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Theorem respreima 5656
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 5285 . . 3  |-  ( Fun 
F  <->  F  Fn  dom  F )
2 elin 3360 . . . . . . . . 9  |-  ( x  e.  ( B  i^i  dom 
F )  <->  ( x  e.  B  /\  x  e.  dom  F ) )
3 ancom 437 . . . . . . . . 9  |-  ( ( x  e.  B  /\  x  e.  dom  F )  <-> 
( x  e.  dom  F  /\  x  e.  B
) )
42, 3bitri 240 . . . . . . . 8  |-  ( x  e.  ( B  i^i  dom 
F )  <->  ( x  e.  dom  F  /\  x  e.  B ) )
54anbi1i 676 . . . . . . 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 5544 . . . . . . . . . 10  |-  ( x  e.  B  ->  (
( F  |`  B ) `
 x )  =  ( F `  x
) )
76eleq1d 2351 . . . . . . . . 9  |-  ( x  e.  B  ->  (
( ( F  |`  B ) `  x
)  e.  A  <->  ( F `  x )  e.  A
) )
87adantl 452 . . . . . . . 8  |-  ( ( x  e.  dom  F  /\  x  e.  B
)  ->  ( (
( F  |`  B ) `
 x )  e.  A  <->  ( F `  x )  e.  A
) )
98pm5.32i 618 . . . . . . 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 240 . . . . . 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 10 . . . . 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 773 . . . . 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 252 . . . 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 5343 . . . . . . . 8  |-  ( F  Fn  dom  F  ->  Fun  F )
15 funres 5295 . . . . . . . 8  |-  ( Fun 
F  ->  Fun  ( F  |`  B ) )
1614, 15syl 15 . . . . . . 7  |-  ( F  Fn  dom  F  ->  Fun  ( F  |`  B ) )
17 dmres 4978 . . . . . . 7  |-  dom  ( F  |`  B )  =  ( B  i^i  dom  F )
1816, 17jctir 524 . . . . . 6  |-  ( F  Fn  dom  F  -> 
( Fun  ( F  |`  B )  /\  dom  ( F  |`  B )  =  ( B  i^i  dom 
F ) ) )
19 df-fn 5260 . . . . . 6  |-  ( ( F  |`  B )  Fn  ( B  i^i  dom  F )  <->  ( Fun  ( F  |`  B )  /\  dom  ( F  |`  B )  =  ( B  i^i  dom 
F ) ) )
2018, 19sylibr 203 . . . . 5  |-  ( F  Fn  dom  F  -> 
( F  |`  B )  Fn  ( B  i^i  dom 
F ) )
21 elpreima 5647 . . . . 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 15 . . . 4  |-  ( F  Fn  dom  F  -> 
( x  e.  ( `' ( F  |`  B ) " A
)  <->  ( x  e.  ( B  i^i  dom  F )  /\  ( ( F  |`  B ) `  x )  e.  A
) ) )
23 elin 3360 . . . . 5  |-  ( x  e.  ( ( `' F " A )  i^i  B )  <->  ( x  e.  ( `' F " A )  /\  x  e.  B ) )
24 elpreima 5647 . . . . . 6  |-  ( F  Fn  dom  F  -> 
( x  e.  ( `' F " A )  <-> 
( x  e.  dom  F  /\  ( F `  x )  e.  A
) ) )
2524anbi1d 685 . . . . 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 248 . . . 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 276 . . 3  |-  ( F  Fn  dom  F  -> 
( x  e.  ( `' ( F  |`  B ) " A
)  <->  x  e.  (
( `' F " A )  i^i  B
) ) )
281, 27sylbi 187 . 2  |-  ( Fun 
F  ->  ( x  e.  ( `' ( F  |`  B ) " A
)  <->  x  e.  (
( `' F " A )  i^i  B
) ) )
2928eqrdv 2283 1  |-  ( Fun 
F  ->  ( `' ( F  |`  B )
" A )  =  ( ( `' F " A )  i^i  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1625    e. wcel 1686    i^i cin 3153   `'ccnv 4690   dom cdm 4691    |` cres 4693   "cima 4694   Fun wfun 5251    Fn wfn 5252   ` cfv 5257
This theorem is referenced by:  paste  17024  respreimaOLD  26380
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-fv 5265
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