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Theorem fnreseql 5808
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 5519 . . . 4  |-  ( ( F  Fn  A  /\  X  C_  A )  -> 
( F  |`  X )  Fn  X )
213adant2 1007 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A  /\  X  C_  A )  -> 
( F  |`  X )  Fn  X )
3 fnssres 5519 . . . 4  |-  ( ( G  Fn  A  /\  X  C_  A )  -> 
( G  |`  X )  Fn  X )
433adant1 1006 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A  /\  X  C_  A )  -> 
( G  |`  X )  Fn  X )
5 fneqeql 5806 . . 3  |-  ( ( ( F  |`  X )  Fn  X  /\  ( G  |`  X )  Fn  X )  ->  (
( F  |`  X )  =  ( G  |`  X )  <->  dom  ( ( F  |`  X )  i^i  ( G  |`  X ) )  =  X ) )
62, 4, 5syl2anc 661 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A  /\  X  C_  A )  -> 
( ( F  |`  X )  =  ( G  |`  X )  <->  dom  ( ( F  |`  X )  i^i  ( G  |`  X ) )  =  X ) )
7 resindir 5122 . . . . . 6  |-  ( ( F  i^i  G )  |`  X )  =  ( ( F  |`  X )  i^i  ( G  |`  X ) )
87dmeqi 5036 . . . . 5  |-  dom  (
( F  i^i  G
)  |`  X )  =  dom  ( ( F  |`  X )  i^i  ( G  |`  X ) )
9 dmres 5126 . . . . 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 3337 . . 3  |-  ( X 
C_  dom  ( F  i^i  G )  <->  ( X  i^i  dom  ( F  i^i  G ) )  =  X )
1311, 12bitr4i 252 . 2  |-  ( dom  ( ( F  |`  X )  i^i  ( G  |`  X ) )  =  X  <->  X  C_  dom  ( F  i^i  G ) )
146, 13syl6bb 261 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 965    = wceq 1369    i^i cin 3322    C_ wss 3323   dom cdm 4835    |` cres 4837    Fn wfn 5408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-fv 5421
This theorem is referenced by:  lspextmo  17114  evlseu  17577  hauseqcn  26277
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