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Theorem fnreseql 5606
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 5311 . . . 4  |-  ( ( F  Fn  A  /\  X  C_  A )  -> 
( F  |`  X )  Fn  X )
213adant2 1011 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A  /\  X  C_  A )  -> 
( F  |`  X )  Fn  X )
3 fnssres 5311 . . . 4  |-  ( ( G  Fn  A  /\  X  C_  A )  -> 
( G  |`  X )  Fn  X )
433adant1 1010 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A  /\  X  C_  A )  -> 
( G  |`  X )  Fn  X )
5 fneqeql 5604 . . 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 4907 . . . . . 6  |-  ( ( F  i^i  G )  |`  X )  =  ( ( F  |`  X )  i^i  ( G  |`  X ) )
87dmeqi 4812 . . . . 5  |-  dom  (
( F  i^i  G
)  |`  X )  =  dom  ( ( F  |`  X )  i^i  ( G  |`  X ) )
9 dmres 4912 . . . . 5  |-  dom  (
( F  i^i  G
)  |`  X )  =  ( X  i^i  dom  ( F  i^i  G ) )
108, 9eqtr3i 2193 . . . 4  |-  dom  (
( F  |`  X )  i^i  ( G  |`  X ) )  =  ( X  i^i  dom  ( F  i^i  G ) )
1110eqeq1i 2178 . . 3  |-  ( dom  ( ( F  |`  X )  i^i  ( G  |`  X ) )  =  X  <->  ( X  i^i  dom  ( F  i^i  G ) )  =  X )
12 df-ss 3134 . . 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 973    = wceq 1348    i^i cin 3120    C_ wss 3121   dom cdm 4611    |` cres 4613    Fn wfn 5193
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206
This theorem is referenced by: (None)
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