |`  B )
" A )  =  ( ( `' F " A )  i^i  B
) )

Proof of Theorem respreima
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 funfn 5618 . . 3  |-  ( Fun 
F  <->  F  Fn  dom  F )
2 elin 3608 . . . . . . . . 9  |-  ( x  e.  ( B  i^i  dom 
F )  <->  ( x  e.  B  /\  x  e.  dom  F ) )
3 ancom 457 . . . . . . . . 9  |-  ( ( x  e.  B  /\  x  e.  dom  F )  <-> 
( x  e.  dom  F  /\  x  e.  B
) )
42, 3bitri 257 . . . . . . . 8  |-  ( x  e.  ( B  i^i  dom 
F )  <->  ( x  e.  dom  F  /\  x  e.  B ) )
54anbi1i 709 . . . . . . 7  |-  ( ( x  e.  ( B  i^i  dom  F )  /\  ( ( F  |`  B ) `  x
)  e.  A )  <-> 
( ( x  e. 
dom  F  /\  x  e.  B )  /\  (
( F  |`  B ) `
 x )  e.  A ) )
6 fvres 5893 . . . . . . . . . 10  |-  ( x  e.  B  ->  (
( F  |`  B ) `
 x )  =  ( F `  x
) )
76eleq1d 2533 . . . . . . . . 9  |-  ( x  e.  B  ->  (
( ( F  |`  B ) `  x
)  e.  A  <->  ( F `  x )  e.  A
) )
87adantl 473 . . . . . . . 8  |-  ( ( x  e.  dom  F  /\  x  e.  B
)  ->  ( (
( F  |`  B ) `
 x )  e.  A  <->  ( F `  x )  e.  A
) )
98pm5.32i 649 . . . . . . 7  |-  ( ( ( x  e.  dom  F  /\  x  e.  B
)  /\  ( ( F  |`  B ) `  x )  e.  A
)  <->  ( ( x  e.  dom  F  /\  x  e.  B )  /\  ( F `  x
)  e.  A ) )
105, 9bitri 257 . . . . . 6  |-  ( ( x  e.  ( B  i^i  dom  F )  /\  ( ( F  |`  B ) `  x
)  e.  A )  <-> 
( ( x  e. 
dom  F  /\  x  e.  B )  /\  ( F `  x )  e.  A ) )
1110a1i 11 . . . . 5  |-  ( F  Fn  dom  F  -> 
( ( x  e.  ( B  i^i  dom  F )  /\  ( ( F  |`  B ) `  x )  e.  A
)  <->  ( ( x  e.  dom  F  /\  x  e.  B )  /\  ( F `  x
)  e.  A ) ) )
12 an32 815 . . . . 5  |-  ( ( ( x  e.  dom  F  /\  x  e.  B
)  /\  ( F `  x )  e.  A
)  <->  ( ( x  e.  dom  F  /\  ( F `  x )  e.  A )  /\  x  e.  B )
)
1311, 12syl6bb 269 . . . 4  |-  ( F  Fn  dom  F  -> 
( ( x  e.  ( B  i^i  dom  F )  /\  ( ( F  |`  B ) `  x )  e.  A
)  <->  ( ( x  e.  dom  F  /\  ( F `  x )  e.  A )  /\  x  e.  B )
) )
14 fnfun 5683 . . . . . . . 8  |-  ( F  Fn  dom  F  ->  Fun  F )
15 funres 5628 . . . . . . . 8  |-  ( Fun 
F  ->  Fun  ( F  |`  B ) )
1614, 15syl 17 . . . . . . 7  |-  ( F  Fn  dom  F  ->  Fun  ( F  |`  B ) )
17 dmres 5131 . . . . . . 7  |-  dom  ( F  |`  B )  =  ( B  i^i  dom  F )
1816, 17jctir 547 . . . . . 6  |-  ( F  Fn  dom  F  -> 
( Fun  ( F  |`  B )  /\  dom  ( F  |`  B )  =  ( B  i^i  dom 
F ) ) )
19 df-fn 5592 . . . . . 6  |-  ( ( F  |`  B )  Fn  ( B  i^i  dom  F )  <->  ( Fun  ( F  |`  B )  /\  dom  ( F  |`  B )  =  ( B  i^i  dom 
F ) ) )
2018, 19sylibr 217 . . . . 5  |-  ( F  Fn  dom  F  -> 
( F  |`  B )  Fn  ( B  i^i  dom 
F ) )
21 elpreima 6017 . . . . 5  |-  ( ( F  |`  B )  Fn  ( B  i^i  dom  F )  ->  ( x  e.  ( `' ( F  |`  B ) " A
)  <->  ( x  e.  ( B  i^i  dom  F )  /\  ( ( F  |`  B ) `  x )  e.  A
) ) )
2220, 21syl 17 . . . 4  |-  ( F  Fn  dom  F  -> 
( x  e.  ( `' ( F  |`  B ) " A
)  <->  ( x  e.  ( B  i^i  dom  F )  /\  ( ( F  |`  B ) `  x )  e.  A
) ) )
23 elin 3608 . . . . 5  |-  ( x  e.  ( ( `' F " A )  i^i  B )  <->  ( x  e.  ( `' F " A )  /\  x  e.  B ) )
24 elpreima 6017 . . . . . 6  |-  ( F  Fn  dom  F  -> 
( x  e.  ( `' F " A )  <-> 
( x  e.  dom  F  /\  ( F `  x )  e.  A
) ) )
2524anbi1d 719 . . . . 5  |-  ( F  Fn  dom  F  -> 
( ( x  e.  ( `' F " A )  /\  x  e.  B )  <->  ( (
x  e.  dom  F  /\  ( F `  x
)  e.  A )  /\  x  e.  B
) ) )
2623, 25syl5bb 265 . . . 4  |-  ( F  Fn  dom  F  -> 
( x  e.  ( ( `' F " A )  i^i  B
)  <->  ( ( x  e.  dom  F  /\  ( F `  x )  e.  A )  /\  x  e.  B )
) )
2713, 22, 263bitr4d 293 . . 3  |-  ( F  Fn  dom  F  -> 
( x  e.  ( `' ( F  |`  B ) " A
)  <->  x  e.  (
( `' F " A )  i^i  B
) ) )
281, 27sylbi 200 . 2  |-  ( Fun 
F  ->  ( x  e.  ( `' ( F  |`  B ) " A
)  <->  x  e.  (
( `' F " A )  i^i  B
) ) )
2928eqrdv 2469 1  |-  ( Fun 
F  ->  ( `' ( F  |`  B )
" A )  =  ( ( `' F " A )  i^i  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904    i^i cin 3389   `'ccnv 4838   dom cdm 4839    |` cres 4841   "cima 4842   Fun wfun 5583    Fn wfn 5584   ` cfv 5589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-fv 5597
This theorem is referenced by:  paste  20387  restmetu  21663  eulerpartlemt  29277
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