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Theorem fnresin 28231
Description: Restriction of a function with a subclass of its domain. (Contributed by Thierry Arnoux, 10-Oct-2017.)
Assertion
Ref Expression
fnresin  |-  ( F  Fn  A  ->  ( F  |`  B )  Fn  ( A  i^i  B
) )

Proof of Theorem fnresin
StepHypRef Expression
1 fnresin1 5708 . 2  |-  ( F  Fn  A  ->  ( F  |`  ( A  i^i  B ) )  Fn  ( A  i^i  B ) )
2 resindi 5139 . . . 4  |-  ( F  |`  ( A  i^i  B
) )  =  ( ( F  |`  A )  i^i  ( F  |`  B ) )
3 fnresdm 5703 . . . . . 6  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
43ineq1d 3663 . . . . 5  |-  ( F  Fn  A  ->  (
( F  |`  A )  i^i  ( F  |`  B ) )  =  ( F  i^i  ( F  |`  B ) ) )
5 incom 3655 . . . . . 6  |-  ( ( F  |`  B )  i^i  F )  =  ( F  i^i  ( F  |`  B ) )
6 resss 5147 . . . . . . 7  |-  ( F  |`  B )  C_  F
7 df-ss 3450 . . . . . . 7  |-  ( ( F  |`  B )  C_  F  <->  ( ( F  |`  B )  i^i  F
)  =  ( F  |`  B ) )
86, 7mpbi 211 . . . . . 6  |-  ( ( F  |`  B )  i^i  F )  =  ( F  |`  B )
95, 8eqtr3i 2453 . . . . 5  |-  ( F  i^i  ( F  |`  B ) )  =  ( F  |`  B )
104, 9syl6eq 2479 . . . 4  |-  ( F  Fn  A  ->  (
( F  |`  A )  i^i  ( F  |`  B ) )  =  ( F  |`  B ) )
112, 10syl5eq 2475 . . 3  |-  ( F  Fn  A  ->  ( F  |`  ( A  i^i  B ) )  =  ( F  |`  B )
)
1211fneq1d 5684 . 2  |-  ( F  Fn  A  ->  (
( F  |`  ( A  i^i  B ) )  Fn  ( A  i^i  B )  <->  ( F  |`  B )  Fn  ( A  i^i  B ) ) )
131, 12mpbid 213 1  |-  ( F  Fn  A  ->  ( F  |`  B )  Fn  ( A  i^i  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    i^i cin 3435    C_ wss 3436    |` cres 4855    Fn wfn 5596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-br 4424  df-opab 4483  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-res 4865  df-fun 5603  df-fn 5604
This theorem is referenced by:  signstres  29473
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