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Theorem funsseq 27580
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 3408 . 2  |-  ( F  =  G  ->  F  C_  G )
2 simpl3 993 . . . . 5  |-  ( ( ( Fun  F  /\  Fun  G  /\  dom  F  =  dom  G )  /\  F  C_  G )  ->  dom  F  =  dom  G
)
32reseq2d 5110 . . . 4  |-  ( ( ( Fun  F  /\  Fun  G  /\  dom  F  =  dom  G )  /\  F  C_  G )  -> 
( G  |`  dom  F
)  =  ( G  |`  dom  G ) )
4 funssres 5458 . . . . 5  |-  ( ( Fun  G  /\  F  C_  G )  ->  ( G  |`  dom  F )  =  F )
543ad2antl2 1151 . . . 4  |-  ( ( ( Fun  F  /\  Fun  G  /\  dom  F  =  dom  G )  /\  F  C_  G )  -> 
( G  |`  dom  F
)  =  F )
6 simpl2 992 . . . . 5  |-  ( ( ( Fun  F  /\  Fun  G  /\  dom  F  =  dom  G )  /\  F  C_  G )  ->  Fun  G )
7 funrel 5435 . . . . 5  |-  ( Fun 
G  ->  Rel  G )
8 resdm 5148 . . . . 5  |-  ( Rel 
G  ->  ( G  |` 
dom  G )  =  G )
96, 7, 83syl 20 . . . 4  |-  ( ( ( Fun  F  /\  Fun  G  /\  dom  F  =  dom  G )  /\  F  C_  G )  -> 
( G  |`  dom  G
)  =  G )
103, 5, 93eqtr3d 2483 . . 3  |-  ( ( ( Fun  F  /\  Fun  G  /\  dom  F  =  dom  G )  /\  F  C_  G )  ->  F  =  G )
1110ex 434 . 2  |-  ( ( Fun  F  /\  Fun  G  /\  dom  F  =  dom  G )  -> 
( F  C_  G  ->  F  =  G ) )
121, 11impbid2 204 1  |-  ( ( Fun  F  /\  Fun  G  /\  dom  F  =  dom  G )  -> 
( F  =  G  <-> 
F  C_  G )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    C_ wss 3328   dom cdm 4840    |` cres 4842   Rel wrel 4845   Fun wfun 5412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-br 4293  df-opab 4351  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-res 4852  df-fun 5420
This theorem is referenced by: (None)
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