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Theorem funsseq 24125
Description: Given two functions with equal domains, equality only requires one direction of the subset relationship. (Contributed by Scott Fenton, 24-Apr-2012.) (Proof shortened by Mario Carneiro, 3-May-2015.)
Assertion
Ref Expression
funsseq  |-  ( ( Fun  F  /\  Fun  G  /\  dom  F  =  dom  G )  -> 
( F  =  G  <-> 
F  C_  G )
)

Proof of Theorem funsseq
StepHypRef Expression
1 eqimss 3230 . 2  |-  ( F  =  G  ->  F  C_  G )
2 simpl3 960 . . . . 5  |-  ( ( ( Fun  F  /\  Fun  G  /\  dom  F  =  dom  G )  /\  F  C_  G )  ->  dom  F  =  dom  G
)
32reseq2d 4955 . . . 4  |-  ( ( ( Fun  F  /\  Fun  G  /\  dom  F  =  dom  G )  /\  F  C_  G )  -> 
( G  |`  dom  F
)  =  ( G  |`  dom  G ) )
4 funssres 5294 . . . . 5  |-  ( ( Fun  G  /\  F  C_  G )  ->  ( G  |`  dom  F )  =  F )
543ad2antl2 1118 . . . 4  |-  ( ( ( Fun  F  /\  Fun  G  /\  dom  F  =  dom  G )  /\  F  C_  G )  -> 
( G  |`  dom  F
)  =  F )
6 simpl2 959 . . . . 5  |-  ( ( ( Fun  F  /\  Fun  G  /\  dom  F  =  dom  G )  /\  F  C_  G )  ->  Fun  G )
7 funrel 5272 . . . . 5  |-  ( Fun 
G  ->  Rel  G )
8 resdm 4993 . . . . 5  |-  ( Rel 
G  ->  ( G  |` 
dom  G )  =  G )
96, 7, 83syl 18 . . . 4  |-  ( ( ( Fun  F  /\  Fun  G  /\  dom  F  =  dom  G )  /\  F  C_  G )  -> 
( G  |`  dom  G
)  =  G )
103, 5, 93eqtr3d 2323 . . 3  |-  ( ( ( Fun  F  /\  Fun  G  /\  dom  F  =  dom  G )  /\  F  C_  G )  ->  F  =  G )
1110ex 423 . 2  |-  ( ( Fun  F  /\  Fun  G  /\  dom  F  =  dom  G )  -> 
( F  C_  G  ->  F  =  G ) )
121, 11impbid2 195 1  |-  ( ( Fun  F  /\  Fun  G  /\  dom  F  =  dom  G )  -> 
( F  =  G  <-> 
F  C_  G )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    C_ wss 3152   dom cdm 4689    |` cres 4691   Rel wrel 4694   Fun wfun 5249
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-res 4701  df-fun 5257
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