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Theorem fneqeql2 5483
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 5482 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  dom  ( F  i^i  G
)  =  A ) )
2 inss1 3262 . . . . . 6  |-  ( F  i^i  G )  C_  F
3 dmss 4698 . . . . . 6  |-  ( ( F  i^i  G ) 
C_  F  ->  dom  ( F  i^i  G ) 
C_  dom  F )
42, 3ax-mp 7 . . . . 5  |-  dom  ( F  i^i  G )  C_  dom  F
5 fndm 5180 . . . . . 6  |-  ( F  Fn  A  ->  dom  F  =  A )
65adantr 272 . . . . 5  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  F  =  A )
74, 6sseqtrid 3113 . . . 4  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  i^i  G )  C_  A )
87biantrurd 301 . . 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 3078 . . 3  |-  ( dom  ( F  i^i  G
)  =  A  <->  ( dom  ( F  i^i  G ) 
C_  A  /\  A  C_ 
dom  ( F  i^i  G ) ) )
108, 9syl6rbbr 198 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( dom  ( F  i^i  G )  =  A  <->  A  C_  dom  ( F  i^i  G ) ) )
111, 10bitrd 187 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    /\ wa 103    <-> wb 104    = wceq 1314    i^i cin 3036    C_ wss 3037   dom cdm 4499    Fn wfn 5076
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-iota 5046  df-fun 5083  df-fn 5084  df-fv 5089
This theorem is referenced by: (None)
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