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Theorem respreima 5654
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 5283 . . 3  |-  ( Fun 
F  <->  F  Fn  dom  F )
2 elin 3358 . . . . . . . . 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 5542 . . . . . . . . . 10  |-  ( x  e.  B  ->  (
( F  |`  B ) `
 x )  =  ( F `  x
) )
76eleq1d 2349 . . . . . . . . 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 5341 . . . . . . . 8  |-  ( F  Fn  dom  F  ->  Fun  F )
15 funres 5293 . . . . . . . 8  |-  ( Fun 
F  ->  Fun  ( F  |`  B ) )
1614, 15syl 15 . . . . . . 7  |-  ( F  Fn  dom  F  ->  Fun  ( F  |`  B ) )
17 dmres 4976 . . . . . . 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 5258 . . . . . 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 5645 . . . . 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 3358 . . . . 5  |-  ( x  e.  ( ( `' F " A )  i^i  B )  <->  ( x  e.  ( `' F " A )  /\  x  e.  B ) )
24 elpreima 5645 . . . . . 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 2281 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 1623    e. wcel 1684    i^i cin 3151   `'ccnv 4688   dom cdm 4689    |` cres 4691   "cima 4692   Fun wfun 5249    Fn wfn 5250   ` cfv 5255
This theorem is referenced by:  paste  17022  respreimaOLD  26369
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-sbc 2992  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-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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