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Theorem fneqeql2 5997
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 5996 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  dom  ( F  i^i  G
)  =  A ) )
2 inss1 3714 . . . . . 6  |-  ( F  i^i  G )  C_  F
3 dmss 5212 . . . . . 6  |-  ( ( F  i^i  G ) 
C_  F  ->  dom  ( F  i^i  G ) 
C_  dom  F )
42, 3ax-mp 5 . . . . 5  |-  dom  ( F  i^i  G )  C_  dom  F
5 fndm 5686 . . . . . 6  |-  ( F  Fn  A  ->  dom  F  =  A )
65adantr 465 . . . . 5  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  F  =  A )
74, 6syl5sseq 3547 . . . 4  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  i^i  G )  C_  A )
87biantrurd 508 . . 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 3514 . . 3  |-  ( dom  ( F  i^i  G
)  =  A  <->  ( dom  ( F  i^i  G ) 
C_  A  /\  A  C_ 
dom  ( F  i^i  G ) ) )
108, 9syl6rbbr 264 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( dom  ( F  i^i  G )  =  A  <->  A  C_  dom  ( F  i^i  G ) ) )
111, 10bitrd 253 1  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <-> 
A  C_  dom  ( F  i^i  G ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    i^i cin 3470    C_ wss 3471   dom cdm 5008    Fn wfn 5589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602
This theorem is referenced by:  evlseu  18311  hauseqcn  28030
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