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Theorem fnreseql 5579
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 5285 . . . 4  |-  ( ( F  Fn  A  /\  X  C_  A )  -> 
( F  |`  X )  Fn  X )
213adant2 1001 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A  /\  X  C_  A )  -> 
( F  |`  X )  Fn  X )
3 fnssres 5285 . . . 4  |-  ( ( G  Fn  A  /\  X  C_  A )  -> 
( G  |`  X )  Fn  X )
433adant1 1000 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A  /\  X  C_  A )  -> 
( G  |`  X )  Fn  X )
5 fneqeql 5577 . . 3  |-  ( ( ( F  |`  X )  Fn  X  /\  ( G  |`  X )  Fn  X )  ->  (
( F  |`  X )  =  ( G  |`  X )  <->  dom  ( ( F  |`  X )  i^i  ( G  |`  X ) )  =  X ) )
62, 4, 5syl2anc 409 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A  /\  X  C_  A )  -> 
( ( F  |`  X )  =  ( G  |`  X )  <->  dom  ( ( F  |`  X )  i^i  ( G  |`  X ) )  =  X ) )
7 resindir 4884 . . . . . 6  |-  ( ( F  i^i  G )  |`  X )  =  ( ( F  |`  X )  i^i  ( G  |`  X ) )
87dmeqi 4789 . . . . 5  |-  dom  (
( F  i^i  G
)  |`  X )  =  dom  ( ( F  |`  X )  i^i  ( G  |`  X ) )
9 dmres 4889 . . . . 5  |-  dom  (
( F  i^i  G
)  |`  X )  =  ( X  i^i  dom  ( F  i^i  G ) )
108, 9eqtr3i 2180 . . . 4  |-  dom  (
( F  |`  X )  i^i  ( G  |`  X ) )  =  ( X  i^i  dom  ( F  i^i  G ) )
1110eqeq1i 2165 . . 3  |-  ( dom  ( ( F  |`  X )  i^i  ( G  |`  X ) )  =  X  <->  ( X  i^i  dom  ( F  i^i  G ) )  =  X )
12 df-ss 3115 . . 3  |-  ( X 
C_  dom  ( F  i^i  G )  <->  ( X  i^i  dom  ( F  i^i  G ) )  =  X )
1311, 12bitr4i 186 . 2  |-  ( dom  ( ( F  |`  X )  i^i  ( G  |`  X ) )  =  X  <->  X  C_  dom  ( F  i^i  G ) )
146, 13bitrdi 195 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 104    /\ w3a 963    = wceq 1335    i^i cin 3101    C_ wss 3102   dom cdm 4588    |` cres 4590    Fn wfn 5167
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-br 3968  df-opab 4028  df-mpt 4029  df-id 4255  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-res 4600  df-iota 5137  df-fun 5174  df-fn 5175  df-fv 5180
This theorem is referenced by: (None)
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