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Theorem fnreseql 5843
Description: Two functions are equal on a subset iff their equalizer contains that subset. (Contributed by Stefan O'Rear, 7-Mar-2015.)
Assertion
Ref Expression
fnreseql  |-  ( ( F  Fn  A  /\  G  Fn  A  /\  X  C_  A )  -> 
( ( F  |`  X )  =  ( G  |`  X )  <->  X 
C_  dom  ( F  i^i  G ) ) )

Proof of Theorem fnreseql
StepHypRef Expression
1 fnssres 5561 . . . 4  |-  ( ( F  Fn  A  /\  X  C_  A )  -> 
( F  |`  X )  Fn  X )
213adant2 977 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A  /\  X  C_  A )  -> 
( F  |`  X )  Fn  X )
3 fnssres 5561 . . . 4  |-  ( ( G  Fn  A  /\  X  C_  A )  -> 
( G  |`  X )  Fn  X )
433adant1 976 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A  /\  X  C_  A )  -> 
( G  |`  X )  Fn  X )
5 fneqeql 5841 . . 3  |-  ( ( ( F  |`  X )  Fn  X  /\  ( G  |`  X )  Fn  X )  ->  (
( F  |`  X )  =  ( G  |`  X )  <->  dom  ( ( F  |`  X )  i^i  ( G  |`  X ) )  =  X ) )
62, 4, 5syl2anc 644 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A  /\  X  C_  A )  -> 
( ( F  |`  X )  =  ( G  |`  X )  <->  dom  ( ( F  |`  X )  i^i  ( G  |`  X ) )  =  X ) )
7 resindir 5166 . . . . . 6  |-  ( ( F  i^i  G )  |`  X )  =  ( ( F  |`  X )  i^i  ( G  |`  X ) )
87dmeqi 5074 . . . . 5  |-  dom  (
( F  i^i  G
)  |`  X )  =  dom  ( ( F  |`  X )  i^i  ( G  |`  X ) )
9 dmres 5170 . . . . 5  |-  dom  (
( F  i^i  G
)  |`  X )  =  ( X  i^i  dom  ( F  i^i  G ) )
108, 9eqtr3i 2460 . . . 4  |-  dom  (
( F  |`  X )  i^i  ( G  |`  X ) )  =  ( X  i^i  dom  ( F  i^i  G ) )
1110eqeq1i 2445 . . 3  |-  ( dom  ( ( F  |`  X )  i^i  ( G  |`  X ) )  =  X  <->  ( X  i^i  dom  ( F  i^i  G ) )  =  X )
12 df-ss 3336 . . 3  |-  ( X 
C_  dom  ( F  i^i  G )  <->  ( X  i^i  dom  ( F  i^i  G ) )  =  X )
1311, 12bitr4i 245 . 2  |-  ( dom  ( ( F  |`  X )  i^i  ( G  |`  X ) )  =  X  <->  X  C_  dom  ( F  i^i  G ) )
146, 13syl6bb 254 1  |-  ( ( F  Fn  A  /\  G  Fn  A  /\  X  C_  A )  -> 
( ( F  |`  X )  =  ( G  |`  X )  <->  X 
C_  dom  ( F  i^i  G ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ w3a 937    = wceq 1653    i^i cin 3321    C_ wss 3322   dom cdm 4881    |` cres 4883    Fn wfn 5452
This theorem is referenced by:  lspextmo  16137  evlseu  19942  hauseqcn  24298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-fv 5465
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