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Theorem fnreseql 5305
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 5040 . . . 4  |-  ( ( F  Fn  A  /\  X  C_  A )  -> 
( F  |`  X )  Fn  X )
213adant2 934 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A  /\  X  C_  A )  -> 
( F  |`  X )  Fn  X )
3 fnssres 5040 . . . 4  |-  ( ( G  Fn  A  /\  X  C_  A )  -> 
( G  |`  X )  Fn  X )
433adant1 933 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A  /\  X  C_  A )  -> 
( G  |`  X )  Fn  X )
5 fneqeql 5303 . . 3  |-  ( ( ( F  |`  X )  Fn  X  /\  ( G  |`  X )  Fn  X )  ->  (
( F  |`  X )  =  ( G  |`  X )  <->  dom  ( ( F  |`  X )  i^i  ( G  |`  X ) )  =  X ) )
62, 4, 5syl2anc 397 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A  /\  X  C_  A )  -> 
( ( F  |`  X )  =  ( G  |`  X )  <->  dom  ( ( F  |`  X )  i^i  ( G  |`  X ) )  =  X ) )
7 resindir 4656 . . . . . 6  |-  ( ( F  i^i  G )  |`  X )  =  ( ( F  |`  X )  i^i  ( G  |`  X ) )
87dmeqi 4564 . . . . 5  |-  dom  (
( F  i^i  G
)  |`  X )  =  dom  ( ( F  |`  X )  i^i  ( G  |`  X ) )
9 dmres 4660 . . . . 5  |-  dom  (
( F  i^i  G
)  |`  X )  =  ( X  i^i  dom  ( F  i^i  G ) )
108, 9eqtr3i 2078 . . . 4  |-  dom  (
( F  |`  X )  i^i  ( G  |`  X ) )  =  ( X  i^i  dom  ( F  i^i  G ) )
1110eqeq1i 2063 . . 3  |-  ( dom  ( ( F  |`  X )  i^i  ( G  |`  X ) )  =  X  <->  ( X  i^i  dom  ( F  i^i  G ) )  =  X )
12 df-ss 2959 . . 3  |-  ( X 
C_  dom  ( F  i^i  G )  <->  ( X  i^i  dom  ( F  i^i  G ) )  =  X )
1311, 12bitr4i 180 . 2  |-  ( dom  ( ( F  |`  X )  i^i  ( G  |`  X ) )  =  X  <->  X  C_  dom  ( F  i^i  G ) )
146, 13syl6bb 189 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 102    /\ w3a 896    = wceq 1259    i^i cin 2944    C_ wss 2945   dom cdm 4373    |` cres 4375    Fn wfn 4925
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-res 4385  df-iota 4895  df-fun 4932  df-fn 4933  df-fv 4938
This theorem is referenced by: (None)
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