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Theorem respreima 5514
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 5121 . . 3  |-  ( Fun 
F  <->  F  Fn  dom  F )
2 elin 3227 . . . . . . . . 9  |-  ( x  e.  ( B  i^i  dom 
F )  <->  ( x  e.  B  /\  x  e.  dom  F ) )
3 ancom 264 . . . . . . . . 9  |-  ( ( x  e.  B  /\  x  e.  dom  F )  <-> 
( x  e.  dom  F  /\  x  e.  B
) )
42, 3bitri 183 . . . . . . . 8  |-  ( x  e.  ( B  i^i  dom 
F )  <->  ( x  e.  dom  F  /\  x  e.  B ) )
54anbi1i 451 . . . . . . 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 5411 . . . . . . . . . 10  |-  ( x  e.  B  ->  (
( F  |`  B ) `
 x )  =  ( F `  x
) )
76eleq1d 2184 . . . . . . . . 9  |-  ( x  e.  B  ->  (
( ( F  |`  B ) `  x
)  e.  A  <->  ( F `  x )  e.  A
) )
87adantl 273 . . . . . . . 8  |-  ( ( x  e.  dom  F  /\  x  e.  B
)  ->  ( (
( F  |`  B ) `
 x )  e.  A  <->  ( F `  x )  e.  A
) )
98pm5.32i 447 . . . . . . 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 183 . . . . . 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 9 . . . . 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 534 . . . . 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 195 . . . 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 5188 . . . . . . . 8  |-  ( F  Fn  dom  F  ->  Fun  F )
15 funres 5132 . . . . . . . 8  |-  ( Fun 
F  ->  Fun  ( F  |`  B ) )
1614, 15syl 14 . . . . . . 7  |-  ( F  Fn  dom  F  ->  Fun  ( F  |`  B ) )
17 dmres 4808 . . . . . . 7  |-  dom  ( F  |`  B )  =  ( B  i^i  dom  F )
1816, 17jctir 309 . . . . . 6  |-  ( F  Fn  dom  F  -> 
( Fun  ( F  |`  B )  /\  dom  ( F  |`  B )  =  ( B  i^i  dom 
F ) ) )
19 df-fn 5094 . . . . . 6  |-  ( ( F  |`  B )  Fn  ( B  i^i  dom  F )  <->  ( Fun  ( F  |`  B )  /\  dom  ( F  |`  B )  =  ( B  i^i  dom 
F ) ) )
2018, 19sylibr 133 . . . . 5  |-  ( F  Fn  dom  F  -> 
( F  |`  B )  Fn  ( B  i^i  dom 
F ) )
21 elpreima 5505 . . . . 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 14 . . . 4  |-  ( F  Fn  dom  F  -> 
( x  e.  ( `' ( F  |`  B ) " A
)  <->  ( x  e.  ( B  i^i  dom  F )  /\  ( ( F  |`  B ) `  x )  e.  A
) ) )
23 elin 3227 . . . . 5  |-  ( x  e.  ( ( `' F " A )  i^i  B )  <->  ( x  e.  ( `' F " A )  /\  x  e.  B ) )
24 elpreima 5505 . . . . . 6  |-  ( F  Fn  dom  F  -> 
( x  e.  ( `' F " A )  <-> 
( x  e.  dom  F  /\  ( F `  x )  e.  A
) ) )
2524anbi1d 458 . . . . 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 191 . . . 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 219 . . 3  |-  ( F  Fn  dom  F  -> 
( x  e.  ( `' ( F  |`  B ) " A
)  <->  x  e.  (
( `' F " A )  i^i  B
) ) )
281, 27sylbi 120 . 2  |-  ( Fun 
F  ->  ( x  e.  ( `' ( F  |`  B ) " A
)  <->  x  e.  (
( `' F " A )  i^i  B
) ) )
2928eqrdv 2113 1  |-  ( Fun 
F  ->  ( `' ( F  |`  B )
" A )  =  ( ( `' F " A )  i^i  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463    i^i cin 3038   `'ccnv 4506   dom cdm 4507    |` cres 4509   "cima 4510   Fun wfun 5085    Fn wfn 5086   ` cfv 5091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-fv 5099
This theorem is referenced by: (None)
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