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Theorem respreima 5624
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 5228 . . 3  |-  ( Fun 
F  <->  F  Fn  dom  F )
2 elin 3310 . . . . . . . . 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 455 . . . . . . 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 5520 . . . . . . . . . 10  |-  ( x  e.  B  ->  (
( F  |`  B ) `
 x )  =  ( F `  x
) )
76eleq1d 2239 . . . . . . . . 9  |-  ( x  e.  B  ->  (
( ( F  |`  B ) `  x
)  e.  A  <->  ( F `  x )  e.  A
) )
87adantl 275 . . . . . . . 8  |-  ( ( x  e.  dom  F  /\  x  e.  B
)  ->  ( (
( F  |`  B ) `
 x )  e.  A  <->  ( F `  x )  e.  A
) )
98pm5.32i 451 . . . . . . 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 557 . . . . 5  |-  ( ( ( x  e.  dom  F  /\  x  e.  B
)  /\  ( F `  x )  e.  A
)  <->  ( ( x  e.  dom  F  /\  ( F `  x )  e.  A )  /\  x  e.  B )
)
1311, 12bitrdi 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 5295 . . . . . . . 8  |-  ( F  Fn  dom  F  ->  Fun  F )
15 funres 5239 . . . . . . . 8  |-  ( Fun 
F  ->  Fun  ( F  |`  B ) )
1614, 15syl 14 . . . . . . 7  |-  ( F  Fn  dom  F  ->  Fun  ( F  |`  B ) )
17 dmres 4912 . . . . . . 7  |-  dom  ( F  |`  B )  =  ( B  i^i  dom  F )
1816, 17jctir 311 . . . . . 6  |-  ( F  Fn  dom  F  -> 
( Fun  ( F  |`  B )  /\  dom  ( F  |`  B )  =  ( B  i^i  dom 
F ) ) )
19 df-fn 5201 . . . . . 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 5615 . . . . 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 3310 . . . . 5  |-  ( x  e.  ( ( `' F " A )  i^i  B )  <->  ( x  e.  ( `' F " A )  /\  x  e.  B ) )
24 elpreima 5615 . . . . . 6  |-  ( F  Fn  dom  F  -> 
( x  e.  ( `' F " A )  <-> 
( x  e.  dom  F  /\  ( F `  x )  e.  A
) ) )
2524anbi1d 462 . . . . 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 2168 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 1348    e. wcel 2141    i^i cin 3120   `'ccnv 4610   dom cdm 4611    |` cres 4613   "cima 4614   Fun wfun 5192    Fn wfn 5193   ` cfv 5198
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206
This theorem is referenced by: (None)
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