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Theorem fneqeql2 5650
Description: Two functions are equal iff their equalizer contains the whole domain. (Contributed by Stefan O'Rear, 9-Mar-2015.)
Assertion
Ref Expression
fneqeql2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <-> 
A  C_  dom  ( F  i^i  G ) ) )

Proof of Theorem fneqeql2
StepHypRef Expression
1 fneqeql 5649 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  dom  ( F  i^i  G
)  =  A ) )
2 inss1 3402 . . . . . 6  |-  ( F  i^i  G )  C_  F
3 dmss 4894 . . . . . 6  |-  ( ( F  i^i  G ) 
C_  F  ->  dom  ( F  i^i  G ) 
C_  dom  F )
42, 3ax-mp 8 . . . . 5  |-  dom  ( F  i^i  G )  C_  dom  F
5 fndm 5359 . . . . . 6  |-  ( F  Fn  A  ->  dom  F  =  A )
65adantr 451 . . . . 5  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  F  =  A )
74, 6syl5sseq 3239 . . . 4  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  i^i  G )  C_  A )
87biantrurd 494 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( A  C_  dom  ( F  i^i  G )  <-> 
( dom  ( F  i^i  G )  C_  A  /\  A  C_  dom  ( F  i^i  G ) ) ) )
9 eqss 3207 . . 3  |-  ( dom  ( F  i^i  G
)  =  A  <->  ( dom  ( F  i^i  G ) 
C_  A  /\  A  C_ 
dom  ( F  i^i  G ) ) )
108, 9syl6rbbr 255 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( dom  ( F  i^i  G )  =  A  <->  A  C_  dom  ( F  i^i  G ) ) )
111, 10bitrd 244 1  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <-> 
A  C_  dom  ( F  i^i  G ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    i^i cin 3164    C_ wss 3165   dom cdm 4705    Fn wfn 5266
This theorem is referenced by:  evlseu  19416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279
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